Consider a body whose temperature is increasing from 1,000 K to 1,000,000 K. The correct statements among the given options are: The peak wavelength of electromagnetic radiation from the body decreases, and the color of the body changes from dark (or dark red) to bright blue. The total intensity of electromagnetic radiation from the body increases. The radiation from the body is called Blackbody radiation. The color of a black body refers to the light emitted by the black body when it is heated. As the temperature of the blackbody increases, it emits radiation with a shorter wavelength and more energy.
Thus, the peak wavelength of the electromagnetic radiation from the body decreases, and the body's color changes from dark red to bright blue. This is because the color perceived by human eyes is due to the peak wavelength of the electromagnetic radiation emitted by the body, and as the temperature increases, the peak wavelength decreases. Therefore, option C is the correct statement. The total intensity of electromagnetic radiation from the body also increases. This is because the energy emitted by the blackbody is directly proportional to the fourth power of the absolute temperature (Stefan-Boltzmann law). Therefore, as the temperature of the blackbody increases, the energy emitted by it increases as well, and so does the total intensity of electromagnetic radiation from the body.
Therefore, option D is the correct statement. The peak wavelength of electromagnetic radiation from the body remains the same is an incorrect statement because the peak wavelength of the radiation emitted by the body is directly proportional to the temperature, and so, as the temperature increases, the peak wavelength decreases. Therefore, option A is an incorrect statement. The total intensity of electromagnetic radiation from the body remains the same is also an incorrect statement. It is because the total intensity of electromagnetic radiation from the body is proportional to the fourth power of the temperature.
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When the Venera 14 probe landed on Venus's surface, its barometer measured an air pressure of 9.5 MPa. The surface acceleration due to gravity was measured to be 8.87 m/s2. If Earth's atmosphere with a pressure of 101 kPa raises mercury 0.760 m where gravitational acceleration is 9.81 m/s2. To what height in m to two significant digits would Venus's atmosphere raise liquid mercury?
The height to which Venus's atmosphere would raise liquid mercury is determined based on the given air pressure and surface acceleration due to gravity. The calculation involves comparing the pressure in Venus's atmosphere to Earth's atmosphere and using the difference to determine the height of the mercury column.
To calculate the height to which Venus's atmosphere would raise liquid mercury, we can use the principle of hydrostatic pressure. The pressure difference between two points in a fluid column is directly proportional to the difference in height.Given that Earth's atmosphere raises mercury to a height of 0.760 m when the pressure is 101 kPa and the acceleration due to gravity is 9.81 m/s^2, we can set up a proportion to find the height in Venus's atmosphere.
The ratio of pressure to height is constant, so we can write:
(9.5 MPa / 101 kPa) = (8.87 m/s^2 / 9.81 m/s^2) * (h / 0.760 m)
Solving for h, we can find the height to which Venus's atmosphere would raise liquid mercury.
By rearranging the equation and substituting the given values, we can calculate the height to two significant digits.
Therefore, the height to which Venus's atmosphere would raise liquid mercury can be determined using the given air pressure and surface acceleration due to gravity.
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X-rays with an energy of 339 keV undergo Compton scattering with a target. If the scattered X-rays are detected at 57.7^{\circ}∘relative to the incident X-rays, what is the wavelength of the scattered photon?
Answer:
The
wavelength
of the scattered photon is approximately 1.11 × 10^(-11) meters.
Explanation:
Compton scattering is a phenomenon where X-rays interact with electrons, resulting in a shift in wavelength. To determine the wavelength of the scattered photon, we can use the Compton scattering formula:
Δλ = λ' - λ = λ_c * (1 - cos(θ))
Where:
Δλ is the change in wavelength
λ' is the wavelength of the scattered photon
λ is the wavelength of the incident X-ray photon
λ_c is the Compton wavelength (approximately 2.43 × 10^(-12) m)
θ is the scattering angle
Given:
Energy of the incident X-ray photon (E) = 339 keV = 339 * 10^3 eV
Scattering angle (θ) = 57.7 degrees
First, let's calculate the wavelength of the incident X-ray photon using the energy-wavelength relationship:
E = hc / λ
Where:
h is Planck's constant (approximately 6.63 × 10^(-34) J·s)
c is the speed of light (approximately 3.00 × 10^8 m/s)
Converting the energy to joules:
E = 339 * 10^3 eV * (1.60 × 10^(-19) J/eV) = 5.424 × 10^(-14) J
Rearranging the equation to solve for λ:
λ = hc / E
Substituting the values:
λ = (6.63 × 10^(-34) J·s * 3.00 × 10^8 m/s) / (5.424 × 10^(-14) J) ≈ 1.22 × 10^(-11) m
Now, let's calculate the change in wavelength using the Compton scattering formula:
Δλ = λ_c * (1 - cos(θ))
Substituting the values:
Δλ = (2.43 × 10^(-12) m) * (1 - cos(57.7 degrees))
Calculating cos(57.7 degrees):
cos(57.7 degrees) ≈ 0.551
Δλ = (2.43 × 10^(-12) m) * (1 - 0.551) ≈ 1.09 × 10^(-12) m
Finally, we can calculate the wavelength of the scattered photon by subtracting the change in wavelength from the wavelength of the incident X-ray photon:
λ' = λ - Δλ
Substituting the values:
λ' = (1.22 × 10^(-11) m) - (1.09 × 10^(-12) m) ≈ 1.11 × 10^(-11) m
Therefore, the wavelength of the scattered photon is approximately 1.11 × 10^(-11) meters.
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A home run is hit in such a way that the baseball just clears a wall 16.0 m high, located 116 m from home plate. The ball is hit at an angle of 37.0° to the horizontal, and air resistance is negligible. (Assume that the ball is hit at a height of 1.0 m above the ground.) (a) Find the initial speed of the ball.
The initial speed of the ball is 36.7 m/s.
* Height of the wall: 16.0 m
* Distance to the wall: 116 m
* Angle of the ball: 37.0°
* Initial height of the ball: 1.0 m
We need to find the initial speed of the ball.
To do this, we can use the following equations:
y = v_y t + 0.5 a t^2
where:
* y is the height of the ball
* v_y is the vertical velocity of the ball
* t is the time it takes the ball to reach the wall
* a is the acceleration due to gravity (9.8 m/s^2)
x = v_x t
where:
* x is the distance the ball travels
* v_x is the horizontal velocity of the ball
We can solve for v_y and v_x using the above equations. Then, we can use the Pythagorean theorem to find the initial speed of the ball.
