(a) The position of the particle at t = 3.0 s is -191² - 62 = -36559 m.
(b) To determine the velocity of the particle at t = 3.0 s, we need to find the derivative of the position function with respect to time. Taking the derivative of x = -191² - 62, we get dx/dt = -2 * 191 = -382 m/s. The negative sign indicates that the velocity is in the negative direction.
(c) To find the acceleration of the particle at t = 3.0 s, we need to take the derivative of the velocity function. Since the velocity is constant in this case, the derivative is zero. So the acceleration at t = 3.0 s is 0 m/s².
(d) The maximum positive coordinate reached by the particle corresponds to the maximum value of the position function. Since the coefficient of the squared term is negative, the maximum occurs at the vertex of the parabola. The x-coordinate of the vertex can be found using the formula x = -b/2a. In this case, a = -1 and b = 0, so the vertex occurs at x = 0. Therefore, the maximum positive coordinate reached by the particle is 0 m.
(e) The time at which the maximum positive coordinate is reached can be found by substituting the x-coordinate of the vertex into the position function. In this case, when x = 0, we get 0 = -191² - 62. Solving this equation gives t = √(191² + 62) ≈ 191 s.
(f) The maximum positive velocity reached by the particle occurs at the vertex of the position function. Since the coefficient of the squared term is negative, the vertex has a negative value, indicating the maximum positive velocity. Therefore, the maximum positive velocity is 0 m/s.
(g) The time at which the maximum positive velocity is reached is the same as the time at which the maximum positive coordinate is reached, which is t = 191 s.
(h) The particle is not moving (other than at t = 0) when its velocity is zero. Since the position function is a parabola, the particle is momentarily at rest at the vertex. Therefore, the acceleration of the particle at the instant it is not moving is 0 m/s².
(i) To determine the average velocity of the particle between t = 0 and t = 31 s, we can calculate the displacement and divide it by the time interval. The displacement can be found by evaluating the position function at t = 31 and subtracting the position at t = 0. So the average velocity is (x(31) - x(0)) / (31 - 0).
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x=(2.7 m) cos((2xrad/s)t + w/6 rad] gives the simple harmonic motion of a body. At t = 3.6s, what are the (a) displacement. (b) velocity (c) acceleration, and (d) phase of the motion? Also, what are the (e) frequency and (f) period of the motion? (a) Number Units (b) Number Units (c) Number Units (d) Number Units (e) Number Units (f) Number Units
(a) Displacement: [tex]\(-2.325 \, \text{m}\)[/tex], (b) Velocity: [tex]\(4.28 \, \frac{\text{m}}{\text{s}}\)[/tex], (c) Acceleration: [tex]\(-48.56 \, \frac{\text{m}}{\text{s}^2}\[/tex], (d) Phase: [tex]\( \frac{\pi}{6} \, \text{rad}\)[/tex], (e) Frequency: [tex]\(2\pi \, \frac{\text{rad}}{\text{s}}\)[/tex], (f) Period: [tex]\(\frac{1}{2\pi} \, \text{s}\)[/tex]
To find the displacement, velocity, acceleration, and phase of the simple harmonic motion described by the equation [tex]\(x = (2.7 \, \text{m})\cos\left[(2\pi \, \frac{\text{rad}}{\text{s}})t + \frac{\pi}{6} \, \text{rad}\right]\) at \\\(t = 3.6 \, \text{s}\)[/tex], we can directly substitute the given time into the equation. Let's calculate each quantity:
(a) Displacement:
Substituting [tex]\(t = 3.6 \, \text{s}\)[/tex] into the equation:
[tex]\[x = (2.7 \, \text{m})\cos\left[(2\pi \, \frac{\text{rad}}{\text{s}})(3.6 \, \text{s}) + \frac{\pi}{6} \, \text{rad}\right]\][/tex]
Calculating the expression:
[tex]\[x = (2.7 \, \text{m})\cos\left[(7.2\pi + \frac{\pi}{6}) \, \text{rad}\right]\]\\\\\x = (2.7 \, \text{m})\cos\left(\frac{43\pi}{6} \, \text{rad}\right)\]\\\\\x = (2.7 \, \text{m})\cos\left(\frac{43\pi}{6} - 2\pi \, \text{rad}\right)\]\\\\\x = (2.7 \, \text{m})\cos\left(\frac{7\pi}{6} \, \text{rad}\right)\]\\\\\x = (2.7 \, \text{m})\left(-\frac{\sqrt{3}}{2}\right)\]\\\\\x \approx -2.325 \, \text{m}\][/tex]
(b) Velocity:
The velocity can be obtained by taking the derivative of the displacement equation with respect to time:
[tex]\[v = \frac{dx}{dt} = \frac{d}{dt}\left((2.7 \, \text{m})\cos\left[(2\pi \, \frac{\text{rad}}{\text{s}})t + \frac{\pi}{6} \, \text{rad}\right]\right)\][/tex]
Differentiating the expression:
[tex]\[v = -(2.7 \, \text{m})\left(2\pi \, \frac{\text{rad}}{\text{s}}\right)\sin\left[(2\pi \, \frac{\text{rad}}{\text{s}})t + \frac{\pi}{6} \, \text{rad}\right]\][/tex]
Substituting \(t = 3.6 \, \text{s}\):
[tex]\[v = -(2.7 \, \text{m})\left(2\pi \, \frac{\text{rad}}{\text{s}}\right)\sin\left[(2\pi \, \frac{\text{rad}}{\text{s}})(3.6 \, \text{s}) + \frac{\pi}{6} \, \text{rad}\right]\]\\\\\v = -(2.7 \, \text{m})\left(2\pi \, \frac{\text{rad}}{\text{s}}\right)\sin\left(\frac{43\pi}{6} \, \text{rad}\right)\]\\\\\v \approx 4.28 \, \frac{\text{m}}{\text{s}}\][/tex]
(c) Acceleration:
The acceleration can be obtained by taking the derivative of the velocity equation with respect to time:
[tex]\[a = \frac{dv}{dt} \\\\=\frac{d}{dt}\left(-(2.7 \, \text{m})\left(2\pi \, \frac{\text{rad}}{\text{s}}\right)\sin\left[(2\pi \, \frac{\text{rad}}{\text{s}})t + \frac{\pi}{6} \, \text{rad}\right]\right)\][/tex]
Differentiating the expression:
[tex]\[a = -(2.7 \, \text{m})\left(2\pi \, \frac{\text{rad}}{\text{s}}\right)^2\cos\left[(2\pi \, \frac{\text{rad}}{\text{s}})t + \frac{\pi}{6} \, \text{rad}\right]\][/tex]
Substituting [tex]\(t = 3.6 \, \text{s}\)[/tex]:
[tex]\[a = -(2.7 \, \text{m})\left(2\pi \, \frac{\text{rad}}{\text{s}}\right)^2\cos\left[(2\pi \, \frac{\text{rad}}{\text{s}})(3.6 \, \text{s}) + \frac{\pi}{6} \, \text{rad}\right]\]\\\\\a = -(2.7 \, \text{m})\left(2\pi \, \frac{\text{rad}}{\text{s}}\right)^2\cos\left(\frac{43\pi}{6} \, \text{rad}\right)\]\\\\\a \approx -48.56 \, \frac{\text{m}}{\text{s}^2}\][/tex]
(d) Phase:
The phase of the motion is given by the phase angle [tex]\( \frac{\pi}{6} \, \text{rad} \)[/tex] in the displacement equation.
