The value of current in a 73- mH inductor as a function of time is: I=7t 2
−5t+13 where I is in amperes and t is in seconds. Find the magnitude of the induced emf at t=6 s. Write your answer as the magnitude of the emf in volts. Question 7 1 pts The circuit shows an R-L circuit in which a battery, switch, inductor and resistor are in series. The values are: resistor =52Ω, inductor is 284mH, battery is 20 V. Calculate the time after connecting the switch after which the current will reach 42% of its maximum value. Write your answer in millseconds.

The voltage that is generated in 3 seconds will be N × πr²/2 × (4πRB - 60 / 3) where r → r' and the given magnetic field flux through a circular wire is 60 Wb.

The magnetic field flux through a circular wire is 60 Wb.

Radius of wire is duplicated over the course of 3 seconds.i.e, Radius initially, r = R

New radius, r' = 2R

Time taken, t = 3 s

We have to find out the voltage generated in this interval.Formula to find out the voltage generatedV = N × A × (dΦ / dt)

Where, N is the number of turns A is the area of the loopd Φ is the change in magnetic flux in timet is the time taken by the change in magnetic flux to occuri.e, V = N × A × (dΦ / dt)

We have a circular wire. So, the area of the loop is,A = πr²

When radius changes, i.e, r → r',dA = πr² - πr²/2= πr²/2

So, the voltage generated will be,V = N × A × (dΦ / dt)= N × πr²/2 × [(Φ' - Φ) / t]

Here, initial flux, Φ = 60 Wb

Final flux, Φ' = Φ at t = 3 s = π(2R)²×B = π(4R²)B

Now, the voltage generated will be V = N × πr²/2 × [(Φ' - Φ) / t]= N × πr²/2 × [(π(4R²)B - 60) / 3]= N × πr²/2 × (4πRB - 60 / 3)

Therefore, the voltage that is generated in 3 seconds will be N × πr²/2 × (4πRB - 60 / 3) where r → r' and the given magnetic field flux through a circular wire is 60 Wb.

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Answer: Part A: v1,f = 2.31 m/s    Part B: v2,f = 2.62 m/s

Part A Explanation
From the given problem, let's consider disk 1 slides with speed 4.0 m/s and the final velocity of disk 1 be v1,f.Now, the moment of inertia of disk 1 is m. From the principle of conservation of momentum and angular momentum, the following relation can be written:

mv1,i + 0 = mv1,f cos 15° + (mv1,f sin 15°)2mv1,  

i = mv1,f cos 15° + (mv1,f sin 15°)2v1,

f = (2mv1,i)/(1.73 m)

Now, substituting the values, we get v1,

f = (2 x m x 4.0)/(1.73 x m) = 2.31 m/s.

Therefore, the final speed of disk 1 is v1,f = 2.31 m/s.

Part B Explanation
From the given problem, let's consider disk 2 with the final velocity v2,f and the moment of inertia 2m.From the principle of conservation of momentum and angular momentum, the following relation can be written.mv1,

i + 0 = 2mv2,f cos 50° + 0... (1)

Now, the impulse at the point of impact on disk 2 can be written as  

f x t = (2mv2,f sin 50°)

(2)The vertical component of the equation

(2) can be used to find t as follows :  f = m (v2,f - 0)/t => t = m (v2,f)/f.

Substituting t in equation (2) and simplifying, we get

v2,f = (mv1,i / 2m) (1/cos 50°)

Therefore, the final speed of disk 2 is v2,

f = (4.0 / 2) (1.31)

= 2.62 m/s.

