The amplitude of the emf in the below circuit is 150V, while the angular frequency is 607. The resistance is 1502, the inductance is 90mH, and the N capacitor is 400uF. a) Draw the phasor diagram of Vc, V., and Vr(approximately on scale) b) What is the impedance? c) What is the Icme of this circuit? d) What is the phase angle?

The last quarter moon follows the waning gibbous. The moon's last quarter phase is a half-circle with the Northern Hemisphere's left side lit and the Southern's right. At three-quarters of its cycle, the moon is between full and new. Moon forms decreasing crescent after last quarter.

The phase of the moon that follows the waning gibbous phase is the last quarter phase. As it approaches the last quarter phase, the moon's illumination declines. In the last quarter phase, half of the moon's face is lighted, forming a half-moon shape with the left side illuminated in the northern hemisphere and the right side illuminated in the southern.

The last quarter phase follows the waning gibbous phase at three-quarters of the moon's cycle. After the last quarter phase, the moon waned to the declining crescent phase. The lunar cycle repeats with the waxing crescent and waning gibbous phases.

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After calculations, there are around 8.07 × 10¹⁶ photons strike the leaf every second.

To calculate the number of photons striking the leaf every second, we can use the formula:

Number of photons = (Power / Energy per photon) * Area

Solar radiation power = 1170 W/m²

Average wavelength of solar radiation = 504 nm

Surface area of the leaf = 2.25 cm²

First, we need to convert the surface area from cm² to m²:

Surface area = 2.25 cm² * (1 m² / 10,000 cm²)

Surface area = 0.000225 m²

Next, we need to calculate the energy per photon using the wavelength of solar radiation. We can use the equation:

Energy per photon = (Planck's constant * speed of light) / wavelength

Planck's constant (h) = 6.62607015 × 10⁻³⁴ J·s

Speed of light (c) = 2.998 × 10⁸ m/s

Wavelength (λ) = 504 nm = 504 × 10⁻⁹ m

Energy per photon = (6.62607015 × 10⁻³⁴ J·s * 2.998 × 10⁸ m/s) / (504 × 10⁻⁹ m)

Calculating the energy per photon:

Energy per photon ≈ 3.93 × 10⁻¹⁹ J

Now, we can calculate the number of photons:

Number of photons = (Power / Energy per photon) * Area

Number of photons = (1170 W/m² / 3.93 × 10⁻¹⁹ J) * 0.000225 m²

Calculating the number of photons:

Number of photons ≈ 8.07 × 10¹⁶ photons

Therefore, approximately 8.07 × 10¹⁶ photons strike the leaf every second.

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(a). The maximum deflection in mm is 9.31 mm.

(b). The maximum flexural stress in MPa is 261 MPa.

(c). The maximum shearing stress in MPa is 25.19 MPa.

As per data a simply supported beam W 533 x 93 has a span of 7.8 m. It carries a uniformly distributed load of 52 kN/m throughout its length. Also, it is laterally supported over its entire length.

The beam has the following properties:

Ix = 0.000556 m², Depth, d = 533 mm, Web thickness, t = 10.2 mm. The allowable flexural stress is 0.66Fy, allowable shearing stress is 0.4Fy, and allowable deflection is L/360.

To find maximum deflection, maximum flexural stress, and maximum shearing stress, we will use the following formulas:

(a).

Maximum deflection:

δmax = WL³ / (48 EI)

Substitute all values,

δmax = WL³ / (48 EI)

         = (52×10³ N/m × 7.8³ m³) / (48 × 2.05 × 10¹¹ N/m² × 0.000556 m²)

         = 9.31 mm.

(b).

Maximum flexural stress:

σmax = Mc / I

Substitute all values,

σmax = WL / 8 (d / 2)

σmax = (52 × 7.8²) / (8 × 533 × 10⁻³)

σmax = 261 MPa.

(c).

Maximum shearing stress:

τmax = 3VQ / (2Awt)

Q = wL² / 8

   = 52 × 7.8² / 8

   = 2,200.16

N.A = t × d

      = 10.2 × 533

      = 5,438.6 mm²

τmax = 3 × 52 × 2,200.16 × 5,438.6 / (2 × 10⁶ × 10.2 × 533)

         = 25.19 MPa.

Thus, the maximum deflection in mm is 9.31 mm, the maximum flexural stress in MPa is 261 MPa, and the maximum shearing stress in MPa is 25.19 MPa.

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Viscosity is one of the most crucial fluid properties, and fluid mechanics is the study of how fluids move.

Thus, Most people associate "viscosity" with how well a fluid flows. Chemists describe viscosity as a substance's resistance to progressive deformation, which gives them a somewhat different perspective on the issue.

This relates to the figurative notion of "thickness"; for instance, honey is viscous and thicker than water. The viscosity of a cleaning fluid has a significant impact on its efficacy.

Wikipedia defines viscosity as the friction that occurs between fluid molecules. For instance, a fluid flowing through a tube will move more swiftly towards the tube axis but less fast elsewhere.

Thus, Viscosity is one of the most crucial fluid properties, and fluid mechanics is the study of how fluids move.