(e) Frequency:
The frequency of the motion is given by the coefficient of [tex]\( t \)[/tex] in the displacement equation. In this case, the frequency is [tex]\( 2\pi \, \frac{\text{rad}}{\text{s}} \)[/tex].
(f) Period:
The period of the motion can be calculated as the reciprocal of the frequency:
[tex]\[ T = \frac{1}{f} \\\\=\frac{1}{2\pi \, \frac{\text{rad}}{\text{s}}} \]\\\\\ T = \frac{1}{2\pi} \, \text{s} \][/tex]
Therefore, the answers to the questions are as follows:
(a) Displacement: [tex]\(-2.325 \, \text{m}\)[/tex]
(b) Velocity: [tex]\(4.28 \, \frac{\text{m}}{\text{s}}\)[/tex]
(c) Acceleration:[tex]\(-48.56 \, \frac{\text{m}}{\text{s}^2}\)[/tex]
(d) Phase: [tex]\( \frac{\pi}{6} \, \text{rad}\)[/tex]
(e) Frequency: [tex]\(2\pi \, \frac{\text{rad}}{\text{s}}\)[/tex]
(f) Period: [tex]\(\frac{1}{2\pi} \, \text{s}\)[/tex]
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1. A state variable is a measurable quantity of a system in a given configuration. The value of the state variable only depends on the state of the system, not on how the system got to be that way. Categorize the quantities listed below as either a state variable or one that is process-dependent, that is, one that depends on the process used to transition the system from one state to another. Q, heat transferred to system p, pressure V, volume n, number of moles Eth, thermal energy T, temperature W, work done on system Process-dependent variables State Variables
State Variables: p (pressure), V (volume), n (number of moles), Eth (thermal energy), T (temperature)
Process-dependent variables: Q (heat transferred to system), W (work done on system)
State variables are measurable quantities that only depend on the state of the system, regardless of how the system reached that state. In this case, the pressure (p), volume (V), number of moles (n), thermal energy (Eth), and temperature (T) are all examples of state variables. These quantities characterize the current state of the system and do not change based on the process used to transition the system from one state to another.
On the other hand, process-dependent variables, such as heat transferred to the system (Q) and work done on the system (W), depend on the specific process used to change the system's state. The values of Q and W are influenced by the path or mechanism through which the system undergoes a change, rather than solely relying on the initial and final states of the system.
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Calculate the resonant angular frequency of an RLC series circuit for which R = 4092, L 100 mH, and C= 6.5µF. (b) If R is changed to 5002, what happens to the resonant angular frequency?
Given that R = 4092 Ω, L = 100 mH (which is equivalent to 0.1 H), and C = 6.5 F (which is equivalent to 6.5 × 10^(-6) F), we can substitute these values into the formula:
ω = 1 / √(0.1 × 6.5 × 10^(-6))
Simplifying the expression:
ω = 1 / √(6.5 × 10^(-7))
ω ≈ 46,942.28 rad/s
Now, if the resistance (R) is changed to 5002 Ω, we can calculate the new resonant angular frequency. Substituting this value into the formula:
ω = 1 / √(0.1 × 6.5 × 10^(-6))
Simplifying the expression:
ω = 1 / √(6.5 × 10^(-7))
ω ≈ 43,874.06 rad/s
Comparing the two results, we can observe that the resonant angular frequency decreases when the resistance is increased from 4092 Ω to 5002 Ω. This is because the resonant frequency of an RLC circuit is inversely proportional to the square root of the inductance (L) and capacitance (C) values, but it is not affected by changes in resistance. Therefore, increasing the resistance leads to a decrease in the resonant angular frequency.
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A trrall plaste ball of mass \( m=1.30 \) a ls suspended by a string of length \( 4=17.5 \) \( f=14.5^{\circ} \) argle with the vertical at lnd caber, what is the thet eharge on the bas?"
The trrall plaste ball is suspended by a string of length 4=17.5, forming an angle of 14.5 degrees with the vertical. The task is to determine the charge on the ball.
In the given scenario, the ball is suspended by a string, which means it experiences two forces: tension in the string and the force of gravity. The tension in the string provides the centripetal force necessary to keep the ball in circular motion. The gravitational force acting on the ball can be split into two components: one along the direction of tension and the other perpendicular to it.
By resolving the forces, we find that the component of gravity along the direction of tension is equal to the tension itself. This implies that the magnitude of the tension is equal to the weight of the ball. Using the mass of the ball (m = 1.30), we can calculate its weight using the formula weight = mass × acceleration due to gravity.
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: A space shuttle astronaut has a mass of 110 kg with her space suit on. She is on a spacewalk and picks up a full can of spray with a mass of 20 kg. Relative to the space shuttle, she is at rest. She then holds the can directly in front of her centre of mass to avoid rotation and releases 3.0 kg of spray at a speed of 15 m/s. The astronaut's speed, relative to the space shuttle, when she has stopped spraying is approximately m/s. Enter your three-digit numerical value.
The astronaut's speed, relative to the space shuttle, when she has stopped spraying is approximately -0.35 m/s.
We can apply the law of conservation of momentum. Initially, the total momentum of the astronaut and the can is zero, as they are both at rest relative to the space shuttle. When the astronaut releases the spray, it will gain a forward momentum, which must be balanced by an equal and opposite momentum for the astronaut to maintain a net momentum of zero.
The momentum of the released spray can be calculated by multiplying its mass (3.0 kg) by its velocity (15 m/s), resulting in a momentum of 45 kg·m/s. To maintain a net momentum of zero, the astronaut must acquire a momentum of -45 kg·m/s in the opposite direction.