Solving for v_y:
16 = v_y t + 0.5 * 9.8 * t^2
16 = v_y t + 4.9 t^2
0 = v_y t + 4.9 t^2 - 16
t (v_y + 4.9 t) = 16
t = 16 / (v_y + 4.9)
We can now solve for v_x:
116 = v_x t
116 = v_x * (16 / (v_y + 4.9))
v_x = (116 * (v_y + 4.9)) / 16
Now that we have v_y and v_x, we can use the Pythagorean theorem to find the initial speed of the ball:
v^2 = v_y^2 + v_x^2
v^2 = (v_y + 4.9)^2 + v_x^2
v = sqrt((v_y + 4.9)^2 + v_x^2)
Plugging in the known values, we get:
v = sqrt((4.9 + 4.9)^2 + (116 * (4.9 + 4.9)) / 16)^2)
v = 36.7 m/s
Therefore, the initial speed of the ball is 36.7 m/s.
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Part A int Sharing What in the approximate radius of a 'Od nucleus? Express your answer to two significant Higures and include the appropriate unite ting NA ? Tools Value Units Submit Hequest A Part B Approximately what is the value of A for a nucles whose ridius i 30-10-15 m Express your answer as an integer 2 θα ΑΣφ.
The radius of a nucleus is determined by measuring the energies of alpha or other particles that are scattered by it. The radius of a nucleus, in general, is determined by determining the nuclear density.
The density of the nucleus is roughly constant, implying that the radius is proportional to the cube root of the nucleon number.For example, the radius of a 208Pb nucleus is given by the following equation:r = r0A1/3, whereA is the mass number of the nucleus,r0 is a constant equal to 1.2 × 10−15 m.Using this equation.
Thus, the approximate radius of a 208Pb nucleus is 6.62 × 10−15 m.Part B:What is the value of A for a nucleus whose radius is 3.0 × 10−15 m?The radius of a nucleus, in general, is determined by determining the nuclear density. The density of the nucleus is roughly constant, implying that the radius is proportional to the cube root of the nucleon number.
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Two identical, 2.6-F capacitors are placed in series with a 17-V battery. How much energy is stored in each capacitor? (in J)
The formula to calculate energy stored in a capacitor is given as E = (1/2) CV²
Where, E = energy stored in capacitor
C = capacitance
V = voltage
Substitute C and V values to get the answer, The potential difference (V) across each capacitor is
V = V₁ + V₂
Where V₁ = voltage across the first capacitor
V₂ = voltage across the second capacitor
The formula to calculate voltage across each capacitor is given as
V = Q/C
C = Q/V
Also,C₁ = C₂ = C = 2.6 F
The equivalent capacitance (Ceq) in a series connection is given by
1/Ceq = 1/C₁ + 1/C₂ + ...
1/Ceq = 1/C + 1/C...
1/Ceq= 2/Ceq
1/Ceq= 1.3 F
Charge (Q) across each capacitor can be calculated as
Q = Ceq * V
Substitute Q and C values to get the voltage across each capacitor,
V = Q/C
C = Q/V = 17
V/2 = 8.5 V
Substitute C and V values to calculate energy stored in each capacitor,
E = (1/2) * C * V²
E = (1/2) * 2.6 F * (8.5 V)²
E = 976.75 J
Therefore, each capacitor stores 976.75 J of energy.
In conclusion Two identical, 2.6-F capacitors placed in series with a 17-V battery stores 976.75 J of energy in each capacitor.
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27 (-/4 Points DETAILS SERCP114.6.OP.639. The pure below shows and feeder that weighs 1977. The feeder i uported by a vertical cable, which is turned to the cables, each of which is attached to a horizontal post. The test cable makes a 60 angle with the post the right cable makes a 30 angle What is the tensionin each cable (in ? w bottomcat
The magnitude of the tension in the right cable is:|T₂| = 828.5 lbWhile the magnitude of the tension in the test cable is:|T₁| = 1148.5 lbThe tension in each cable is 988.5 lb.
The tension in each cable is 988.5 lb. Given that the feeder weighs 1977 lb and is supported by a vertical cable which is attached to two horizontal posts and two cables. One of the horizontal cables makes an angle of 30° with the vertical cable while the other makes an angle of 60° with the same cable.Using the principles of vectors, the weight of the feeder is resolved into two components.
One of these components is perpendicular to the angle made by the right cable with the vertical cable while the other is perpendicular to the angle made by the test cable with the vertical cable.The weight of the feeder perpendicular to the right cable is:W₁ = 1977 lb × cos 30° = 1709.2 lbThe weight of the feeder perpendicular to the test cable is:W₂ = 1977 lb × cos 60° = 988.5 lbBy considering the horizontal and vertical components of the tension in each cable, the tension in each cable can be expressed as:T1 = T₁ cos 30° + T₂ cos 60°andT2 = T₁ sin 30° + T₂ sin 60°Since the tension in the vertical cable is the weight of the feeder, we can write:T₁ + T₂ = 1977 lbSubstituting the expressions above in the equation above:T₁ cos 30° + T₂ cos 60° + T₁ sin 30° + T₂ sin 60° = 1977 lbSimplifying and substituting T₂ with T₁ - 1977 lb:T₁ = 1977 lb ÷ (cos 30° + sin 30° + cos 60° + sin 60°)T₁ = 1148.5 lbUsing the expression for T₂ above:T₂ = T₁ - 1977 lbT₂ = -828.5 lbThe negative sign means that the tension in the cable is acting in the opposite direction to the one assumed.
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A man is pulling a box with a rope attached to it which is making an angle of 60° with the horizontal surface. If the force applied by the man is 3.8 N and the box is displaced by 7.1 m along the horizontal surface while frictional force is 1.1 N, find the net work done on the box. Answer: Choose... Check
The net work done on the box is 21.225 Joules. Displacement is the magnitude of the displacement along the horizontal surface (7.1 m).
Work = Force * Displacement * cos(theta)
Force is the magnitude of the force applied (3.8 N).
Displacement is the magnitude of the displacement along the horizontal surface (7.1 m).
theta is the angle between the force vector and the displacement vector (60°).
Work_applied = 3.8 N * 7.1 m * cos(60°)
To calculate the work done against friction, we use the formula:
Work_friction = Force_friction * Displacement * cos(180°)
Since the frictional force acts opposite to the direction of motion, we take the cosine of 180°.
Work_friction = 1.1 N * 7.1 m * cos(180°)
Net work = Work_applied - Work_friction
Net work = (3.8 N * 7.1 m * cos(60°)) - (1.1 N * 7.1 m * cos(180°))
cos(60°) = 0.5
cos(180°) = -1
Net work = (3.8 N * 7.1 m * 0.5) - (1.1 N * 7.1 m * -1)
= 13.415 J + 7.81 J
= 21.225 J
Therefore, the net work done on the box is 21.225 Joules.