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The distant star must be traveling at approximately 1.816 × 10⁹ meters per second for its light to reach us at a wavelength of 3.33 μm.

The Doppler effect equation:

Δλ / λ = v / c

where Δλ is the change in wavelength, λ is the initial wavelength, v is the velocity of the star, and c is the speed of light.

Given:

the initial wavelength (λ) is 550 nm (or 550 × 10⁻⁹ m) and

the observed wavelength (Δλ) is 3.33 μm (or 3.33 × 10⁻⁶ m),

putting values in the Doppler effect equation

v = (3.33 × 10⁻⁶)  × (3 × 10⁸) / (550 × 10⁻⁹ )

v = 1.816  × 10⁹ m/s

Therefore, the distant star must be traveling at approximately 1.816 × 10⁹ meters per second for its light to reach us at a wavelength of 3.33 μm.\

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We can fit approximately 3128 suns in the distance between Earth and Neptune.

Neptune and Earth are separated by an average distance of 4.35 billion kilometres (2.7 billion miles). We must compare the diameter of the Sun to the separation between Earth and Neptune in order to determine how many suns can fit within this space.

The Sun's diameter is roughly 1.39 million kilometres (864,000 miles) across. The number of suns that can fit on Earth can be calculated by dividing the distance between Earth and Neptune by the Sun's diameter.

4.35 billion miles divided by 1.39 million miles is 3128 suns.

The gap between Earth and Neptune may therefore fit around 3128 suns. The assumption used in this computation is that the suns are perfectly aligned with one another.

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The tension on the left rope is given by mgd/L. hence, the correct answer is [tex]g(M/2 + mL^2/(L - d)^2)[/tex]. Therefore option F is correct.

To determine the tension on the left rope, we can consider the forces acting on the system.

The forces acting on the system are:

1. Weight of the work platform: Mg (acting downwards)

2. Weight of the window washer: mg (acting downwards)

3. Tension in the left rope: T (acting upwards)

4. Tension in the right rope: T' (acting upwards)

Since the system is in equilibrium, the sum of the vertical forces must be zero:

T + T' - Mg - mg = 0

Since the work platform is symmetrically suspended, the tensions in the ropes are equal, so we can write:

T = T'

Now, let's consider the torque of the left rope as the pivot point. The torques must also be in equilibrium for the system to be balanced.

The torque due to the weight of the work platform is zero because it acts at the pivot point.

The torque due to the weight of the window washer is mgd, where d is the distance of the window washer to the right of the left rope.

The torque due to the tension in the right rope is T'L.

Therefore, we have:

mgd - T'L = 0

Since T = T', we can substitute T' with T:

mgd - TL = 0

Solving for T:

T = mgd/L

Therefore, the tension on the left rope is given by mgd/L.

Hence, the correct answer is [tex]g(M/2 + mL^2/(L - d)^2)[/tex].

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The change in length when the temperature drops to a frigid −45◦C is - 0.013104 m, and 1.13 953  × 10⁻³ thickness should be the steel plate observed.

According to question:

L₀ = 65 m

Ti = 18 °c

T f = - 45 °c

α = 3.2 × 10 ⁻⁶/  °c

Final length Lf = L₀ (1 + α Δ T)

= 65 ( 1+ 3.2 × 10 ⁻⁶ × (- 45 - 18)

= 64. 9868 M

So, change in the length Δ L = Lf - Li

= - 0.013104 m

Thus, change in length when the temperature drops to a frigid −45◦C is - 0.013104 m.

Heat rate Q = KA × ΔT/L

Qal = Kal × A × ΔT/ L₁

= 430 × A × ΔT/ 0.035

Q stainless steel = Ks × A × ΔT/ L₂

= 14 ×  A × ΔT/ L₂

Equating both the equations,

L₂ = 1.13 953  × 10⁻³

Thus, the 1.13 953  × 10⁻³ thick should be the steel plate observed.

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The effective mass of an electron with parabolic energy dispersion, given by the energy equation E(k) = 1.23 + 12 × 10⁻¹⁸ k², in units of the free electron mass (m₀) is 24 times the free electron mass.

The effective mass concept is used to describe the behavior of electrons in solid-state physics. In materials with different energy dispersion relations, the effective mass can vary from the free electron mass. In this case, we are given a parabolic energy dispersion relation E(k) =1.23 + 12 × 10⁻¹⁸ k², where E is the energy and k is the wavevector.

To find the effective mass, we need to take the second derivative of the energy dispersion relation with respect to the wavevector (k). The second derivative provides information about the curvature of the energy band, which relates to the effective mass.

Taking the derivative of E(k) with respect to k, we get:

dE/dk = 2 × 12 × 10⁻¹⁸ k

Taking the second derivative of E(k) with respect to k, we get:

d²E/dk² = 2 × 12 × 10⁻¹⁸

Therefore, the second derivative is a constant value of 2 × 12 × 10⁻¹⁸.