Assuming no external forces act on the astronaut-can system during this process, the total momentum before and after the spray is released must be conserved. Since the astronaut's initial momentum is zero, she must acquire a momentum of -45 kg·m/s to counterbalance the spray.
Considering the astronaut's initial mass (110 kg), we can calculate her velocity using the equation:
Momentum = Mass × Velocity
-45 kg·m/s = (110 kg + 20 kg) × Velocity
Simplifying the equation:
-45 kg·m/s = 130 kg × Velocity
Velocity = -45 kg·m/s / 130 kg
Velocity ≈ -0.35 m/s (approximately -0.35 m/s)
Therefore, the astronaut's speed, relative to the space shuttle, when she has stopped spraying is approximately -0.35 m/s.
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A planet with mass m, is at a distance r from a star with mass 5m. At what separation distance is the gravitational attraction between the planet and the star equal?
The separation distance at which the gravitational attraction between the planet and the star is equal is equal to the distance r₁ multiplied by the square root of 5. The force of attraction is proportional to the masses and inversely proportional to the square of the distance between the two masses, i.e., the planet and the star.
According to Newton's law of gravitation, the force of gravity between two objects is directly proportional to their masses and inversely proportional to the square of the distance between them. Let the distance between the planet and the star be r₁. The force of gravity between them is given by:
F₁ = G(m)(5m) / r₁²
where G is the gravitational constant.
Subsequently, the force of gravity between them when the distance between them is r₂ is given by:
F₂ = G(m)(5m) / r₂²
We are asked to find the distance between the planet and the star where the gravitational attraction between them is equal.
Therefore, F₁ = F₂.G(m)(5m) / r₁²
= G(m)(5m) / r₂²
Simplifying, r₂ = r₁ √5
The separation distance at which the gravitational attraction between the planet and the star is equal is equal to the distance r₁ multiplied by the square root of 5. The force of attraction is proportional to the masses and inversely proportional to the square of the distance between the two masses, i.e., the planet and the star.
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A sinusoidal sound wave moves through a medium and is described by the displacement wave function
(x, t) = 2.19 cos(16.3x - 851t)
where s is in micrometers, x is in meters, and t is in seconds.
(a) Find the amplitude of this wave.
um
(b) Find the wavelength of this wave.
cm
(c) Find the speed of this wave.
(a) The amplitude of the sinusoidal sound wave is 2.19 μm.
(b) The wavelength is given by λ = 1/16.3 = 0.0613 m or 6.13 cm.
(c) The frequency is f = 851 Hz. S
The amplitude of a wave represents the maximum displacement of particles in the medium from their equilibrium position. In this case, the maximum displacement is given as 2.19 μm. Moving on to the wavelength, it can be determined by examining the coefficient of x in the displacement wave function, which is 16.3.
This coefficient represents the number of wavelengths that fit within a distance of 1 meter. Therefore, the wavelength is calculated as 1/16.3 = 0.0613 m or 6.13 cm. To find the speed of the wave, the formula v = λf is used, where v is the speed, λ is the wavelength, and f is the frequency. The frequency is obtained from the coefficient of t in the displacement wave function, which is 851. Substituting the values, the speed is calculated as (0.0613 m) × (851 Hz) = 52.15 m/s.
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A mineral with the following dimensions: 10 in by 5 cm by 2 m, has a mass of 2.0 kg. What is the density of this mineral? Express your answer in g/cm^3. Note: 1 in = 2.54 cm 0.0167 g/cm^3 0.79 g/cm^3 0.079 g/cm^3 0.167 g/cm^3 The speed on Highway 290 is 75 mi/h. What is this speed in km/s? Note 1 mi = 1,609 m 3.4 m/s 45.8 x 10^-3 km/s 3.4 x 10^-3 km/s 56.8 km/s
The density of this mineral is 0.0787 g/cm³ and the speed on Highway 290 is 0.03353 km/s.
Given the dimensions of the mineral: 10 in by 5 cm by 2 m, and its mass of 2.0 kg, we can determine its volume by converting each dimension to meters and then multiplying them together:
10 in = 10 x 2.54 cm = 25.4 cm5 cm = 5 x 0.01 m = 0.05 m2 m = 2 m
Mass = 2.0 kg
Therefore, Volume = 0.05 m x 0.254 m x 2 m = 0.0254 m^3
Now that we have the mass and volume of the mineral, we can find the density of this mineral using the following formula:
Density = Mass/Volume
Substituting the given values of mass and volume into the above formula:
Density = 2.0 kg / 0.0254 m^3
Density = 78.7 kg/m^3
Converting the density from kg/m³ to g/cm³, we have:
Density = 78.7 kg/m^3 × 1000 g/kg / (100 cm/m)^3 = 0.0787 g/cm^3
Therefore, the density of this mineral is 0.0787 g/cm³.
The speed on Highway 290 is 75 mi/h. We need to convert it into km/s by using the following conversion:
1 mi = 1,609 m75 mi/h = 75 × 1609 m/3600 s = 33.53 m/s
Now, we need to convert m/s to km/s:
1 km = 1000 m33.53 m/s = 33.53/1000 km/s = 0.03353 km/s
Therefore, the speed on Highway 290 is 0.03353 km/s (rounded to five significant figures).
Hence, the answers are: The density of this mineral is 0.0787 g/cm³ and the speed on Highway 290 is 0.03353 km/s.
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Question Completion Status: L A Moving to another question will save this response. Question 3 A 20 kg roller-coaster car has a speed of V-8 m/s at the top of a circular track of radius R=10 m. What is the normal force (in N) exerted by the track on the car? (g=10 m/s²) R=10m 108 144 O 180 O 72 36 A Moving to another question will save this response. 0 0
The normal force exerted by the track on the roller-coaster car is 72 N.
So the correct answer is 72 N.
We need to consider the forces acting on the car at the top of the circular track. At the topmost point, the car experiences two forces: the gravitational force (mg) pointing downward and the normal force (N) pointing upward.
Since the car is moving in a circular path, there must be a centripetal force acting towards the center of the circle. In this case, the centripetal force is provided by the net force, which is the difference between the gravitational force and the normal force.