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A parallel plate capacitor has plates 0.142 m2 in area and a separation of 14.2 mm. A battery charges the plates to a potential difference of 120 V and is then disconnected. A sheet of dielectric material 4 mm thick and with a dielectric constant of 6.1 is then placed symmetrically between the plates. With the sheet in position, what is the potential difference between the plates? Answer in Volts and two decimal
The potential difference between the plates with the dielectric in place is 384.22 V (rounded to two decimal places). The potential difference between the plates of a parallel plate capacitor before and after a dielectric material is placed between the plates can be calculated using the formula:V = Ed.
where V is the potential difference between the plates, E is the electric field between the plates, and d is the distance between the plates. The electric field E can be calculated using the formula:E = σ / ε0,where σ is the surface charge density of the plates, and ε0 is the permittivity of free space. The surface charge density σ can be calculated using the formula:σ = Q / A,where Q is the charge on the plates, and A is the area of the plates.The charge Q on the plates can be calculated using the formula:
Q = CV,where C is the capacitance of the capacitor, and V is the potential difference between the plates. The capacitance C can be calculated using the formula:
C = ε0 A / d,where ε0 is the permittivity of free space, A is the area of the plates, and d is the distance between the plates.
1. Calculate the charge Q on the plates before the dielectric is placed:
Q = CVQ = (ε0 A / d) VQ
= (8.85 × [tex]10^-12[/tex] F/m) (0.142 m²) (120 V) / (14.2 × [tex]10^-3[/tex] m)Q
= 1.2077 × [tex]10^-7[/tex]C
2. Calculate the surface charge density σ on the plates before the dielectric is placed:
σ = Q / Aσ = 1.2077 × [tex]10^-7[/tex] C / 0.142 m²
σ = 8.505 ×[tex]10^-7[/tex] C/m²
3. Calculate the electric field E between the plates before the dielectric is placed:
E = σ / ε0E
= 8.505 × [tex]10^-7[/tex]C/m² / 8.85 × [tex]10^-12[/tex]F/m
E = 96054.79 N/C
4. Calculate the potential difference V between the plates after the dielectric is placed:
V = EdV
= (96054.79 N/C) (4 × [tex]10^-3[/tex]m)V
= 384.22 V
Therefore, the potential difference between the plates with the dielectric in place is 384.22 V (rounded to two decimal places).
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Episode 2: Tom uses his owner's motorcycle to chase Jerry (with an ax). The motorcy- cle has a 95 hp engine, that is, the rate it does work at is 95 hp. It has an efficiency of 23%. a) How much energy in the form of heat from burning gasoline) enters the engine every second? b) Assume that engine has half the efficiency of a Carnot engine running between the same high and low temperatures. If the low temperature is 360 K. what is the high tem- perature? c) Assume the temperature of the inside of the engine is 360 K. One part of the engine is a steel rectangle. 0.0400 m by 0.0500 m and 0.0200 m thick. Heat flows from that temper- ature through the thickness of the steel to a temperature of 295 K. What is the rate of heat flow?
The engine receives 79.85 hp of energy per second from burning gasoline at a high temperature of 639.22 K. Approximately 5.60W of heat flows through the steel rectangle.
a) To determine the amount of energy entering the engine every second from burning gasoline, we need to calculate the power input. The power input can be obtained by multiplying the engine's horsepower (95 hp) by its efficiency (23%). Therefore, the power input is:
Power input = [tex]95 hp * \frac{23}{100}[/tex]= 21.85 hp.
However, power is commonly measured in watts (W), so we need to convert horsepower to watts. One horsepower is approximately equal to 746 watts. Therefore, the power input in watts is:
Power input = 21.85 hp * 746 W/hp = 16287.1 W.
This represents the total power entering the engine every second.
b) Assuming the engine has half the efficiency of a Carnot engine running between the same high and low temperatures, we can use the Carnot efficiency formula to find the high temperature. The Carnot efficiency is given by:
Carnot efficiency =[tex]1 - (T_{low} / T_{high}),[/tex]
where[tex]T_{low}[/tex] and[tex]T_{high}[/tex] are the low and high temperatures, respectively. We are given the low-temperature [tex]T_{low }= 360 K[/tex].
Since the engine has half the efficiency of a Carnot engine, its efficiency would be half of the Carnot efficiency. Therefore, the engine's efficiency can be written as:
Engine efficiency = (1/2) * Carnot efficiency.
Substituting this into the Carnot efficiency formula, we have:
(1/2) * Carnot efficiency = 1 - ( [tex]T_{low[/tex] / [tex]T_{high[/tex]).
Rearranging the equation, we can solve for T_high:
[tex]T_{high[/tex] =[tex]T_{low}[/tex] / (1 - 2 * Engine efficiency).
Substituting the values, we find:
[tex]T_{high[/tex]= 360 K / (1 - 2 * (23/100)) ≈ 639.22 K.
c) To calculate the rate of heat flow through the steel rectangle, we can use Fourier's law of heat conduction:
Rate of heat flow = (Thermal conductivity * Area * ([tex]T_{high[/tex] - [tex]T_{low}[/tex])) / Thickness.
We are given the dimensions of the steel rectangle: length = 0.0400 m, width = 0.0500 m, and thickness = 0.0200 m. The temperature difference is [tex]T_{high[/tex] -[tex]T_{low}[/tex] = 360 K - 295 K = 65 K.
The thermal conductivity of steel varies depending on the specific type, but for a general estimate, we can use a value of approximately 50 W/(m·K).
Substituting the values into the formula, we have:
Rate of heat flow =[tex]\frac{ (50 W/(m·K)) * (0.0400 m * 0.0500 m) * (65 K)}{0.0200m}[/tex] = 5.60 W.
Therefore, the rate of heat flow through the steel rectangle is approximately 5.60 W.
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A ladder with a length of 12.3 m and weight of 591.0 N rests against a frictionless wall, making an angle of 61.0° with the horizontal. Find the horizontal force exerted on the base of the ladder by Earth when a firefighter weighing 898.0 N is 3.91 m from the bottom of the ladder. Answer in units of N.
The horizontal force exerted on the base of the ladder by Earth is approximately 50.9 N.
To find the horizontal force exerted on the base of the ladder by Earth, we need to consider the torque equilibrium of the ladder.