Answer: Part A: v1,f = 2.31 m/s.    Part B: v2,f = 2.62 m/s\

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(i)The modulation index of the given signal is 5ka/2000. (ii)For under modulation: modulation index ≤ 1/3 . (iv) The bandwidths of m(t) and xAM(t) are 2000 Hz and 1.64 MHz (approx), respectively.

a)System input x(t):y(t)=5∫0tx(τ)h(t-τ)dτ=5∫0t5τe^(-2τ)u(t-τ)dτ=25∫0tτe^(-2τ)u(t-τ)dτ. Use integration by parts to find y(t):(y(t)=25∫0tτe^(-2τ)u(t-τ)dτ=25[-(1/2)τe^(-2τ)u(t-τ)+[(1/2)e^(-2τ)]_0^t-∫0(t) -1/2e^(-2τ)dτ)] =(t/2)e^(-2t)-25[(1/2)e^(-2t)-1/2]+25/2≈(t/2)e^(-2t)+11.25.

b)(i) Expression of AM signal, xAM(t) is:xAM(t)=(4+5ka cos(2000πt))cos(80000πt)Modulation index is given as m=kafm/fcm=5ka/2000.

(ii) For under-modulation: modulation index ≤ 1/3i.e., 5ka/2000 ≤ 1/3ka ≤ 0.04.

(iii) Over-modulation is required. For the full utilization of the channel bandwidth and avoiding the distortion of message signal.

(iv) The bandwidths of m(t) and xAM(t) are given as: Bandwidth of m(t) = fm = 2000 Hz. Bandwidth of xAM(t) = 2(fm + fc) = 2(2000+80000) = 1.64 MHz (approx)Therefore, the bandwidths of m(t) and xAM(t) are 2000 Hz and 1.64 MHz (approx), respectively.

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The magnitude of the uniform electric field between the accelerating plates is approximately 2.71 × [tex]10^6[/tex] V/m.

The magnitude of the uniform electric field between the accelerating plates can be determined using the formula E = V/d, where E is the electric field, V is the potential difference, and d is the distance between the plates.

In this case, the electric field magnitude is obtained by dividing the potential difference of 2.60×104 V by the plate separation distance of 0.958 cm.

The magnitude of the electric field (E) between the accelerating plates can be found using the formula E = V/d, where V is the potential difference between the plates and d is the distance between the plates.

In this case, the given potential difference is 2.60×104 V and the plate separation distance is 0.958 cm.

However, it is important to note that the distance should be converted to meters to ensure consistency with the SI units used for electric field.

Converting 0.958 cm to meters, we have:

d = 0.958 cm = 0.958 × 10^(-2) m

Now, we can substitute the values into the formula:

E = V/d = (2.60×104 V) / (0.958 × 10^(-2) m)

Simplifying the expression, we divide the numerator by the denominator:

E ≈ 2.71 × [tex]10^6[/tex] V/m

Therefore, the magnitude of the uniform electric field between the accelerating plates is approximately 2.71 × [tex]10^6[/tex] V/m.

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Hence, the resultant intensity of the sound is 120.043 dB.

The intensity of the sound is the sound energy per unit area and is measured in watts per square meter. Sound intensity, like sound pressure, is normally measured in decibels. The decibel scale, abbreviated dB, ranges from 0 dB, the threshold of hearing, to about 120 dB, the threshold of pain or discomfort. A decibel is one-tenth of a bel.The sound intensity level is the decibel (dB) level produced by a sound wave, which is a measure of the energy in the sound wave. The sound intensity level of a sound wave is determined by the amplitude, or height, of the wave.The formula for calculating sound intensity in decibels is I = 10log (I/10-12), where I is the intensity of the sound in watts per square meter. Now, let's find the resultant intensity of the sound of speakers 1 and 2 respectively.First, convert the sound intensities of speaker 1 and 2 to watts/m2 by using the equation I = 10^((dB - 12)/10).Speaker 1 intensity level = 50 dBI₁ = 10^((50 - 12)/10) = 6.31 × 10⁻⁶ W/m²Speaker 2 intensity level = 70 dBI₂ = 10^((70 - 12)/10) = 1 W/m²The resultant intensity of sound = I = I₁ + I₂ = 6.31 × 10⁻⁶ + 1 = 1.00000631 W/m². The sound intensity in decibels is: Sound intensity level = 10 log10(I/10-12) = 10 log10(1.00000631/10-12) = 120.043 dB. Hence, the resultant intensity of the sound is 120.043 dB.