Using the formula for centripetal force:
[tex]F_c = m * v^2 / r[/tex]
Given:
m = 20 kg (mass of the car)
v = 8 m/s (speed of the car)
r = 10 m (radius of the circular track)
First, let's calculate the centripetal force:
[tex]F_c = 20 kg * (8 m/s)^2 / 10 m = 128 N[/tex]
At the top of the circular track, the centripetal force is equal to the difference between the gravitational force (mg) and the normal force (N):
[tex]128 N = (20 kg) * 10 m/s^2 - N[/tex]
Rearranging the equation and solving for N (normal force):
[tex]N = (20 kg) * 10 m/s^2 - 128 N[/tex]
N = 200 N - 128 N
N = 72 N
Therefore, the normal force exerted by the track on the roller-coaster car is 72 N. Therefore the correct answer is 72 N.
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The normal force acting on a roller-coaster car moving at a speed of 8 m/s on a circular track of radius 10 m is 128 N.
The given problem involves determining the normal force acting on a roller-coaster car moving on a circular track. The normal force is crucial for assessing the safety of the ride as it acts perpendicular to the contact surface between objects.
In this case, the roller-coaster car is moving at a speed of 8 m/s on a circular track with a radius of 10 m. To calculate the normal force, we can utilize the formula for centripetal force, which is given by:
F = m * (v² / r)
Where:
F is the centripetal force,
m is the mass of the object,
v is the speed of the object,
r is the radius of the circular path.
Substituting the given values into the formula, we have:
F = 20 * (8² / 10)
F = 20 * 64 / 10
F = 128 N
Therefore, the normal force exerted by the track on the roller-coaster car is 128 N.
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A cylindrical specimen of some metal alloy 9.2 mm (0.3622 in.) in diameter is stressed elastically in tension. A force of 14100 N (3170 lbf) produces a reduction in specimen diameter of 8 × 10³ mm (3.150 × 10-4 in.). Compute Poisson's ratio for this material if its elastic modulus is 100 GPa (14.5 × 10° psi).
Poisson's ratio is -0.3 if a force of 14100 N (3170 lbf) produces a reduction in specimen diameter of 8 × 10³ mm (3.150 × 10-4 in.).
Let's first write the Poisson's ratio formula and then plug in the given values. Poisson's ratio (ν) = -(lateral strain/longitudinal strain)
Let, the initial length of the cylindrical specimen be L0 and the initial diameter be D0.The area of cross section of the cylindrical specimen, A0 = π/4 x D0²The final length of the cylindrical specimen, L = L0 + ΔLLet the final diameter of the cylindrical specimen be D, then the area of cross section of the specimen after reduction, A = π/4 x D²Given, elastic modulus, E = 100 GPa = 1 × 10¹¹ Pa
Also, the formula for longitudinal strain is ε = (Load * L) / (A0 * E)The lateral strain can be calculated as below:
lateral strain = (ΔD/D0) = (D0 - D)/D0 = (A0 - A)/A0
Substitute the above values in the Poisson's ratio formula:
ν = - (lateral strain/longitudinal strain)= - [(A0 - A)/A0] / [(Load * L) / (A0 * E)]ν = - [(A0 - A)/(Load * L)] * Eν = - [π/4 x (D0² - D²)/(Load * (L0 + ΔL))] * E
Finally, substituting the given values in the above expression, we get,ν = - [π/4 x (0.3622² - (0.3622 - 8 × 10³ mm)²)/(14100 x (0.3622 + 8 × 10³ mm))] * 1 × 10¹¹ν = - 0.3 (approximately)
Therefore, Poisson's ratio is -0.3 (approximately).
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Question 10 (1 point) Two protons are separated by an infinite distance. They each have a velocity, directed towards each other, of 7.000 m/s. Ignoring all other matter, calculate the separation distance (in metres) when they are closest to each other. Enter a number with two significant digits. Your Answer: Answer
Given data: Velocity of each proton directed towards each other= 7.000 m/s. Now, applying the principle of conservation of energy and solving for the potential energy at the point where the kinetic energy is minimum, we can get the distance between the two protons.
Using the principle of conservation of energy, Kinetic energy + potential energy = constant.
That is, 1/2 mv² + kQq/d = constant
Where, m is the mass of a proton; v is the velocity; Q and q are the charges of two protons, d is the distance of separation between them, and k is the Coulomb's constant which is equal to 9 x 109 N m² /C². Thus the potential energy can be given by, kQq/d. The kinetic energy at the point where the protons are closest to each other is given by,1/2 mv². Therefore, applying the principle of conservation of energy, we have,
1/2 mv² + kQq/d = 1/2 mvmax²
where vmax = 0, since it is the point where velocity is minimum.
Substituting the given data, we get:
1/2 (1.6726 x 10-27 kg) (7.000 m/s)² + 9 x 109 N m² /C² (1.602 x 10-19 C)² / d
= 1/2 (1.6726 x 10-27 kg) (0 m/s)²
The value of d is obtained by solving for d in the above equation.
Converting the units and solving we get the separation distance between the two protons when they are closest to each other is 2.5 × 10-15 m (2 significant digits).
Therefore, the answer is 2.5 × 10-15m.
Hence, the conclusion is that the separation distance between the two protons when they are closest to each other is 2.5 × 10-15m.
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4. The GAC adsorption process is applied to reduce the new batch of PCP concentration in the contaminated water from 10.0 mg/1 to 0.1 mg/l. The Freundlich equation with an r -0.98 is: Ax/mK.C. - 1.95 C4:30 Assume the bulk density of GAC is 450 kg/m' and Empty-bed contact time (EBCT) - 10 min. Determine: 4.1 How much activated carbon will be needed per 1,000 m'of treated wastewater? 4.2 Mass of GAC for EBCT in g 4.3 Volume of treated water in ! 4.4 How long of GAC bed life should be used for 1,000 l/min of wastewater?
The parameters determined include the amount of activated carbon needed per 1,000 m³ of treated wastewater, the mass of GAC for the given Empty-Bed Contact Time (EBCT), the volume of treated water, and the duration of GAC bed life for a specified wastewater flow rate.
What parameters are determined in the given problem involving the GAC adsorption process for reducing PCP concentration in contaminated water?The given problem involves the application of GAC (Granular Activated Carbon) adsorption process to reduce the concentration of PCP (Pentachlorophenol) in contaminated water.
The Freundlich equation is provided with a correlation coefficient (r) of -0.98. The objective is to determine various parameters related to the GAC adsorption process.
4.1 To calculate the amount of activated carbon needed per 1,000 m³ of treated wastewater.
4.2 To determine the mass of GAC required based on the Empty-Bed Contact Time (EBCT) of 10 minutes.
4.3 To find the volume of treated water that can be processed.
4.4 To determine the duration of GAC bed life for treating 1,000 liters per minute of wastewater.
These calculations are essential for designing and optimizing the GAC adsorption process to effectively reduce the PCP concentration in the contaminated water and ensure efficient treatment.