First, let's determine the vertical and horizontal components of the ladder's weight. The weight of the ladder is given as 591.0 N. The vertical component is given by:
Vertical Component = Weight of Ladder × sin(61.0°)
= 591.0 N × sin(61.0°)
≈ 505.0 N
The horizontal component of the ladder's weight is given by:
Horizontal Component = Weight of Ladder × cos(61.0°)
= 591.0 N × cos(61.0°)
≈ 299.7 N
Next, we need to consider the weight of the firefighter. The weight of the firefighter is given as 898.0 N. The vertical component of the firefighter's weight does not exert any torque because it passes through the point of contact. Therefore, we only need to consider the horizontal component of the firefighter's weight, which is:
Horizontal Component of Firefighter's Weight = Weight of Firefighter × cos(61.0°)
= 898.0 N × cos(61.0°)
≈ 453.7 N
Now, let's consider the torque equilibrium. The torques exerted by the ladder and the firefighter must balance each other out. The torque exerted by the ladder is given by the product of the vertical component of the ladder's weight and its distance from the bottom:
Torque by Ladder = Vertical Component of Ladder's Weight × Distance from Bottom
= 505.0 N × 3.91 m
≈ 1976.6 N·m
The torque exerted by the firefighter is given by the product of the horizontal component of the firefighter's weight and its distance from the bottom:
Torque by Firefighter = Horizontal Component of Firefighter's Weight × Distance from Bottom
= 453.7 N × 3.91 m
≈ 1775.7 N·m
Since the ladder is in equilibrium, the torques exerted by the ladder and the firefighter must balance each other out:
Torque by Ladder = Torque by Firefighter
To maintain equilibrium, the horizontal force exerted on the base of the ladder by Earth must balance out the torques. Therefore, the horizontal force exerted on the base of the ladder by Earth is:
Horizontal Force = (Torque by Ladder - Torque by Firefighter) / Distance from Bottom
= (1976.6 N·m - 1775.7 N·m) / 3.91 m
≈ 50.9 N
Therefore, the horizontal force exerted on the base of the ladder by Earth is approximately 50.9 N.
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4)Consider a charge +q, located at the origin. We observe the potential V and the electric field E at distance r. Here we are interested in quantities along the z-axis. Find or, if you know it, simply write, the electric potential V along the z-axis (x =y=0), for [z| not equal to zero. (Izis the absolute value of z.) Can you make any arguments by symmetry about the electric field? For instance, are any components of the electric field equal to zero along the z-axis? Use the standard expression, E; = -av/axi, where the E; are the x, y, z components of E, and Xi refer to the coordinates x, y, z. Using the expression you found above for V, what is the vector E as a function of position along the positive z-axis (lz| not equal zero). = Recall the differential form of Gauss' law, V. E=p/€0. What does this say about the component Ex at points Ax away from the z-axis, as a function of z? -
The electric potential, V along the z-axis (x=y=0) is as follows: Let r = (x² + y² + z²)¹/² Thus,
V = kq/r. When
x=y=0,
V = kq/z,
provided z is not equal to zero. By symmetry, the components of the electric field E along the x and y-axes are zero since the charge +q at the origin does not produce any component of E along these axes.
Hence E; = (0,0, Ez). It follows that Ex = 0 and Ey = 0 because of symmetry along the x- and y-axes. The electric field E can be found using
E= -av/axi
= - (dV/dx)i - (dV/dy)j - (dV/dz)k.
Using V = kq/z, it follows that:
E = -d/dz(kq/z)k
= kq/z²k.
Hence E has only a z-component, and its magnitude is given by E = kq/z² along the positive z-axis.
The differential form of Gauss' law, V. E=p/€0. If z > Ax, then we can draw a Gaussian surface that is cylindrical and coaxial with the z-axis. By symmetry, Ex = 0, so that p = 0. Thus, V. E = 0, and since V is non-zero, it follows that E must be zero.
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When an aluminum bar is connected between a hot reservoir at 725K and a cold reservoir at 310K, 2.50kj of energy is transferred by heat from the hot reservoir to the cold reservoir. In this irreversible process, calculate the change in entropy of(a) the hot reservoir
The change in entropy of the hot reservoir is 3.45 J/K.
When an aluminum bar is connected between a hot reservoir at 725K and a cold reservoir at 310K, heat is transferred from the hot reservoir to the cold reservoir. In this irreversible process, we are asked to calculate the change in entropy of the hot reservoir.
To calculate the change in entropy, we can use the formula:
[tex]ΔS = Q/T[/tex]
where [tex]ΔS[/tex] represents the change in entropy, Q represents the amount of heat transferred, and T represents the temperature at which the heat is transferred.
In this case, we are given that 2.50 kJ of energy is transferred by heat from the hot reservoir. To convert this to Joules, we multiply by 1000:
Q = 2.50 kJ * 1000 J/kJ
= 2500 J
The temperature of the hot reservoir is given as 725K. Plugging these values into the formula, we get:
[tex]ΔS = 2500 J / 725K[/tex]
= 3.45 J/K
Therefore, the change in entropy of the hot reservoir is 3.45 J/K.
In summary, when an aluminum bar is connected between a hot reservoir at 725K and a cold reservoir at 310K, and 2.50 kJ of energy is transferred from the hot reservoir to the cold reservoir, the change in entropy of the hot reservoir is 3.45 J/K.
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If we could measure the overall curvature of cosmic space and found it to be negative, then we would conclude that the universe ____.
A. will expand forever
B. is expanding faster than we thought
C. is neither expanding nor contracting now
D. is actually contracting now
The correct option for the following question is A. will expand forever. If we could measure the overall curvature of cosmic space and found it to be negative, then we would conclude that the universe will expand forever.
The curvature of cosmic space is determined by the amount of matter and energy present in the universe. There are three possible curvatures: positive curvature (closed or spherical), negative curvature (open or hyperbolic), and zero curvature (flat).
In the case of a negative curvature, the geometry of space is open and extends infinitely. This indicates that the gravitational pull of matter and energy is not strong enough to halt the expansion of the universe. Thus, the universe will continue to expand indefinitely. Therefore, if the overall curvature of cosmic space is measured to be negative, we would conclude that the universe will expand forever.
If the overall curvature of cosmic space is negative, it indicates that the universe will expand forever. The negative curvature implies an open geometry where the expansion will continue indefinitely due to the lack of sufficient gravitational forces to stop it.
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6. a. A space ship moves away from the earth at a speed of 0.850. The ship launches a small probe that moves with a speed of 0.780c with respect to the ship. From the point of view of the crew on the ship, the probe is moving back directly towards the earth. Compute the speed of the probe as seen from the earth. Is the probe moving towards the carth or away from the earth (as viewed from the earth)?
b. From the earth, the ship described above appears to have a length of 4.50 m. What would be the length of the ship as measured by the ship's crew?
(a.)The speed of the probe as seen from the earth is approximately 0.970c. (b.) The length of the ship as measured by the ship's crew is approximately 6.15 m.
a. To calculate the speed of the probe as seen from the earth, we need to use the relativistic velocity addition formula:
v' = (v + u) / (1 + (vu/c^2)),
where v' is the velocity of the probe as seen from the earth, v is the velocity of the ship (relative to the earth), u is the velocity of the probe (relative to the ship), and c is the speed of light.
v = 0.850c (speed of the ship relative to the earth),
u = 0.780c (speed of the probe relative to the ship).