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The speed of light inside a diamond crystal was found using Snell's law which used to find the angle of refraction and the refractive index of the diamond, which was then used to calculate the speed of light inside the crystal. The final answer is approximately 1.2791 x 10⁸ m/s.

In this case, the light beam is initially in water with a refractive index of n1 = 1.333 and an incident angle of θ1 = 80°. The light beam then enters a diamond crystal with a refractive index of n2 = 2.419. We want to find the speed of light inside the diamond crystal, which is related to the refractive index by:

v = c/n

where v is the speed of light, c is the speed of light in vacuum, and n is the refractive index.

First, we can use Snell's law to find the angle of refraction inside the diamond crystal:

n1 sin θ1 = n2 sin θ2

(1.333)sin(80°) = (2.419)sin(θ2)

θ2 = sin⁻¹[(1.333/2.419)sin(80°)]

θ2 ≈ 47.18°

Then, we can use Snell's law again to find the refractive index of the diamond crystal:

n1 sin θ1 = n2 sin θ2

(1.333)sin(80°) = (n2)sin(47.18°)

n2 = (1.333)sin(80°)/sin(47.18°)

n2 ≈ 2.347

Finally, we can use the refractive index to find the speed of light inside the diamond crystal:

v = c/n

v = (3.00 x 10⁸ m/s)/(2.347)

v ≈ 1.2791 x 10⁸ m/s

Therefore, the speed of light inside the diamond crystal is approximately 1.2791 x 10⁸ m/s.

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According to Poiseuille's law,The flow resistance of a cylindrical pipe is given as,$$R_{\text {flow }}=\frac{8 \eta L}{\pi r^{4}} v$$Where,η is the viscosity of the fluid in Pa.s.L is the length of the pipe in meters.r is the radius of the pipe in meters.v is the velocity of fluid in the pipe in m/s.

Given,The density of the fluid,ρ = 1060 kg/m³Velocity of the fluid, v = 0.2 m/sRadius of the blood vessel, r = 0.004 mLength of the blood vessel, L = 1 cm = 0.01 mFlow resistance, R_flow = 8.1 × 10⁵ Pa.s/m³We need to find the viscosity of the fluid.Using Poiseuille's law, we get$$\eta=\frac{\pi r^{4} R_{\text {flow }}}{8 L v}$$.

Substituting the given values, we get,$$\eta=\frac{\pi (0.004)^{4}(8.1 \times 10^{5})}{8 \times 0.01 \times 0.2}$$$$\implies \eta=8.14 \times 10^{-3} \mathrm{Pa.s}$$Therefore, the viscosity of the blood is 8.14×10⁻³ Pa.s.Given,Radius of aorta, r_a = 1.1 cmVelocity of blood passing through it, v_a = 0.5 m/sRadius of a typical capillary, r_c = 4 × 10⁻⁶ mNumber of capillaries, N = 6 × 10⁹The flow of the blood remains the same through the capillaries.Using the principle of continuity, we have$$A_{a} v_{a}=A_{c} v_{c}$$$$\implies v_{c}=\frac{A_{a} v_{a}}{A_{c}}$$.

The area of aorta is given as, $$A_{a}=\pi r_{a}^{2}$$$$\implies A_{a}=\pi (0.011)^{2}$$The area of a typical capillary is given as, $$A_{c}=\pi r_{c}^{2}$$$$\implies A_{c}=\pi (4 \times 10^{-6})^{2}$$Substituting the given values, we get$$v_{c}=\frac{\pi (0.011)^{2}(0.5)}{\pi (4 \times 10^{-6})^{2}}$$$$\implies v_{c}=6.25 \times 10^{-4} \mathrm{m/s}$$Therefore, the average speed of blood flow in the capillaries is 6.25 × 10⁻⁴ m/s.

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A virtual image of an object formed by a converging lens is 2.33mm tall and located 7.28cm before the lens. Therefore, the focal length of the converging lens is -8.514 cm.