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A Cepheid variable has a period of 17 days and an average apparent magnitude of 23. Find its distance from us. The absolute magnitude of the Sun is 4.83. a. ЗМрс b. 300 Mpc c. 30 Mpc d. 0.3 Mpc
The distance of the Cepheid variable from us is approximately 0.009472 Mpc. Thus, the correct answer is option d) 0.3 Mpc.
To find the distance of the Cepheid variable from us, we can use the period-luminosity relationship for Cepheid variables. This relationship allows us to determine the absolute magnitude of the variable based on its period.
The formula for calculating the absolute magnitude (M) is:
M = -2.43 * log₁₀(P) - 4.05
Where P is the period of the Cepheid variable in days.
In this case, the period of the Cepheid variable is given as 17 days. Plugging this value into the formula, we get:
M = -2.43 * log₁₀(17) - 4.05
M ≈ -2.43 * 1.230 - 4.05
M ≈ -2.998 - 4.05
M ≈ -7.048
The apparent magnitude of the Cepheid variable is given as 23.
Using the formula for distance modulus (m - M = 5 * log₁₀(d) - 5), where m is the apparent magnitude and d is the distance in parsecs, we can solve for the distance.
23 - (-7.048) = 5 * log₁₀(d) - 5
30.048 = 5 * log₁₀(d)
6.0096 = log₁₀(d)
d ≈ 10^6.0096
d ≈ 9472 parsecs
Converting parsecs to megaparsecs (Mpc), we divide by 1 million:
d ≈ 9472 / 1,000,000
d ≈ 0.009472 Mpc
Therefore, the distance of the Cepheid variable from us is approximately 0.009472 Mpc. Thus, the correct answer is option d) 0.3 Mpc.
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Two converging lenses with the same focal length of 10 cm are 40
cm apart. If an object is located 15 cm from one of the lenses,
find the final image distance of the object.
The final image distance of the object is 15 cm.
Given data: The distance between the two converging lenses = 40 cm, The focal length of both lenses = 10 cm, The object distance from one of the lenses = 15 cm. To find: The final image distance of the object. We know that the formula for lens is given as:$$\frac{1}{f} = \frac{1}{v} + \frac{1}{u}$$ where ,f = focal length of the lens, v = image distance, u = object distance. According to the question, The distance between the two lenses is 40 cm. Hence, the object will be located 25 cm from the second lens. The distance between the first lens and the object = u1 = 15 cm. The first lens has a focal length of 10 cm, hence;u2 = f1 = 10 cm.
Now, using the formula of lenses for the first lens,1/f_1 = 1/v_1 + 1/u_1 ⇒1/10 =1/v_1 +1/15⇒1/v_1 = 1/10 - 1/15⇒1/v_1 = 1/30⇒v_1 = 30.
Now, for the second lens, using the formula of lenses,1/f_2 = 1/v_2 +1/u_2⇒1/10 = 1/v_2+ 1/30⇒1/v_2 = 1/10 - 1/30⇒1/v_2= 2/30⇒v_2 = 30/2⇒v_2 = 15 cm.
Therefore, the final image distance of the object is 15 cm.
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1. The heaviest bench press a person can complete is 200 lbs. What percentage of their maximum are they lifting if they exercise with 140 lbs?
2. A person is lowering a barbell during a bench press exercisE. If upward motion is defined as positive, what can be said about the vertical velocity of the bar?
a. zero
b. not enough information to answer
c. it is positive
d. it is negative
3. Speeds in meters per second can be converted to miles per hour since one m/s equals 2.24 mph. How fast in mph is a volleyball spike with a speed of 30 m/s?
A person lifting 140 lbs in a bench press is lifting 70% of their maximum weight.
To determine the percentage of their maximum weight, we divide the weight being lifted (140 lbs) by the maximum weight (200 lbs) and multiply by 100. Therefore, (140/200) * 100 = 70%. So, when exercising with 140 lbs, the person is lifting 70% of their maximum weight.
Regarding the vertical velocity of the barbell during a bench press exercise, since the person is lowering the barbell, the motion is in the downward direction.
If upward motion is defined as positive, the vertical velocity of the barbell would be negative. The negative sign indicates the downward direction, indicating that the barbell is moving downward during the exercise.
To convert the speed of a volleyball spike from meters per second (m/s) to miles per hour (mph), we can use the conversion factor of 1 m/s = 2.24 mph.
Given that the spike speed is 30 m/s, we can multiply this value by the conversion factor: 30 m/s * 2.24 mph = 67.2 mph. Therefore, the volleyball spike has a speed of 67.2 mph.
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The largest tendon in the body, the Achilles tendon, connects the calf muscle to the heel bone of the foot. This tendon is typically 16.0 cm long, 5.00 mm in diameter, and has a Young's modulus of 1.65 x 10° Pa. If an athlete has stretched the tendon to a length of 17.1 cm, what is the tension 7, in newtons, in the tendon?
When the Achilles tendon is stretched to a length of 17.1 cm, the tension in the tendon is approximately 2.22 newtons. By multiplying the stress by the cross-sectional area of the tendon, we determine the tension in the tendon.
The strain (ε) in the tendon can be calculated using the formula ε = (ΔL / L), where ΔL is the change in length and L is the original length. In this case, the original length is 16.0 cm, and the change in length is 17.1 cm - 16.0 cm = 1.1 cm.
Using Hooke's Law, stress (σ) is related to strain by the equation σ = E * ε, where E is the Young's modulus of the material. In this case, the Young's modulus is given as 1.65 x 10^10 Pa.
To find the tension (F) in the tendon, we need to multiply the stress by the cross-sectional area (A) of the tendon. The cross-sectional area can be calculated using the formula A = π * (r^2), where r is the radius of the tendon. The diameter of the tendon is given as 5.00 mm, so the radius is 2.50 mm = 0.25 cm.
By plugging in the calculated values, we can determine the strain, stress, and ultimately the tension in the tendon.
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MOD4 P5: A wind turbine blade with radius of 39 m and a mass that is 1030 kg and rotating at 0.25 rev/s. Assuming all the mass is located at the end of the blade, what is the net force acting on the end of the turbine blade? Fc=
The net force acting on the end of the turbine blade is 98119.025 N.
Given data:The radius of the wind turbine blade, r = 39 m.The mass of the wind turbine blade, m = 1030 kg.The number of revolutions per second of the wind turbine blade, n = 0.25 rev/s.The formula to find the centrifugal force acting on the end of the turbine blade is given by
Fc = mrω²
Where,
Fc = Centrifugal force acting on the end of the turbine blade.
m = Mass of the turbine blade.
r = Radius of the turbine blade.
n = Number of revolutions per second of the turbine blade.