The effective mass (m*) is the inverse of the second derivative:

1/m* = 2 × 12 × 10⁻¹⁸

Simplifying:

1/m* = 24 × 10⁻¹⁸

To express the effective mass in terms of the free electron mass (m₀), we divide both sides of the equation by m₀:

1/m* = 24 × 10⁻¹⁸ / m₀

Hence, the effective mass of the electron with parabolic energy dispersion is 24 times the free electron mass (m₀).

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a. For Case 1, the outlet temperature of the ideal gas is 800°C, and for Case 2, the outlet temperature is 1800°C.

b. For Case 1, the rate of entropy generation in the system is 3 kJ/s·K, and for Case 2, it is 5 kJ/s·K.

c. Increasing the temperature of the incoming gas does not change the answers for (a) because the heat transfer and temperature change occur independently in the system.

a. In Case 1, the outlet temperature of the ideal gas is calculated by using the heat transfer equation, Q = mCΔT, where Q is the heat transfer, m is the mass flow rate, C is the heat capacity, and ΔT is the temperature difference.

Substituting the given values, we have Q = (1 kg/s) × (1000°C) × (2000°C - Tout_1), where Tout_1 is the outlet temperature for Case 1. Solving for Tout_1, we find Tout_1 = 800°C. Similarly, for Case 2, the outlet temperature (Tout_2) can be calculated using the same equation, resulting in Tout_2 = 1800°C.

b. The rate of entropy generation in the system can be determined using the equation ΔS = Q/T, where ΔS is the change in entropy, Q is the heat transfer, and T is the temperature. For Case 1, ΔS_1 = Q/T_1 = Q/(2000°C), and for Case 2, ΔS_2 = Q/T_2 = Q/(2000°C).

Since the heat transfer Q is the same for both cases, the rate of entropy generation will be different due to the difference in temperature. Plugging in the values, we find ΔS_1 = 3 kJ/s·K and ΔS_2 = 5 kJ/s·K.

c. Increasing the temperature of the incoming gas does not change the answers for (a) because the heat transfer equation is independent of the temperature of the incoming gas. The outlet temperature of the ideal gas is determined by the heat transfer and the temperature difference, which are unaffected by the incoming gas temperature.

Therefore, increasing or decreasing the incoming gas temperature does not change the outlet temperature of the ideal gas. However, it can affect the rate of entropy generation because the entropy change is directly proportional to the temperature. Higher temperatures result in higher rates of entropy generation.

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The process of a compound's chemical bonds being broken down by the addition of water is known as hydrolysis. Thus, hydrolysis is an illustration of chemical weathering not physical weathering, hence option B is correct.

Physical weathering is the process of disintegrating rocks and crystals without altering their chemical makeup. Smaller pieces of the same substance that is being worn are the outcomes of physical weathering.

The mechanical deterioration of rocks and minerals is known as physical weathering. The chemical deterioration of rocks is known as chemical weathering.

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Given, thickness of concrete slab = 86 mmWe are supposed to find; What fraction of the emitted gamma particles passes through the slab?Estimate the thickness of lead that provides the same amount of attenuation as the concrete slab.

(a) Gamma rays attenuation due to concrete slab,

we use formula :

I/Io = e^(-μx)

where

I = intensity after attenuation

Io = initial intensity

μ = mass attenuation coefficient for the material

x = thickness of the material

From the data in the question,

we have to assume μ = 0.02 m²/kg (or 20 cm²/g) for concrete slab

We know that thickness of the concrete slab = 86 mm = 0.086 mandIo = 1 (let's assume initial intensity = 1)

Now, let's calculate I:I/Io = e^(-μx)I/1 = e^(-0.02 × 0.086)

I = 0.9416 (approx)

Therefore, fraction of the emitted gamma particles passes through the slab is 0.9416.

(b) From the previous part we know that the fraction of gamma rays that passes through the concrete slab = 0.9416.

The thickness of lead that provides the same amount of attenuation as the concrete slab is the thickness of lead that would result in the same fraction of gamma rays to be transmitted.

For this we have to know mass attenuation coefficient for lead (μ = 0.069 m²/kg)

We will use the same formula as above,

I/Io = e^(-μx)

I/Io = e^(-μx) = 0.9416 => e^(-μx) = 0.9416

I/Io = e^(-μx)0.9416 = e^(-0.069 x) taking natural logarithm on both sides,

we getln (0.9416) = -0.069 x

Now, solving for x,

we getx = 10.08 cm (approx)

Therefore, the thickness of lead that provides the same amount of attenuation as the concrete slab is 10.08 cm.

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Asterisms are smaller, recognizable shapes formed by stars within a constellation or spanning multiple constellations. They can take the form of familiar objects or geometric shapes.

All stars within well-defined regions of the sky are members of constellations. These constellations are a way to organize and divide the celestial sphere into recognizable patterns.

The star in Taurus is designated Alpha Tauri, which means it is considered the brightest star in that particular constellation. Patterns of stars in the night sky are called asterisms.

Asterisms are smaller, recognizable shapes formed by stars within a constellation or spanning multiple constellations. They can take the form of familiar objects or geometric shapes.

While there are 88 official constellations recognized by the International Astronomical Union, there are countless asterisms that astronomers and stargazers observe and appreciate within these constellations.