Substituting the values into the formula:
v' = (0.850c + 0.780c) / (1 + (0.850c)(0.780c)/(c^2))
= (1.63c) / (1 + 0.663)
≈ 0.970c.
Therefore, the speed of the probe as seen from the earth is approximately 0.970c. Since the speed is greater than the speed of light, it implies that the probe is moving away from the earth (as viewed from the earth).
b. The length of the ship as measured by the ship's crew can be calculated using the relativistic length contraction formula:
L' = L * √(1 - (v^2/c^2)),
where L' is the length of the ship as measured by the crew, L is the length of the ship as measured by an observer at rest (in this case, the earth), v is the velocity of the ship (relative to the earth), and c is the speed of light.
L = 4.50 m (length of the ship as measured from the earth),
v = 0.850c (speed of the ship relative to the earth).
Substituting the values into the formula:
L' = 4.50 m * √(1 - (0.850c)^2/c^2)
= 4.50 m * √(1 - 0.7225)
= 4.50 m * √(0.2775)
≈ 6.15 m.
Therefore, the length of the ship as measured by the ship's crew is approximately 6.15 m.
a. The speed of the probe as seen from the earth is approximately 0.970c. The probe is moving away from the earth (as viewed from the earth).
b. The length of the ship as measured by the ship's crew is approximately 6.15 m.
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Suppose the yellow clip in the above image is attached to the G+ input on your iOLab, and the black clip is attached to the G-input, and that the High Gain sensor was being recorded during the flip. Describe what you think the High Gain data chart looks like. You will need to design your Lab 9 setup so that Δ∅ is as big as possible when the loop is rotated, which means you need to think about ways to make the product of N and A and B1 as big as possible. Faraday's Law states that the magnitude of the emf is given by Δ∅/Δt, so you should also take into. account the time it takes you to flip the loop. Take some time to discuss this with one of your classmates so you can design an experimental setup that maximizes the emf generated using the wires in your E\&M accessory kit and the Earth's magnetic field. 4. In the space below, summarize your thoughts and reasoning from your discussion with your classmate. Some things you might discuss include: - What is the best initial orientation of the loop? - What ' $ best axis of rotation and speed with which to flip or rotate the loop? - Is it best to have a big loop with fewer turns of wire or a smaller loop with more turns of wire? (Some examples for different sizes of loops are shown under the 'Help' button) N. Faraday's law: Moving the Loop: In Lab 9 you will be using the wires in your E\&M Accessory pack and the Earth's magnetic field to create the largest emf you can create. This activity will help you start thinking about how to maximize the emf you generate. To make a loop your group can use any or all of the wire from one E\&M Accessory Pack: Hookup wires with clips Magnet wire Important Note: Connecting to the Magnet Wire at both ends. You will be using the Earth itself as the magnet. Since moving the magnet is not so easy in this scenario we need to review how we can move a loop in a constant magnetic field to induce an emf. As you learned in your textbook and homework on Faraday's Law, the flux ∅ through a loop with N turns and area A in a constant magnetic field B is given by ∅=NA⋅B. As illustrated below, if the loop is flipped by 180∘ the change in flux is given by △∅=2NAB⊥. where B⊥ is the component of the magnetic field that is perpendicular to the plane of the loop:
The goal is to design an experimental setup that maximizes the electromotive force (emf) generated by flipping a loop in a constant magnetic field.
Factors to consider include the initial orientation of the loop, the axis of rotation, the speed of flipping, and the size of the loop. By maximizing the product of the number of turns (N) and the area of the loop (A) while ensuring a perpendicular magnetic field (B), the change in flux (∆∅) and subsequently the emf can be increased.
To maximize the emf generated, several considerations need to be made. Firstly, the loop should have an initial orientation that maximizes the change in flux when flipped by 180 degrees (∆∅). This can be achieved by ensuring the loop is perpendicular to the magnetic field at the start.
Secondly, the axis of rotation and the speed of flipping should be optimized. A quick and smooth flipping motion is desirable to minimize the time it takes to complete the rotation, thus maximizing the rate of change of flux (∆t).
Lastly, the size of the loop should be considered. Increasing the number of turns of wire (N) and the area of the loop (A) will result in a larger product of N and A, leading to a greater change in flux and higher emf. However, practical constraints such as available wire length and the physical limitations of the setup should also be taken into account.
By carefully considering these factors and optimizing the setup, it is possible to design an experimental configuration that maximizes the emf generated by flipping the loop in the Earth's magnetic field.
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Question 12 Standing Waves As the tension in the string is increased, the frequency of the n-1 standing wave should: O increase O decrease O stay the same Question 13 1 pts Standing Waves If your micr
As the tension in the string is increased, the frequency of the (n-1) standing wave should increase.
In a string under tension, the frequency of a standing wave is directly proportional to the tension in the string. This means that as the tension increases, the frequency of the standing wave also increases.
Therefore, the correct answer is: Increase.
When a string is under tension and forms standing waves, the frequency of the standing waves depends on various factors, including the tension in the string.
The fundamental frequency (n = 1) of a standing wave on a string is determined by the length of the string, its mass per unit length, and the tension in the string.
As we increase the tension in the string while keeping other factors constant, such as the length and mass per unit length, the frequency of the fundamental (n = 1) standing wave increases.
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X Х Two identical balls of putty moving perpendicular to each other, both moving at 10.06 m/s, experience a perfectly inelastic collision. What is the speed of the combined ball after the collision? Give your answer to two decimal places
The speed of the combined ball after the perfectly inelastic collision remains the same at 10.06 m/s.
In a perfectly inelastic collision, two objects stick together and move as one mass after the collision. To calculate the speed of the combined ball after the collision, we can use the principle of conservation of momentum.
Given:
- Two identical balls of putty
- Both moving at 10.06 m/s
- Perfectly inelastic collision
Let's denote the initial velocity of each ball as v1 and v2, and the final velocity of the combined ball as vf.
According to the conservation of momentum:
(m1 * v1) + (m2 * v2) = (m1 + m2) * vf
Since the balls are identical, their masses (m1 and m2) are the same, so we can rewrite the equation as:
(2 * m * v1) = (2 * m) * vf
The masses cancel out, leaving us with:
2 * v1 = 2 * vf
Simplifying further:
v1 = vf
Since both balls are moving at the same speed before the collision, the speed of the combined ball after the collision is also equal to 10.06 m/s.
Therefore, the speed of the combined ball after the collision is 10.06 m/s.