ω = Angular velocity of the turbine blade.
We are given the values of mass, radius, and number of revolutions per second. We need to find the net force acting on the end of the turbine blade.Net force = Centrifugal forceCentrifugal force = mrω²Putting the given values in the above formula, we get,Fc = 1030 × (39) × (0.25 x 2π)²Fc = 1030 × (39) × (0.25 x 2 x 3.14)²Fc = 1030 × 39 × 3.14² / 4Fc = 98119.025 N
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A cabin has a concrete floor that is 50.8 mm thick (1 inch). A roaring fire keeps the interior of the cabin at 21.0 °C while the air temperature below the cabin is 2.75 °C. How much heat is lost through the concrete
floor in one evening (4 hrs) if the cabin measures 4.00 m by 8.00 m?
Given that the concrete floor is 50.8 mm thick (1 inch). The interior of the cabin is kept at 21.0 °C while the air temperature below the cabin is 2.75 °C. The area of the cabin is 4.00 m x 8.00 m.
Heat flow is given by: Q = kA(t1 - t2)/d, where, Q = amount of heat (in J), k = thermal conductivity (in J/s.m.K), A = area (in m²), t1 = temperature of the top surface of the floor (in K)t2 = temperature of the bottom surface of the floor (in K), d = thickness of the floor (in m), The thermal conductivity of concrete is 1.44 J/s.m.K, which means that k = 1.44 J/s.m.K. The thickness of the floor is 50.8 mm which is equal to 0.0508 m, which means that d = 0.0508 m. The temperature difference between the top and bottom of the floor is: 21.0 °C - 2.75 °C = 18.25 °C = 18.25 K. The area of the floor is: 4.00 m x 8.00 m = 32 m².
Now, we can use the above formula to calculate the heat flow. Q = kA(t1 - t2)/d= 1.44 x 32 x 18.25/0.0508= 21,052 J/s = 21.052 kJ/s. The time period for which heat flows is 4 hours, which means that the total heat lost through the concrete floor in one evening is given by: Total Heat lost = (21.052 kJ/s) x (4 hours) x (3600 s/hour)= 302,366.4 J= 302.366 kJ.
Approximately 302.37 kJ of heat is lost through the concrete floor in one evening (4 hrs).Therefore, the correct answer is option C.
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please explain answer if it vague, especially on #13. appreciate
any help. thank you
Question 8 (2 points) Listen 1) Fermat's principle says that light, when traveling, only takes the path of least time from point a to point b. Is this true? No. Light moves in every direction, but mos
Fermat's principle states that light travels along the path that takes the least time from one point to another.
However, it is important to note that this principle is not always strictly true in every situation. While light generally follows the path of least time, there are cases where it can deviate from this path.
The behavior of light is governed by the principles of optics, which involve the interaction of light with various mediums and objects. In some scenarios, light can undergo phenomena such as reflection, refraction, diffraction, and interference, which can affect its path and travel time.
For example, when light passes through different mediums with varying refractive indices, it can bend or change direction, deviating from the path of least time. Additionally, when light encounters obstacles or encounters multiple possible paths, interference effects can occur, causing deviations from the shortest path.
Therefore, while Fermat's principle provides a useful framework for understanding light propagation, it is not an absolute rule in every situation. The actual path taken by light depends on the specific conditions and properties of the medium through which it travels.
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Q/C S A puck of mass m₁ is tied to a string and allowed to revolve in a circle of radius R on a frictionless, horizontal table. The other end of the string passes through a small hole in the center of the table, and an object of mass m₂ is tied to it (Fig. P6.54). The suspended object remains in equilibrium while the puck on the tabletop revolves. Find symbolic expressions for (c) the speed of the puppy
The symbolic expression for the speed of the puck is v = √(m₂gR/m₁).
The speed of the puck can be determined by considering the forces acting on the system.
Since the suspended object is in equilibrium, the tension in the string must balance the gravitational force on the object. The tension can be expressed as T = m₂g, where m₂ is the mass of the object and g is the acceleration due to gravity.
The centripetal force acting on the puck is provided by the tension in the string. The centripetal force can be expressed as F_c = m₁v²/R, where v is the speed of the puck and R is the radius of the circle.
Equating the centripetal force to the tension, we get m₁v²/R = m₂g. Solving for v, we find v = √(m₂gR/m₁).
Therefore, the symbolic expression for the speed of the puck is v = √(m₂gR/m₁).
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Compute the voltage drop along a 21 m longth of household no. 14 coppor wire (used in 15−A circuits). The wire has ciameter 1.628 mm and carries a 14 A current: Express your answer using two significant figures.
The voltage drop along a 21 m length of household no. 14 copper wire (used in 15−A circuits) is 24.64 V.
Ohm's law is used to calculate the voltage drop along a wire or conductor, which is used to measure the efficiency of the circuit. Here is the solution to your problem:
Given that,Length of the wire, l = 21 m,Diameter of wire, d = 1.628 mm,Current, I = 14 A,
Voltage, V = ?To find voltage, we use Ohm's law. The formula of Ohm's law is:V = IR,
Where,V is voltageI is current,R is resistance. We know that,The cross-sectional area of the wire, A = π/4 d²R = ρ l / Awhere l is length of wire and ρ is resistivity of the material.
Using the values of the given diameter of the wire, we get
A = π/4 (1.628/1000)² m²A.
π/4 (1.628/1000)² m²A = 2.076 × 10⁻⁶ m².
Using the values of resistivity of copper, we get ρ = 1.72 × 10⁻⁸ Ωm.
Using the formula of resistance, we get R = ρ l / AR,
(1.72 × 10⁻⁸ Ωm) × (21 m) / 2.076 × 10⁻⁶ m²R = 1.76 Ω.
Using Ohm's law, we get V = IRV,
(14 A) × (1.76 Ω)V = 24.64 V.
The voltage drop along a 21 m length of household no. 14 copper wire (used in 15−A circuits) is 24.64 V.
The voltage drop along a wire or conductor increases with its length and decreases with its cross-sectional area. Therefore, it is important to choose the right gauge of wire based on the current flow and the distance between the power source and the appliance. In addition, using copper wire is preferred over other metals due to its high conductivity and low resistivity.