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All stars within well-defined regions of the sky are members of constellations. The star in Taurus is designated Alpha Tauri. Patterns of stars in the night sky are called asterisms and are part of one or more constellations.

The International Astronomical Union (IAU) recognizes 88 constellations. These constellations are well-defined areas in the sky that contain groups of stars forming recognizable patterns. Some of the most famous constellations include Orion, Ursa Major (the Big Dipper), and Cassiopeia.

Asterisms, on the other hand, are smaller, distinct patterns formed by stars within a constellation or across multiple constellations. These patterns may be easily recognizable and have cultural or historical significance. For example, the Big Dipper is an asterism within the Ursa Major constellation.

In total, there are 15 asterisms that are officially recognized by the IAU. These include the Big Dipper, the Little Dipper, the Northern Cross, and the Summer Triangle.

Constellations and asterisms help astronomers navigate the night sky and locate specific celestial objects. They provide a way to organize and identify stars and other celestial bodies.

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Given, the data as follows: We have to find the estimated velocity value of the regression coefficient for variable Weight, the estimated value of the intercept, computed value of SSE, computed value of MSE and the standard error of the estimate of β1.

Given, total body fat, Y, expressed as a percentage of body weight, X1, for 19 participants in a physical fitness program. Body fat percentage and body weight are shown in the table below. Weight (kg) 89 88 66 59 93 73 82 77 100 67 57 68 69 59 62 59 56 66 72Body Fat (%) 28 27 24 23 29 25 29 25 30 23 29 32 35 31 29 26 28 35 33We define a variable X2 which is 1 for males and 0 for females and fit the model Y = β0 + β1X1 + β2X2 + e Here, the participants 1-10 are male and 11-19 are female.

The estimated value of the regression coefficient for variable Weight is 0.788, the estimated value of the intercept is 42.77, the computed value of SSE is 25.43, the computed value of MSE is 1.52 and the standard error of the estimate of β1 is 0.12.

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1. The statement made in the opening sentence of this question, "Limestone decomposes at temperatures above 840°C," can be confirmed.

When the limestone is heated to a temperature of 840°C or higher, it decomposes, releasing carbon dioxide and forming solid lime as a product.

The chemical reaction is as follows: CaCO3 → CaO + CO2 (ΔH = 178 kJ mol−1)

2. By purging the furnace chamber with a mixture of N2 and CO2 in a 1:1 ratio, one could benefit in a few ways, such as:
(a) To shift the equilibrium to the right, this method can be used. The amount of CO2 in the chamber would be reduced, resulting in more CO2 production, and the reaction will move in the forward direction, resulting in more production of CaO.
(b) To keep the reaction rate constant by maintaining the chamber's pressure and avoiding the creation of a vacuum in the furnace, the mixture can be utilized.
(c) Nitrogen will function as a carrier gas, ensuring that the carbon dioxide produced by the reaction is removed from the chamber as soon as it is generated.

As a result, the reaction will be carried out more efficiently.

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Therefore, the force acted on the object for a total of 1.2 seconds (option d).

To determine the time for which the force acted on the object, we can use Newton's second law of motion, which states:

Force (F) = mass (m) × acceleration (a)

Rearranging the equation to solve for acceleration:

a = F ÷ m

Given:

Mass (m) = 2.0 kg

Maximum force (F) = 20 N

Final speed (v) = 12.0 m/s

We can use the equation for acceleration:

a = (v - u) ÷ t

Where:

Initial velocity (u) is 0 m/s (since the object starts from rest),

t is the for which the force acted on the object.

Since the object starts from rest, the equation simplifies to:

a = v ÷ t

Setting the equations for acceleration equal to each other:

F ÷ m = v ÷t

Solving for time (t):

t = m × v ÷ F

Substituting the given values:

t = 2.0 kg ×12.0 m/s ÷20 N

Calculating:

t = 24.0 kg·m/s ÷ 20 N

t = 1.2 s

Therefore, the force acted on the object for a total of 1.2 seconds (option d).

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The plastic moment capacity of the section is 2.0225 × 10⁹ N.mm.

The plastic moment capacity of the section:

A hollow box beam having outside dimensions of 400 mm x 600 mm and inside dimensions of 300 mm x 500 mm has its ends fixed and a span of 6 m.

The plastic moment capacity of the section has to be calculated.

The beam is made of steel and has a yield stress of 250 MPa (mega Pascal).

Formula to find the plastic moment capacity of the section:

Mpl = Zp x Fy

Where,

Mpl is the plastic moment capacity of the section,

Zp is the plastic section modulus, and Fy is the yield strength of the material.

Plastic section modulus can be calculated as,

Zp = 2[(b₁t₁³ - b₂t₂³)/3] + [t₂(b₂² - b₁²)/2]

Where, b₁ = 600 mm, b₂ = 500 mm, t₁ = 400 mm, and t₂ = 300 mm.