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Why must hospital personnel wear special conducting shoes while working around oxygen in an operating room?What might happen if the personnel wore shoes with rubber soles?
Hospital personnel must wear special conducting shoes in operating rooms to prevent the buildup of static electricity, which could potentially ignite the highly flammable oxygen. Wearing shoes with rubber soles increases the risk of static discharge and should be avoided to ensure the safety of everyone in the operating room.
Hospital personnel must wear special conducting shoes while working around oxygen in an operating room because oxygen is highly flammable and can ignite easily. These special shoes are made of materials that conduct electricity, such as leather, to prevent the buildup of static electricity.
If personnel wore shoes with rubber soles, static electricity could accumulate on their bodies, particularly on their feet, due to the friction between the rubber soles and the floor. This static electricity could then discharge as a spark, potentially igniting the oxygen in the operating room.
By wearing conducting shoes, the static electricity is safely discharged to the ground, minimizing the risk of a spark that could cause a fire or explosion. The conducting materials in these shoes allow any static charges to flow freely and dissipate harmlessly. This precaution is crucial in an environment where oxygen is used, as even a small spark can lead to a catastrophic event.
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The current through a 40 W, 120 V light bulb is:
A.
1/3 A
b.
3A
c.
80 A
d
4,800 A
AND.
None
Comparing the options provided, we see that the current is approximately 0.333 A, which corresponds to option A: 1/3 A. Option A is correct.
We are given a 40 W light bulb with a voltage of 120 V. To find the current, we can rearrange the formula P = VI to solve for I:
I = P / V
Substituting the given values:
I = 40 W / 120 V
Calculating the current:
I ≈ 0.333 A
Comparing the options provided, we see that the current is approximately 0.333 A, which corresponds to option A: 1/3 A. Therefore, the correct answer is A.
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A piece of wood has a mass of 20 g and when placed in water it floats. That is, if totally immersed its buoyant force is more than enough to overcome its weight. Therefore a sinker is attached to the block of wood. Since only the buoyant force of the wood when totally immersed is required and not that of the wood and sinker combination, first the sinker is immersed with the wood out of water as in figure 1 to obtain an apparent mass of 40 g. Then the water in the container is raised to cover the wood as in figure 2 and the apparent mass is 16 g.
What is the specific gravity of the wood?
The specific gravity of the wood is 1
To find the specific gravity of the wood, we can use the concept of buoyancy and the equation:
Specific gravity = Density of the wood / Density of water
First, let's calculate the apparent loss of weight of the wood when submerged. We can use the equation:
Apparent loss of weight = Mass of wood out of water - Mass of wood in water
Given that the mass of the wood out of water is 40g and the mass of the wood in water is 16 g:
Apparent loss of weight = 40 g - 16 g = 24 g
Next, let's calculate the weight of the water displaced by the wood. We know that the buoyant force acting on the wood is equal to the weight of the water displaced by the wood.
Since the wood is floating, the buoyant force is equal to the weight of the wood.
Weight of water displaced = Apparent loss of weight of the wood = 24 g
The density of water is 1 g/cm³ (or 1000 kg/m³).
Density of the wood = (Weight of water displaced) / (Volume of water displaced)
To find the volume of water displaced, we can use the equation:
Volume of water displaced = (Mass of water displaced) / (Density of water)
Since the density of water is 1 g/cm³, the volume of water displaced is equal to the mass of water displaced.
Volume of water displaced = Mass of water displaced = Apparent loss of weight of the wood = 24 g
Now, we can calculate the density of the wood:
Density of the wood = (Weight of water displaced) / (Volume of water displaced) = 24 g / 24 g = 1 g/cm³
Finally, we can calculate the specific gravity of the wood:
Specific gravity = Density of the wood / Density of water = 1 g/cm³ / 1 g/cm³ = 1
Therefore, the specific gravity of the wood is 1.
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A bullet with a mass of 66.3 g is shot into a block of ice at 500 m/s. The bullet fully stops within the block of ice. The block of ice has a temperature of 0°C. Assuming that all the generated he goes in to melting ice, what will be the mass of ice that will melt as a result of this collision. Ignore the small amount of heat absorbed by the bullet.
The mass of ice that will melt as a result of this collision is 24.8 g.
When the bullet hits the ice, the kinetic energy of the bullet will be converted into heat and used to melt the ice. The amount of ice that melts will be determined by the amount of heat generated by the bullet's kinetic energy. The bullet's kinetic energy can be determined using the formula KE = (1/2)mv² where m is the mass of the bullet and v is the velocity of the bullet. Plugging in the values given in the question, we get:
KE = (1/2)(0.0663 kg)(500 m/s)²
KE = 8.2875 kJ
The amount of heat needed to melt ice is given by the formula Q = mLf where Q is the heat required, m is the mass of the ice, and Lf is the latent heat of fusion of ice. The latent heat of fusion of ice is 334 kJ/kg. Solving for m, we get:
m = Q/Lf
m = (8.2875 kJ)/(334 kJ/kg)
m = 0.0248 kg or 24.8 g
Therefore, the mass of ice that will melt as a result of this collision is 24.8 g.
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Question 13 It turns out that -40'C is the same temperature as -40°F. Is there a temperature at which the Kelvin and Celsius scales agree? a yes, at O'C Ob yes, at OK yes at 273°C d No
Yes, there is a temperature at which the Kelvin and Celsius scales agree. the temperature at which the Kelvin and Celsius scales agree is at -273.15°C, which corresponds to 0 Kelvin.
The Kelvin scale is an absolute temperature scale, where 0 Kelvin (0 K) represents absolute zero, the point at which all molecular motion ceases. On the other hand, the Celsius scale is based on the properties of water, with 0 degrees Celsius (0°C) representing the freezing point of water and 100 degrees Celsius representing the boiling point of water at standard atmospheric pressure.
To find the temperature at which the Kelvin and Celsius scales agree, we need to find the temperature at which the Celsius value is numerically equal to the Kelvin value. This occurs when the temperature on the Celsius scale is -273.15°C.
The relationship between the Kelvin (K) and Celsius (°C) scales can be expressed as:
K = °C + 273.15
At -273.15°C, the Celsius value is numerically equal to the Kelvin value:
-273.15°C = -273.15 + 273.15 = 0 K
Therefore, at a temperature of -273.15°C, which is known as absolute zero, the Kelvin and Celsius scales agree.
At temperatures below absolute zero, the Kelvin scale continues to decrease, while the Celsius scale remains positive. This is because the Kelvin scale represents the absolute measure of temperature, while the Celsius scale is based on the properties of water. As such, the Kelvin scale is used in scientific and technical applications where absolute temperature is important, while the Celsius scale is commonly used for everyday temperature measurements.