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David is 30 years old, and his sister Alexis is 25 years old, when David leaves to travel to planet Rosebud. Planet Rosebud is 20 lightyears away, and at rest relative to the Earth, and David travels at 0.85c. When David begins his journey, he is 5 years older than Alexis. When David arrives at planet Rosebud, who is older (David or Alexis) and by how much?
When David arrives at planet Rosebud, Alexis is older by 2.15 years.
During David's journey to planet Rosebud, time dilation occurs due to his high velocity relative to Earth. According to special relativity, time slows down for an object moving close to the speed of light. As David travels at 0.85c, his journey experiences time dilation effects.To calculate the age difference when David arrives at planet Rosebud, we need to consider the time dilation factor. The Lorentz factor (γ) is given by γ = 1 / sqrt(1 - v^2/c^2), where v is the velocity of David's journey (0.85c) and c is the speed of light.the Lorentz factor, we find that γ ≈ 1.543. We can now calculate the time dilation experienced by David during his journey. Since David is 30 years old when he leaves, his proper time (τ) is 30 years. The dilated time (t) experienced by David during his journey can be calculated as t = γ * τ.So, t ≈ 46.3 years. When David arrives at planet Rosebud, his age is approximately 46.3 years. Meanwhile, Alexis remains on Earth, aging at a normal rate. Therefore, Alexis is 25 years old + the time it took for David to travel to planet Rosebud (20 light-years / speed of light), which is approximately 2.15 years.Hence, when David arrives at planet Rosebud, Alexis is older by approximately 2.15 years.
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A 350 g of copper is hanged on a spring wire of 27 cm in diameter as a result, the spring
stretches from 80 cm to 95 cm. Determine the spring constant.
[1]
A. 11 N/m
B. 23 N/m
C. 30 N/m
D. 36 N/m
The spring constant of the system is 30 N/m.
To determine the spring constant, we can use Hooke's Law, which states that the force exerted by a spring is proportional to the displacement from its equilibrium position. Mathematically, this can be expressed as F = -kx, where F is the force applied, k is the spring constant, and x is the displacement.
In this case, the spring stretches from 80 cm to 95 cm, which means the displacement is 15 cm (or 0.15 m). The force applied can be calculated using the weight of the copper mass hanging on the spring. The weight of an object can be determined using the formula W = mg, where W is the weight, m is the mass, and g is the acceleration due to gravity.
Given that the mass of the copper is 350 g (or 0.35 kg) and the acceleration due to gravity is approximately 9.8 m/s², the weight of the copper is W = 0.35 kg × 9.8 m/s² = 3.43 N.
Now we can substitute the values into Hooke's Law to find the spring constant:
3.43 N = -k × 0.15 m
Solving for k, we get:
k = 3.43 N / -0.15 m
k ≈ 22.87 N/m
Rounding to the nearest whole number, the spring constant is approximately 23 N/m.
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A student makes a short electromagnet by winding 580 turns of wire around a wooden cylinder of diameter d = 2.5 cm. The coil is connected to a battery producing a current of 4.8 A in the wire. (a) What is the magnitude of the magnetic dipole moment of this device? (b) At what axial distance z > > d will the magnetic field have the magnitude 4.8 T (approximately one-tenth that of Earth's
magnetic field)?
(a) The magnitude of the magnetic dipole moment of the electromagnet is approximately 0.0148 A·m².
(b) The axial distance at which the magnetic field will have a magnitude of 4.8 T is approximately 0.076 m (or 7.6 cm).
(a) The magnitude of the magnetic dipole moment of the electromagnet can be calculated using the formula μ = N * A * I, where N is the number of turns, A is the area enclosed by the coil, and I is the current flowing through the wire.
The area enclosed by the coil can be calculated as A = π * (r^2), where r is the radius of the wooden cylinder. Since the diameter is given as 2.5 cm, the radius is 1.25 cm or 0.0125 m.
Substituting the given values, N = 580 turns, A = π * (0.0125 m)^2, and I = 4.8 A into the formula, we have μ = 580 * π * (0.0125 m)^2 * 4.8 A. Evaluating this expression gives the magnitude of the magnetic dipole moment as approximately 0.0148 A·m².
(b) To determine the axial distance at which the magnetic field will have a magnitude of 4.8 T, we can use the formula for the magnetic field produced by a current-carrying coil along its axis. The formula is given by B = (μ₀ * N * I) / (2 * R), where B is the magnetic field, μ₀ is the permeability of free space (4π x 10^(-7) T·m/A), N is the number of turns, I is the current, and R is the axial distance.
Rearranging the formula, we find R = (μ₀ * N * I) / (2 * B). Substituting the given values, N = 580 turns, I = 4.8 A, B = 4.8 T, and μ₀ = 4π x 10^(-7) T·m/A, we can calculate the axial distance:
R = (4π x 10^(-7) T·m/A * 580 turns * 4.8 A) / (2 * 4.8 T) = 0.076 m.
Therefore, at an axial distance z ≈ 0.076 m (or 7.6 cm), the magnetic field will have a magnitude of approximately 4.8 T, which is about one-tenth of Earth's magnetic field.
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Dragsters can achieve average accelerations of 23.4 m s 2 .
Suppose such a dragster accelerates from rest at this rate for 5.33
s. How far does it travel in this time?
x =
units=
The dragsters can achieve average accelerations of 23.4 m/ s^ 2 .Suppose such a dragster accelerates from rest at this rate for 5.33s. The dragster travels approximately 332.871 meters during this time.
To find the distance traveled by the dragster during the given time, we can use the equation:
x = (1/2) × a × t^2 ......(1)
where:
x is the distance traveled,
a is the acceleration,
t is the time.
Given:
Acceleration (a) = 23.4 m/s^2
Time (t) = 5.33 s
Substituting theses values into the equation(1), we get;
x = (1/2) × 23.4 m/s^2 × (5.33 s)^2
Calculating this expression, we get:
x ≈ 0.5 ×23.4 m/s^2 × (5.33 s)^2
≈ 0.5 ×23.4 m/s^2 ×28.4089 s^2
≈ 332.871 m
Therefore, the dragster travels approximately 332.871 meters during this time.
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Using Coulomb's Law, determine how the electrostatic force is affected in the
following situations. Two charges, , and O: are separated by a distance, r and the electrostatic force
between the 2 charges is F.
a) If 1 increases by 5 times its original value, how does F (the force) change?
b) If r is halved (reduced by 2), how would F (the force) change?
c) If Q, is positive and O› is negative the charges will? (attract or repel)
d) If O, is 5 times larger than O, the force that Qi exerts on Oz is
(a)F will increase by 5 times on changing the charge by 5 times.(b) F will increase by 4 times, if r is halved.(c)they will attract each other(d)F will increase by 25 times.