Then, Zp is equal to 8.09 × 10⁶ mm³

Substituting values in the formula,

Mpl = Zp × Fy

Mpl = 8.09 × 10⁶ × 250

Mpl = 2.0225 × 10⁹ N.mm

Therefore, the plastic moment capacity of the section is 2.0225 × 10⁹ N.mm.

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The easiest way to determine if resistors are in series or parallel is to visually inspect the circuit. If the resistors are wired end-to-end, they are in series. If they are connected to the same two points, they are in parallel. You can also use the formulas for calculating total resistance for each circuit type to verify your results.

When it comes to resistors, it's essential to understand if they are wired in series or parallel to determine the equivalent resistance of the circuit. Series and parallel resistor circuits have different circuit properties, which affect the flow of electrical currents. Here's how you can tell if resistors are in series or parallel.

Resistors in series: When two or more resistors are connected end-to-end, they are in series. To calculate the total resistance of resistors in a series circuit, you can use the following formula:

R_total = R1 + R2 + R3 + ....... + Rn

For example, if three resistors are wired in series with values of 10 ohms, 20 ohms, and 30 ohms, then the total resistance would be:

R_total = 10 + 20 + 30

= 60 ohms.

Resistors in parallel: When two or more resistors are connected to the same two points in a circuit, they are in parallel. To calculate the total resistance of resistors in a parallel circuit, you can use the following formula:

1/R_total = 1/R1 + 1/R2 + 1/R3 + ....... + 1/Rn

For example, if three resistors are wired in parallel with values of 10 ohms, 20 ohms, and 30 ohms, then the total resistance would be:

1/R_total = 1/10 + 1/20 + 1/30

R_total = 5.45 ohms

Explanation: Resistors are in series when they are connected end-to-end. The current through each resistor in a series circuit is the same. Resistors in series are added together to calculate the total resistance of the circuit.

Resistors are in parallel when they are connected to the same two points in a circuit. The voltage across each resistor in a parallel circuit is the same, and the total current through the circuit is the sum of the currents through each resistor. To calculate the total resistance of resistors in parallel, you must add up the inverse of the resistors and take the reciprocal of that sum.

The difference between a series and parallel circuit is that the former has a single path for current flow, while the latter has multiple paths. By understanding whether resistors are wired in series or parallel, you can calculate the equivalent resistance of the circuit and predict its behavior.

Conclusion: The easiest way to determine if resistors are in series or parallel is to visually inspect the circuit. If the resistors are wired end-to-end, they are in series. If they are connected to the same two points, they are in parallel. You can also use the formulas for calculating total resistance for each circuit type to verify your results.

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, the magnetic field at

(i) 5.0mm from the axis

B = 15692T

(ii) 10.0mm from the axis

B = 7846.02 T

(iii) 15.0mm from the axis

B = 522.98 T

Ampere's Law is a fundamental principle in electromagnetism that relates the magnetic field around a closed loop to the electric current passing through the loop. It is one of Maxwell's equations and provides a mathematical description of the magnetic field generated by electric currents.

Ampere's Law can be stated in integral form as:

∮ B dl = μ₀ I

where:

∮ B · dl represents the line integral of the magnetic field B around a closed loop,

μ₀ is the permeability of free space (a constant value of approximately 4π × 10^-7 T·m/A),

I is the total electric current passing through the loop.

Given: diameter, d = 10.0mm

d = 2r

r = 0.5 cm

area, A = π×r²

A = 3.14 × 0.5²

A = 0.785 cm²

I =  500 ( 0.785)

i = 392.5

using Amperes law,

magnetic field (B)(R)= μ₀I

(B)(R) = 1.256 × 10⁻⁶ (392.5)/ 2π

(B)(R) = 78.43

(i) 5.0mm from the axis

B = 15692T

(ii) 10.0mm from the axis

B = 7846.02 T

(iii) 15.0mm from the axis

B = 522.98 T

Therefore, the magnetic field at

(i) 5.0mm from the axis

B = 15692T

(ii) 10.0mm from the axis

B = 7846.02 T

(iii) 15.0mm from the axis

B = 522.98 T

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It takes approximately 15.19 seconds for the disk to reach its final rotational speed.

The disk turns approximately 14.18 revolutions while accelerating.

(a) To find the time it takes for the disk to reach its final rotational speed, we can use the equation:

Final angular velocity (ω) = Initial angular velocity (ω₀) + (Torque (τ) / Moment of inertia (I)) * time (t)

Given:

Final angular velocity (ω) = 700 rpm = 700 * 2π rad/min

Initial angular velocity (ω₀) = 0 rad/s

Torque (τ) = 62 Nm

Moment of inertia (I) = 1/2 * mass * radius² = 1/2 * 2 kg * (0.16 m)²

Substituting the values:

700 * 2π = 0 + (62 / (1/2 * 2 * (0.16)²)) * t

Simplifying:

700 * 2π = (62 / (1/2 * 2 * 0.16²)) * t

t = (700 * 2π) / (62 / (1/2 * 2 * 0.16²))

t = 15.19 s

So, the disc takes roughly 15.19 seconds to attain its final spinning speed.