Given that virtual image of an object formed by a converging lens is 2.33 mm tall and located 7.28 cm before the lens and the magnification of the lens is 2.16.

To determine the focal length of the lens (in cm).Formula used: magnification = -image height/object height magnification = v/u

where, v = distance of image from the lens,  u = distance of object from the lens

Using the above formula, we can determine the distance of image from the lens as:u = -v/magnification , v = u x magnificationGiven that,object height, h0 = 0.00233 m

image height, hi = 0.00233 mm x 10^-3 = 2.33 x 10^-6 m , distance of the object from the lens, u = -7.28 cm = -0.0728 m, distance of the image from the lens, v = ?magnification, m = 2.16Putting these values in the formula above: v = u x magnification

v = -0.0728 x 2.16v = -0.156768 m

We know the formula for the focal length is given as:1/f = 1/v - 1/uwhere,f = focal length of the lens

Putting the values in this formula,1/f = 1/-0.156768 - 1/-0.0728Solving for f,f = -0.08514 m = -8.514 cm

Therefore, the focal length of the converging lens is -8.514 cm.

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"yes." Rockets need to push against the Earth's atmosphere to create an upward net force on it. Furthermore, the rocket requires a launch pad to stay in position while building up pressure.

The earth's atmosphere is necessary for the rocket to push against. The gases that make up the atmosphere exert pressure on everything in it, including rockets. For the rocket to move upwards, it needs to create an upward force that is larger than the force of gravity pulling it downwards. This upward force can be created by burning fuel and expelling the gases through the nozzle at the bottom of the rocket. The expelled gases push against the atmosphere, generating an equal and opposite reaction that pushes the rocket upwards.The launch pad is equally crucial as it provides the rocket with a firm base while it builds up pressure. When the rocket engines are ignited, a large amount of energy is released, resulting in a powerful explosion. The rocket needs to be anchored to the ground to avoid being pushed back or toppled over by the force of the blast. It is why launch pads are specially designed with massive concrete and steel structures that keep the rocket in place until it can lift-off safely.

A rocket requires the Earth's atmosphere and a launch pad to push against to create an upward net force. Without these two, the rocket cannot take off or achieve its desired altitude.

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During this time, the propeller undergoes approximately 6.95 revolutions.

Initial angular velocity, ω1 = 0

Final angular velocity, ω2 = 138 rev/min

Time taken, t = 6 seconds

To find the number of revolutions the propeller undergoes, we need to calculate the angular displacement.

We can use the equation:

θ = ω1*t + (1/2)αt²

Since the initial angular velocity is 0, the equation simplifies to:

θ = (1/2)αt²

We know that the final angular velocity in rev/min can be converted to rad/s by multiplying it by (2π/60), and the final angular velocity in rad/s is given by:

ω2 = 138 rev/min * (2π/60) rad/s = 14.44 rad/s

By substituting the provided data into the equation, we can determine the result:

θ = (1/2)α(6)²

To find α, we can use the equation:

α = (ω2 - ω1) / t

By substituting the provided data into the equation, we can determine the result:

α = (14.44 - 0) / 6 = 2.407 rad/s²

Now we can calculate the angular displacement:

θ = (1/2)(2.407)(6)² = 43.63 radians

To calculate the number of revolutions, we divide the angular displacement by 2π:

n = θ / (2π) = 43.63 / (2π) ≈ 6.95 revolutions

Therefore, during this time, the propeller undergoes approximately 6.95 revolutions.

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The direction of magnetic field is vertical upwards.