In summary, This temperature, known as absolute zero, represents the point of complete absence of molecular motion.
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Question 24 Review Session 3 In Problem II, we knew the image was virtual because O it was 120 cm from the lens. O it was on the same side as the object. O it was upright O the lens was diverging. Que
In the case of lenses, the image will always be reversed if it is real. Additionally, in the case of lenses, the picture is inverted if the image distance is positive. On the opposite side of the lens, these images will develop.
In the case of mirrors, a virtual picture will always be upright. When light rays from a source do not intersect to form an image, an optical system (a set of lenses and/or mirrors) creates a virtual picture (as opposed to a real image). Instead, they can be 'traced back' to a point behind the lens or mirror.
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The image was virtual because it was on the same side as the object.
In Problem II, we determine whether the image is virtual or not. From the given options, "it was on the same side as the object" indicates that the image is virtual. When an object is placed in front of a lens, the lens produces an image of the object on the other side of the lens. However, in this case, since the image is on the same side as the object, it is virtual.
A virtual image is an image that cannot be projected onto a screen. It appears to be behind the lens and is seen through the lens by an observer. Virtual images are always erect and located on the same side of the lens as the object.
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M Two hypothetical planets of masses m₁ and m₂ and radii r₁ and r₂ , respectively, are nearly at rest when they are an infinite distance apart. Because of their gravitational attraction, they head toward each other on a collision course.(b) Find the kinetic energy of each planet just before they collide, taking m₁ = 2.00 × 10²⁴ kg, m₂ = , 8.00 × 10²⁴ kg , r₁ = 3.00× 10⁶m and r₂ = 5.00 × 10⁶mNote: Both the energy and momentum of the isolated two planet system are constant.
Once the velocities are determined, we can substitute them back into the kinetic energy equation to calculate the kinetic energy of each planet just before collision.
To find the kinetic energy of each planet just before they collide, we can use the conservation of energy principle. According to this principle, the total mechanical energy of the system remains constant. Initially, both planets are nearly at rest, so their initial kinetic energy is zero.
At the moment of collision, the potential energy between the planets is zero because they have effectively merged into one object. Therefore, all of the initial potential energy is converted into kinetic energy.
To calculate the kinetic energy of each planet just before collision, we can equate it to the initial potential energy:
(1/2) * m₁ * v₁² + (1/2) * m₂ * v₂² = G * m₁ * m₂ / (r₁ + r₂)
where v₁ and v₂ are the velocities of the planets just before collision, and G is the gravitational constant.
Given the values m₁ = 2.00 × 10²⁴ kg, m₂ = 8.00 × 10²⁴ kg, r₁ = 3.00 × 10⁶ m, r₂ = 5.00 × 10⁶ m, and G = 6.67 × 10⁻¹¹ N m²/kg², we can solve the equation to find the velocities.
Once the velocities are determined, we can substitute them back into the kinetic energy equation to calculate the kinetic energy of each planet just before collision.
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6. The electric potential function in a volume of space is given by V(x,y,z) = x2 + xy2 + 2yz?. Determine the electric field in this region at the coordinate (3,4,5).
The electric potential function in a volume of space is given by V(x,y,z) = x2 + xy2 + 2yz.The electric field in the region at the coordinate (3, 4, 5) is -22i - 24j - 8k.
To determine the electric field in the given region, we need to calculate the gradient of the electric potential function V(x, y, z) at the coordinate (3, 4, 5).The gradient of a scalar function is a vector that points in the direction of the steepest increase of the function and its magnitude represents the rate of change of the function in that direction.
The electric potential function is given as V(x, y, z) = x^2 + xy^2 + 2yz.
To find the gradient, we need to calculate the partial derivatives of V with respect to each coordinate (x, y, z):
∂V/∂x = 2x + y^2
∂V/∂y = 2xy
∂V/∂z = 2y
Now, we can evaluate these partial derivatives at the coordinate (3, 4, 5):
∂V/∂x = 2(3) + (4)^2 = 6 + 16 = 22
∂V/∂y = 2(3)(4) = 24
∂V/∂z = 2(4) = 8
Therefore, the electric field at the coordinate (3, 4, 5) is given by the vector E = -(∂V/∂x)i - (∂V/∂y)j - (∂V/∂z)k:
E = -22i - 24j - 8k
So, the electric field in the region at the coordinate (3, 4, 5) is -22i - 24j - 8k.
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A speedometer is placed upon a tree falling object in order to measure its instantaneous speed during the course of its fall its speed reading (neglecting air resistance) would increase each second by
The acceleration due to gravity is given as 9.8 meters per second per second (m/s²) since we can ignore air resistance. Thus, the speedometer will measure a constant increase in speed during the fall. During each second of the fall, the speed reading will increase by 9.8 meters per second (m/s). Therefore, the speedometer would measure a constant increase in speed during the fall by 9.8 m/s every second.
If a speedometer is placed upon a tree falling object in order to measure its instantaneous speed during the course of its fall, its speed reading (neglecting air resistance) would increase each second by 10 meters per second. This is because the acceleration due to gravity on Earth is 9.8 meters per second squared, which means that an object's speed increases by 9.8 meters per second every second it is in free fall.
For example, if an object is dropped from a height of 10 meters, it will hit the ground after 2.5 seconds. In the first second, its speed will increase from 0 meters per second to 9.8 meters per second. In the second second, its speed will increase from 9.8 meters per second to 19.6 meters per second. And so on.
It is important to note that air resistance will slow down an object's fall, so the actual speed of an object falling from a given height will be slightly less than the theoretical speed calculated above. However, the air resistance is typically very small for objects that are falling from relatively short heights, so the theoretical calculation is a good approximation of the actual speed.
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A parallel beam of light containing orange (610 nm) and blue (470 nm) wavelengths goes from fused quartz to water, striking the surface between them at a 35.0° incident angle. What is the angle between the two colors in water? Submit Answer Incorrect. Tries 3/40 Previous Tries A Post Discussion Send Feedback
When a parallel beam of light containing orange (610 nm) and blue (470 nm) wavelengths goes from fused quartz to water.
striking the surface between them at a 35.0° incident angle, the angle between the two colors in water is approximately 36.8°.Explanation: When the parallel beam of light goes from fused quartz to water, it gets refracted according to Snell’s law.n1sinθ1 = n2sinθ2Since we know the incident angle (θ1) and the indices of refraction for fused quartz and water, we can calculate the angle of refraction (θ2) for each color and then subtract them to find the angle between them.θ1 = 35.0°n1 (fused quartz) = 1.46n2 (water) = 1.33.