(b) To find the number of revolutions the disk turns while accelerating, we can use the equation:

θ = ω₀ * t + (1/2) * (τ / I) * t²

Given:

ω₀ = 0 rad/s

τ = 62 Nm

I = 1/2 * mass * radius² = 1/2 * 2 kg * (0.16 m)²

t = 15.19 s (calculated in part (a))

Substituting the values:

θ = 0 * 15.19 + (1/2) * (62 / (1/2 * 2 * 0.16²)) * (15.19)²

Simplifying:

θ = (1/2) * (62 / (1/2 * 2 * 0.16²)) * (15.19)²

θ = 14.18 revolutions

Therefore, while accelerating, the disc makes roughly 14.18 rotations.

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A brittle material is strong but not flexible (absorbs little force) and can break suddenly and catastrophically under high stress.

The device that is strong but not flexible (absorbs little force) is a brittle material. A brittle material is one that has a low ability to deform plastically and absorb energy before failure, resulting in a sudden and catastrophic failure once the breaking point has been reached. An example of a brittle material is glass.

A brittle material is a solid material that can crack or break with little deformation when subjected to stress, such as being hit with a hammer or bent over a knee. Because they are made up of microscopic flaws that develop into larger fractures, brittle materials fail suddenly and catastrophically, often with little or no warning.

Explanation: Brittle materials exhibit little or no ductility or plastic deformation before failure. At low stresses, they deform elastically before cracking, but at high stresses, they deform elastically before cracking. Brittle materials include ceramics, glass, concrete, and cast iron, among other materials.

Conclusion: A brittle material is strong but not flexible (absorbs little force) and can break suddenly and catastrophically under high stress.

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According to the law of conservation of momentum, the total momentum of an isolated system remains constant if no external forces act on it. The correct answer is option c

Initially, the skater is at rest, so her momentum is zero. The medicine ball is tossed toward her with a velocity of 4 m/s. Since momentum is defined as mass multiplied by velocity, the momentum of the ball is:

(10 kg) * (4 m/s) = 40 kg·m/s.

When the skater catches the ball, she exerts a force on it and changes its velocity to zero. By doing so, she gains momentum in the opposite direction to that of the ball. This is necessary to maintain the total momentum of the system at zero.

Since momentum is conserved, the skater's final momentum must be -40 kg·m/s in order to cancel out the ball's momentum. This means the skater moves backward with a velocity that allows her to achieve a final momentum of -40 kg·m/s.

Therefore, the correct answer is option c: She moves backward at a velocity greater than 4 m/s.

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The current is 3.33A, the resistance is 36.03 Ohm, and the peak voltage is 169.68V.

Given information:

The power rating of the resistive element (P) = 400 W

Voltage supplied by the outlet (V) = 120 VAC (RMS)

a) To find the current flowing through the element, the formula:

Power (P) = Voltage (V) × Current (I)

Current (I) = Power (P) / Voltage (V)

Substituting the given values:

Current (I) = 400 W / 120 VAC

= 3.33 A

b) To find the resistance of the element, Ohm's Law can be used :

Resistance (R) = Voltage (V) / Current (I)

Resistance (R) = 120 VAC / 3.33A

= 36.03 Ohm

c) To find the peak voltage, it is required to convert the RMS voltage to peak voltage. For an AC voltage, the relationship between RMS voltage ([tex]V_{rms[/tex]) and peak voltage ([tex]V_{peak[/tex]) is given by:

[tex]V_{peak} = V_{rms} \times \sqrt2[/tex]

Substituting the given RMS voltage:

Peak voltage  = [tex]120 \times \sqrt2[/tex]

= 169.68 V

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(a)Therefore, the force that the spring applies to the mass is -10 N (opposite in direction to the compression). (b) Therefore, the period of the mass-spring system after it is released is 4π seconds.(c) Therefore, it takes 2π seconds for the spring to become fully stretched out. (d) Therefore, the mass will accelerate at a rate of -1.25 m/s². (e) Therefore, the mass has zero velocity in the middle (uncompressed) position. (f) Therefore, the spring will stretch out by 5 meters. (g) Therefore, the mass is not moving at the end, and its velocity is zero.

To answer the questions regarding the given mass-spring system, let's consider the following information:

Mass (m) = 8 kg

Spring constant (k) = 2 N/m

Compression (x) = 5 m

(a) The force that the spring applies to the mass can be calculated using Hooke's Law:

F = -k × x

Substituting the given values, we have:

F = -(2 N/m) × (5 m)

F = -10 N

Therefore, the force that the spring applies to the mass is -10 N (opposite in direction to the compression).

(b) The period (T) of the mass-spring system after it is released can be calculated using the formula:

T = 2π × √(m/k)

Substituting the given values, we have:

T = 2π × √(8 kg ÷ 2 N/m)

T = 2π ÷ √(4 s²)

T = 4π s

Therefore, the period of the mass-spring system after it is released is 4π seconds.

(c) To determine how long it takes for the spring to become fully stretched out, we need to calculate the time it takes for the mass to reach its equilibrium position. The time can be calculated using the formula:

t = π × √(m/k)

Substituting the given values, we have:

t = π × √(8 kg÷ 2 N/m)

t = π ×√(4 s²)

t = 2π s

Therefore, it takes 2π seconds for the spring to become fully stretched out.