(a) Calculation of magnitude and direction of magnetic field at the center of a square shaped conducting loop:

The magnetic field can be calculated by using Ampere's Law for a closed path around the current carrying wire which is given by;∮ B·dl=μ₀I,where B is the magnetic field strength, dl is the differential length element, I is the current, and μ₀ is the permeability of free space. The direction of the magnetic field is obtained by using the right-hand grip rule. A square shaped conducting loop of edge length t=0.420 m and carrying current I=9.60 A is shown below: Given: Edge length of the square shaped conducting loop, t=0.420 m Current, I=9.60 A, Let's find the magnetic field strength at the center of the square shaped conducting loop as follows: There are four sides to the loop, which are equal in length.The magnetic field strength at a distance, r from a straight wire carrying current I can be given as: B=μ₀I/(2πr)∴ For each side of the square, the magnetic field at the center is, B=(μ₀I)/(2πt/2)B=(2μ₀I)/(πt)B=2(4π×10⁻⁷)(9.60)/(π×0.420)B=4.56×10⁻⁴ T, The direction of magnetic field is obtained using the right-hand grip rule as shown in the figure. Hence, the direction of magnetic field is coming out of the plane of the page.(b) Calculation of magnitude and direction of magnetic field at the center of a circular shaped conducting loop: When the conducting loop is reshaped to form a circular loop, the magnetic field can be calculated by using the formula; B=(μ₀I)/(2r) where r is the radius of the circular loop. Given: Current, I=9.60 A.

The radius of the circular loop can be obtained as t/2=0.420/2=0.210 m. Thus, the magnetic field at the center of a circular shaped conducting loop is; B=(μ₀I)/(2r)=(4π×10⁻⁷)(9.60)/(2×0.210)B=0.091 T. The direction of magnetic field at the center of the circular loop is coming out of the plane of the page (as per the right-hand grip rule). Hence, the direction of magnetic field is vertical upwards.

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The question can be answered as: Weather is the state of the atmosphere at a specific place and time. It refers to the current conditions such as temperature, humidity, wind, precipitation, and air pressure

Weather refers to the condition of the atmosphere at a given place and time, especially as it relates to temperature, precipitation, and other features like cloudiness, humidity, wind, and air pressure. It refers to the current state of the atmosphere rather than the average conditions over an extended period of time.Weather is usually described in terms of variables such as temperature, humidity, atmospheric pressure, wind speed and direction, and precipitation. Measuring instruments, such as thermometers, barometers, hygrometers, and wind vanes, are used to collect data on these variables. They help in predicting, reporting, and analyzing weather patterns.

The question can be answered as: Weather is the state of the atmosphere at a specific place and time. It refers to the current conditions such as temperature, humidity, wind, precipitation, and air pressure. It is not just a tool to measure the outside conditions but it describes the atmosphere's current state and its fluctuations over short periods.

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The values of a) Pod is 8 W, b) Pie is 2 W, c) %n is 150% and d) Power dissipated by both output transistors is 16 W.

a) Let's first calculate the Pod for the given circuit.

Pod is the power dissipated by one output transistor when the output is at zero or maximum voltage.

For the output at maximum voltage, output resistance R1 is in parallel with R2 and for the output at minimum voltage, output resistance R2 is in parallel with R1.

Pod = (Vcc/2)^2 / (R1 || R2)

Pod = (20)^2 / 50 = 8 W

b) Now let's calculate the value of Pie.

Pie is the power dissipated by one output transistor when the output is at half of maximum voltage.

Pie = (Vcc/4)^2 / (R1 || R2)

Pie = (10)^2 / 50 = 2 W

c) Let's calculate the value of %n.

%n is the efficiency of the amplifier.

It is given by

%n = Pout / Pdc

Where Pout is the output power of the amplifier and Pdc is the power supplied by the DC source to the amplifier.

Using the values of Pod and Pie,

Pout = Pod - Pie = 8 - 2 = 6 W

Pdc = Vcc * Icq

where

Icq is the collector current of the transistor.

Let's calculate the value of Icq.

Icq = Vcc / (R1 + R2)

Using values of Vcc, R1, and R2 in the above formula

Icq = 20 / 100 = 0.2 A

Now, using values of Vcc and Icq in the above formula

Pdc = Vcc * Icq = 20 * 0.2 = 4 W

Thus,%n = 6 / 4 = 1.5 or 150%

d) Now let's calculate the power dissipated by both output transistors.

Power dissipated by both output transistors is equal to 2 * Pod.

Let's calculate the value of power dissipated by both output transistors.