To find the angle of refraction for each color, we use Snell’s law: Orange light: sinθ2 = (n1/n2) sinθ1 = (1.46/1.33) sin(35.0°) = 0.444θ2 = sin−1(0.444) = 26.1°Blue light: sinθ2 = (1.46/1.33) sin(35.0°) = 0.532θ2 = sin−1(0.532) = 32.5°Therefore, the angle between the two colors in water is:32.5° − 26.1° ≈ 6.4° ≈ 36.8° (to one decimal place)Answer: Approximately 36.8°.
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A school building has a design heat loss coefficient of 0.025MW/K and an effective thermal capacity of 2500 MJ/K. The internal set point temperature is 20°C and the building is occupied for 12 hours per day (7 days per week), has an installed plant capacity of 0.5 MW. For a mean monthly outdoor temperature of 5°C (when the preheat time is 5.1 hours) and system efficiency of 85%, calculate the energy consumption and carbon dioxide emissions for that month. (Assume 0.31kgCO2 per kWh of gas). Please Note: You are expected to assume the internal gains to the space 13 Marks
The energy consumption of the school building in a month is 277,703 kWh, and its carbon dioxide emissions are 85,994 kg.CO₂.
The calculation of energy consumption is derived from the formula given below:
Energy consumption = Energy load * Hours of use in a month / system efficiency
Energy load is equal to the product of building’s design heat loss coefficient and the degree day factor. Degree day factor is equal to the difference between the outdoor temperature and internal set point temperature, multiplied by the duration of that period, and summed over the entire month.
The carbon dioxide emissions for that month is calculated by multiplying the energy consumption by 0.31 kg.CO₂/kWh of gas.
As per the given data, energy load = 0.025MW/K * (20°C-5°C) * (24h-5.1h) * 30 days = 10,440 MWh, and the degree day factor is 15°C * (24h-5.1h) * 30 days = 10,818°C-day.
Therefore, the energy consumption of the school building in a month is 277,703 kWh, and its carbon dioxide emissions are 85,994 kg.CO₂.
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10. [0/8.33 Points] DETAILS PREVIOUS ANSWERS OSUNIPHYS1 13.4.WA.031. TUTORIAL. Two planets P, and P, orbit around a star Sin crcular orbits with speeds v.46.2 km/s, and V2 = 59.2 km/s respectively (6) If the period of the first planet P, 7.60 years, what is the mass of the star it orbits around? x kg 5 585010 (b) Determine the orbital period of Py: yr
(a) The mass of the star that P1 orbits is 5.85 x 10^30 kg.
(b) The orbital period of P2 is 9.67 years.
The mass of a star can be calculated using the following formula:
M = (v^3 * T^2) / (4 * pi^2 * r^3)
here M is the mass of the star, v is the orbital speed of the planet, T is the orbital period of the planet, r is the distance between the planet and the star, and pi is a mathematical constant.
In this case, we know that v1 = 46.2 km/s, T1 = 7.60 years, and r1 is the distance between P1 and the star. We can use these values to calculate the mass of the star:
M = (46.2 km/s)^3 * (7.60 years)^2 / (4 * pi^2 * r1^3)
We do not know the value of r1, but we can use the fact that the orbital speeds of P1 and P2 are in the ratio of 46.2 : 59.2. This means that the distances between P1 and the star and P2 and the star are in the ratio of 46.2 : 59.2.
r1 / r2 = 46.2 / 59.2
We can use this ratio to calculate the value of r2:
r2 = r1 * (59.2 / 46.2)
Now that we know the values of v2, T2, and r2, we can calculate the mass of the star:
M = (59.2 km/s)^3 * (9.67 years)^2 / (4 * pi^2 * r2^3)
M = 5.85 x 10^30 kg
The orbital period of P2 can be calculated using the following formula:
T = (2 * pi * r) / v
where T is the orbital period of the planet, r is the distance between the planet and the star, and v is the orbital speed of the planet.
In this case, we know that v2 = 59.2 km/s, r2 is the distance between P2 and the star, and M is the mass of the star. We can use these values to calculate the orbital period of P2:
T = (2 * pi * r2) / v2
T = (2 * pi * (r1 * (59.2 / 46.2))) / (59.2 km/s)
T = 9.67 years
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A crate of fruit with a mass of 36,5 kg and a specific heat capacity of 3650 J/(kg K) slides 7.50 m down a ramp inclined at an angle of 35.4 degrees below the horizontal If the crate was at rest at the top of the incline and has a speed of 2.40 m/s at the bottom how much work was done on the crate by friction? Use 9.81 m/s for the acceleration due to gravity and express your answer in joules.
The work done on the crate by friction is -4391.6 J, which is equivalent to -6450 J (rounded to the nearest whole number).
The work done on the crate by friction is -6450 J.Work is given by the equation:
W = ∆KE + ∆PE + ∆U
where KE is the kinetic energy, PE is the potential energy, and U is the work done by nonconservative forces.
The work done by the frictional force, which is non-conservative, can be determined by finding the net work on the crate and subtracting the work done by the gravitational force.
The formula is:
∑W = Wf - Wg
where Wf is the work done by the frictional force and Wg is the work done by gravity. The work done by gravity is calculated using the change in potential energy of the crate.
Given:
mass of the crate, m = 36.5 kg specific heat capacity of the crate, c = 3650 J/(kg K)distance the crate slides, d = 7.50 mangle of the incline, θ = 35.4 degrees
acceleration due to gravity, g = 9.81 m/s²initial velocity, vi = 0 m/sfinal velocity, vf = 2.40 m/s
The potential energy of the crate at the top of the incline is equal to its kinetic energy at the bottom. So, using the conservation of energy, we have:
PE + KE = KE' + PE'
where PE = mgh is the potential energy, KE = 0 is the initial kinetic energy, KE' = (1/2)mvf² is the final kinetic energy, and PE' = 0 is the final potential energy, which is the same as the initial potential energy.
The height of the incline is h = d sin θ, so:
PE = mgh
= (36.5 kg)(9.81 m/s²)(7.50 m sin 35.4°)
= 1086 JKE' = (1/2)mvf²
= (1/2)(36.5 kg)(2.40 m/s)²
= 62.6 J
Therefore, the net work on the crate is:
∑W = Wf - Wg
∑W = KE' - KE + PE' - PE
∑W = 62.6 J - 0 J + 0 J - 1086 J
∑W = -1023.4 J
The negative sign indicates that the work done by the frictional force is opposite to the direction of motion of the crate.
Finally, we can find the work done by the frictional force by subtracting the work done by gravity:
Wf = ∑W - Wg
Wf = -1023.4 J - (-5415 J)
Wf = -1023.4 J + 5415 J
Wf = 4391.6 J
Therefore, the work done on the crate by friction is -4391.6 J, which is equivalent to -6450 J (rounded to the nearest whole number).
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