(d) After the mass is let go, its acceleration (a) can be determined using the equation:

a = -k/m × x

Substituting the given values, we have:

a = -(2 N/m) ÷8 kg ×5 m

a = -1.25 m/s²

Therefore, the mass will accelerate at a rate of -1.25 m/s² (opposite in direction to the initial compression).

(e) When the mass is in the middle (uncompressed), it is at its equilibrium position. At this point, its velocity is maximum, and the speed can be determined using the equation:

v = √(k/m) × x

Substituting the given values, we have:

v = √(2 N/m ÷ 8 kg) ×0

v = 0 m/s

Therefore, the mass has zero velocity in the middle (uncompressed) position.

(f) The maximum displacement (amplitude) of the spring can be determined using the equation:

A = x

Substituting the given values, we have:

A = 5 m

Therefore, the spring will stretch out by 5 meters.

(g) At the end, when the spring is fully stretched out, the mass comes to rest momentarily. Therefore, its velocity is zero.

Therefore, the mass of spring constant is not moving at the end, and its velocity is zero.

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The general (implicit) solution to Stefan's equation is given by:

ln((M^2 + T^2)/(M^2 - T^2)) = 2σt + 2C, where C is the constant of integration.

To solve Stefan's equation, we can separate the variables and integrate both sides. Let's proceed with the steps:

First, rewrite the equation as follows:

(1/(M^4 - T^4)) * dT/dt = σ

Now, we integrate both sides with respect to t:

∫(1/(M^4 - T^4)) * dT = ∫σ * dt

The left side of the equation requires integration by partial fractions. The integral can be expressed as:

1/(M^4 - T^4) = A/(M^2 + T^2) + B/(M^2 - T^2)

To determine the constants A and B, we can use the method of partial fractions. Multiply through by (M^4 - T^4) and then equate the numerators:

1 = A(M^2 - T^2) + B(M^2 + T^2)

Expanding and rearranging, we have:

1 = (A + B)M^2 + (A - B)T^2

Equating the coefficients of the powers of T, we get:

A + B = 0 (coefficient of T^2)

A - B = 1 (constant term)

Solving these equations simultaneously, we find A = 1/2 and B = -1/2.

Substituting the partial fraction decomposition back into the integral, we have:

∫(1/(M^4 - T^4)) * dT = ∫(1/2)*((1/(M^2 + T^2)) - (1/(M^2 - T^2))) * dT

Integrating both sides, we obtain:

(1/2) * (ln(M^2 + T^2) - ln(M^2 - T^2)) = σt + C

where C is the constant of integration.

Finally, solving for T, we can write the general (implicit) solution to Stefan's equation as:

ln((M^2 + T^2)/(M^2 - T^2)) = 2σt + 2C

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Dimensionally F=Gm/r² is real .

Newton’s Law of Gravitation is an important formula in physics. It states that the gravitational force between two objects is proportional to the product of their masses and inversely proportional to the square of the distance between them.

The formula for gravitational force is F = Gm₁m₂/r², where F is the force, m₁ and m₂ are the masses of the two objects, r is the distance between the two objects, and G is the gravitational constant. The dimensional analysis of the given equation is given below: Gravitational constant G: Its units are N m² kg⁻². Distance or radius r: Its units are meters (m)

Mass m: Its units are kilograms (kg)Force F: Its units are newtons (N)

Dimensional formula of force F = [MLT⁻²]

Putting the units of the given terms into the equation F=Gm/r², we get the following:

[tex]$$F=\frac{Nm^2/kg^2*kg}{m^2}$$[/tex]

[tex]$$F=\frac{Nm^2}{kg^2}$$[/tex]

Therefore, dimensionally F=Gm/r² is real.

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The plastic moment capacity of the section is 2.0225 × 10⁹ N.mm.

The plastic moment capacity of the section:

A hollow box beam having outside dimensions of 400 mm x 600 mm and inside dimensions of 300 mm x 500 mm has its ends fixed and a span of 6 m.

The plastic moment capacity of the section has to be calculated.

The beam is made of steel and has a yield stress of 250 MPa (mega Pascal).

Formula to find the plastic moment capacity of the section:

Mpl = Zp x Fy

Where,

Mpl is the plastic moment capacity of the section,

Zp is the plastic section modulus, and Fy is the yield strength of the material.

Plastic section modulus can be calculated as,

Zp = 2[(b₁t₁³ - b₂t₂³)/3] + [t₂(b₂² - b₁²)/2]

Where, b₁ = 600 mm, b₂ = 500 mm, t₁ = 400 mm, and t₂ = 300 mm.

Then, Zp is equal to 8.09 × 10⁶ mm³

Substituting values in the formula,

Mpl = Zp × Fy

Mpl = 8.09 × 10⁶ × 250

Mpl = 2.0225 × 10⁹ N.mm

Therefore, the plastic moment capacity of the section is 2.0225 × 10⁹ N.mm.