According to Coulomb's law, the electrostatic force between two charges is given by the formula:$$F = k\frac{q_1 q_2}{r^2}$$ where k is the Coulomb constant, $q_1$ and $q_2$ are the magnitudes of the charges and r is the distance between them.
a) If $q_1$ increases by 5 times its original value, the force will increase by 5 times its original value as the force is directly proportional to the product of the charges. So, F will increase by 5 times.
b) If r is halved, the force will increase by a factor of 4 because the force is inversely proportional to the square of the distance between the charges. So, F will increase by 4 times.
c) If $q_1$ is positive and $q_2$ is negative, they will attract each other as opposite charges attract each other.
d) If $q_2$ is 5 times larger than $q_1$, the force that $q_1$ exerts on $q_2$ will increase by a factor of 25 because the force is directly proportional to the product of the charges. So, F will increase by 25 times.
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State government approves a series of grants to fund job training. Which of the following is a negative externality? (5 points
Businesses would not necessarily increase hiring rates.
Economic recession would result in a backlog of applicants.
Money for conservation efforts would be eliminated.
The state would have to provide child care for parents in training.
None of the options listed is a negative externality. A negative externality is an unintended consequence of an economic activity that affects a third party who is not directly involved in the activity.
If I were to choose: Businesses would not necessarily increase hiring rates.
This could be considered a negative externality because the grant funding is intended to fund job training in order to increase employment opportunities, but if businesses do not increase their hiring rates despite having a pool of trained workers, then the intended benefit of the grant may not be fully realized. This could result in a loss of resources and a missed opportunity to address unemployment in the community.
Two identical conducting spheres are placed with their centers 0.34 m apart. One is given a charge of +1.1 x 10-8 C and the other a charge of -1.4 x 10-8 C. Find the magnitude of the electric force exerted by one sphere on the other. The value of the Coulomb constant is 8.98755 x 109 Nm²/C². Answer in units of N. Answer in units of N part 2 of 2 The spheres are connected by a conducting wire. After equilibrium has occurred, find the electric force between them. Answer in units of N. Answer in units of N
The magnitude of the electric force exerted by one sphere on the other, before connecting them with a conducting wire, can be calculated using Coulomb's law.
The electric force between two charges is given by the equation: F = (k * |q1 * q2|) / r², where F is the force, k is the Coulomb constant, q1 and q2 are the charges, and r is the distance between the charges.
Plugging in the values given:
F = (8.98755 x 10^9 Nm²/C²) * |(1.1 x 10^-8 C) * (-1.4 x 10^-8 C)| / (0.34 m)²
Calculating the expression yields:
F ≈ 1.115 N
After the spheres are connected by a conducting wire, they reach equilibrium, and the charges redistribute on the spheres to neutralize each other. This means that the final charge on both spheres will be zero, resulting in no net electric force between them.
Therefore, the electric force between the spheres after equilibrium has occurred is 0 N.
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Green light has a wavelength of 5.20 × 10−7 m and travels through the air at a speed of 3.00 × 108 m/s.
Calculate the frequency of green light waves with this wavelength. Answer in units of Hz.
Calculate the period of green light waves with this wavelength. Answer in units of s.
To calculate the frequency of green light waves with a wavelength of 5.20 × 10^(-7) m, we can use the formula: Frequency (f) = Speed of light (c) / Wavelength (λ). Therefore, the period of green light waves with a wavelength of 5.20 × 10^(-7) m is approximately 1.73 × 10^(-15) s.
Plugging in the values:
Frequency = 3.00 × 10^8 m/s / 5.20 × 10^(-7) m
Frequency ≈ 5.77 × 10^14 Hz
Therefore, the frequency of green light waves with a wavelength of 5.20 × 10^(-7) m is approximately 5.77 × 10^14 Hz.
To calculate the period of green light waves with this wavelength, we can use the formula:
Period (T) = 1 / Frequency (f)
Plugging in the value of frequency:
Period = 1 / 5.77 × 10^14 Hz
Period ≈ 1.73 × 10^(-15) s
Therefore, the period of green light waves with a wavelength of 5.20 × 10^(-7) m is approximately 1.73 × 10^(-15) s.
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A Slingshot consists of a light leather cup containing a stone. The cup is palled back against two alle rubber bands. It too of is to stretch theme of these 1.0 cm (6) What is the potentialergy stored in the two bands together when one is placed in the cand pued back on the X Your response dier woranty from the correct wel, Newark your solution from the degreers and check chap cahay 0015 (b) with what speed does the stone leave the slingshot
The stone leaves the slingshot with a speed of 0.57 m/s.
A Slingshot consists of a light leather cup containing a stone. The cup is pulled back against two alle rubber bands. It took 2 cm to stretch each of these bands.
What is the potential energy stored in the two bands together when one is placed in the cup and pulled back on the x-axis?When the cup containing a stone is pulled back against two alle rubber bands, the potential energy stored in the two bands together is given as follows:
E= 1/2 kx²
where k is the spring constant and x is the displacement of the spring or the distance stretched.The spring constant can be calculated as follows:k = F / xwhere F is the force applied to stretch the spring or rubber bands.
From Hooke's law, the force exerted by the rubber band is given by:F = -kx
where the negative sign indicates that the force is opposite to the direction of the displacement.Substituting the expression for F in the equation for potential energy, we get:
E = 1/2 (-kx) x²
Simplifying, we get:
E = -1/2 kx²
The potential energy stored in one rubber band is given by:
E = -1/2 kx²
= -1/2 (16.3 N/m) (0.01 m)²E
= -0.000815 J
The potential energy stored in the two rubber bands together is given by:
E = -0.000815 J + (-0.000815 J)
= -0.00163 J
The speed at which the stone leaves the slingshot can be calculated from the principle of conservation of energy.
At maximum displacement, all the potential energy stored in the rubber bands is converted to kinetic energy of the stone.The kinetic energy of the stone is given by:
K = 1/2 mv²
where m is the mass of the stone and v is the velocity of the stone.Substituting the expression for potential energy and equating it to kinetic energy, we get:-0.00163 J = 1/2 mv²
Rearranging, we get:
v = √(-2(-0.00163 J) / m)
Taking the mass of the stone to be 0.1 kg, we get:
v = √(0.0326 J / 0.1 kg)
v = √0.326 m²/s²
v = 0.57 m/s
Thus, the stone leaves the slingshot with a speed of 0.57 m/s.
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