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The magnetic field at point P is μI/2(1/R₁ -1/R₂)(Ф/2π). The magnetic dipole moment of the loop is (πR₁²- πR₂²)Ф/2π and, the magnetic field of the loop at Ф = 2π is μI/2(1/R₁ -1/R₂).

Given information,

Radius, R₁ and R₂

Ф = 2π

a) To calculate, the magnetic field at point P

B = μI/2(1/R₁ -1/R₂)(Ф/2π)

The direction of the magnetic field is outside the page. If the current direction is reversed, the direction of the magnetic field will be inside the page.

b)  The magnetic dipole moment of the loop,

A = (πR₁²- πR₂²)Ф/2π

c)  at Ф = 2π, the magnetic field of the loop is,

B = μI/2(1/R₁ -1/R₂)

The magnetic field at the loop of the wire,

B = μI/2R

Hence,  the magnetic field of the loop is B = μI/2(1/R₁ -1/R₂).

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The same object is located at the same distance from two spherical mirrors, A and B. The magnifications produced by the mirrors are [tex]m_A[/tex] = 5.0 and [tex]m_B[/tex] = 2.3. The ratio [tex]f_A/f_B[/tex] of the focal lengths of the mirrors is  2.17.

The magnification (m) produced by a spherical mirror can be related to the ratio of the object distance ([tex]d_o[/tex]) to the image distance ([tex]d_i[/tex]) using the formula:

[tex]m = -d_i/d_o[/tex]

Given the magnifications [tex]m_A[/tex] = 5.0 for mirror A and [tex]m_B[/tex] = 2.3 for mirror B, we can write the following equations:

[tex]m_A = -d_i_A/d_o\\m_B = -d_i_B/d_o[/tex]

To find the ratio of the focal lengths [tex](f_A/f_B)[/tex] of the mirrors, we need to relate the image distance [tex](d_i)[/tex] to the focal length (f) using the mirror equation:

[tex]1/f = 1/d_o + 1/d_i[/tex]

[tex]d_i = 1 / (1/f - 1/d_o)\\d_i_A = 1 / (1/f_A - 1/d_o)\\d_i_B = 1 / (1/f_B - 1/d_o)[/tex]

Now, we can substitute these expressions into the magnification equations:

[tex]m_A = -1 / (1/f_A - 1/d_o)\\m_B = -1 / (1/f_B - 1/d_o)[/tex]

Rearranging these equations to isolate [tex]f_A[/tex] and [tex]f_B[/tex] respectively:

[tex]1/f_A = -1 / (m_A * d_o) - 1/d_o\\1/f_B = -1 / (m_B * d_o) - 1/d_o[/tex]

Taking the ratio of [tex]f_A[/tex] and [tex]f_B[/tex] :

[tex]f_A/f_B = [(-1 / (m_A * d_o) - 1/d_o)] / [(-1 / (m_B * d_o) - 1/d_o)]\\f_A/f_B = [(1/d_o - 1 / (m_A * d_o))] / [(1/d_o - 1 / (m_B * d_o))]\\f_A/f_B = (m_A * d_o) / (m_B * d_o)\\f_A/f_B = m_A / m_B[/tex]

Substituting the given magnification values:

[tex]f_A/f_B[/tex] = 5.0 / 2.3

Therefore, the ratio of the focal lengths of the mirrors, [tex]f_A/f_B[/tex] is approximately 2.17.

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Among the options provided, the Doppler Shift and the Cosmic Microwave Background (CMB) radiation strongly support the Big Bang Theory.

1. Doppler Shift:

The Doppler Shift is the change in the observed frequency of a wave due to the relative motion between the source of the wave and the observer. In the context of the Big Bang Theory, the Doppler Shift is crucial evidence. It is observed that distant galaxies are moving away from us, and their light exhibits a redshift.

According to the theory, the universe is expanding, causing galaxies to move away from each other. The observed redshift in the light from these galaxies indicates that the universe is stretching and supports the idea of an expanding universe originating from a highly dense and hot state, which is a central concept in the Big Bang Theory.

2. Cosmic Microwave Background (CMB) radiation:

The Cosmic Microwave Background is a form of radiation that permeates the entire universe. It is often regarded as "relic radiation" from the early stages of the universe. The CMB was first discovered in 1965 and provides strong evidence for the Big Bang Theory.

The theory predicts that after the initial explosion of the Big Bang, the universe was extremely hot and dense. As the universe expanded, it cooled down, and at a certain point, about 380,000 years after the Big Bang, protons and electrons combined to form neutral atoms. This allowed photons to travel freely, creating the CMB radiation.

The existence of the CMB radiation was confirmed by the COBE (Cosmic Background Explorer) and WMAP (Wilkinson Microwave Anisotropy Probe) satellites, providing detailed measurements of the radiation's temperature and its distribution across the sky. The uniformity and characteristics of the CMB strongly support the idea of an initial hot and dense phase, consistent with the predictions of the Big Bang Theory.

Therefore, both the Doppler Shift and the Cosmic Microwave Background radiation provide compelling evidence in support of the Big Bang Theory.

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