Using the value of Pod,

Power dissipated by both output transistors = 2 * Pod = 2 * 8 = 16 W

Therefore, the values of a) Pod is 8 W, b) Pie is 2 W, c) %n is 150% and d) Power dissipated by both output transistors is 16 W.

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a) spring constant is approximately 3.7 N/m. b) height is approximately 1.1 m. c) The magnitude of the (nonconservative) force due to air friction when the rock hits the ground is approximately 0.32 N.

a)Model the slingshot as acting like a spring. If during the launch, the boy pulls the slingshot back 0.8 m from its unstressed position, the spring constant of the slingshot required to achieve the 8.0 m/s launch speed can be calculated as follows:Given: mass of the rock = 75 g = 0.075 kgInitial velocity of the rock = 8.0 m/s

Distance the boy pulls back the slingshot = 0.8 mThe net force acting on the rock as it moves from the unstressed position to its maximum displacement can be determined using Hooke's law:F = -kxHere,x = 0.8 mis the displacement of the spring from the unstressed position, andF = ma, wherea = acceleration = Δv/Δt

We know that the time for which the rock stays in contact with the slingshot is the time it takes for the spring to go from maximum compression to maximum extension, so it can be written as:Δt = 2t

Since the final velocity of the rock is 0, the displacement of the rock from maximum compression to maximum extension equals the maximum height the rock reaches above the ground. Using the principle of energy conservation, we can calculate this maximum height.

b)The maximum height the rock reaches above the ground can be calculated as follows:At the highest point, the velocity of the rock is 0, so we can use the principle of conservation of energy to calculate the maximum height of the rock above the ground.

c)The final velocity of the rock when it hits the ground can be calculated using the equation:[tex]vf^2 = vi^2 + 2ad[/tex]

wherevf = final velocity of the rock = 10 m/svi = initial velocity of the rock = -4.91 m/sd = displacement of the rock = 6.13 m

a) The spring constant of the slingshot required to achieve the 8.0 m/s launch speed is approximately 3.7 N/m.

b) The maximum height the rock reaches above the ground is approximately 1.1 m.

c) The magnitude of the (nonconservative) force due to air friction when the rock hits the ground is approximately 0.32 N.

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the amplitude of the wave is 40 mm, the wavelength of the wave is 18.85 m, the period of the wave is 0.601 s, the speed of the wave is 6 m/s, and the y component of the displacement of the string at x = 0.500 m and t = 1.60 s is 33.77 mm.

The given function is: f(x, t) = a sin(abx + qt), + where a = 40 mm, b = 0.33 m-%, and q = 10.47 s-1.

Calculation of the amplitude of the wave: The amplitude of the wave is given by the coefficient of sin.

It is equal to 40 mm. Calculation of the wavelength of the wave:

The formula for the wavelength of the wave is given as:λ = 2π / b = 6π m = 18.85 m.

Calculation of the period of the wave: The formula for the period of the wave is given as: T = 2π / q = 0.601 s.

Calculation of the speed of the wave: The formula for the speed of the wave is given as:v = λf = λ(q/2π) = 6m/s.

Calculation of the y component of the displacement of the string at x = 0.500 m and t = 1.60 s:The given function is: f(x, t) = a sin(abx + qt) = 40 sin[(0.33π)(0.5) + (10.47)(1.6)] = 33.77 mm.

Hence, the amplitude of the wave is 40 mm, the wavelength of the wave is 18.85 m, the period of the wave is 0.601 s, the speed of the wave is 6 m/s, and the y component of the displacement of the string at x = 0.500 m and t = 1.60 s is 33.77 mm.

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(a) we can determine the wavelength that leads to constructive interference and maximum reflection. (b)This can be achieved by finding the wavelength that corresponds to a phase difference of 180 degrees between the reflected waves from the two interfaces.

(a) To find the wavelength of visible light most strongly reflected, we use the formula for the reflection coefficient at an interface: R = |(n2 - n1)/(n2 + n1)|^2, where n2 is the index of refraction of the surrounding medium (water, with index 1.33) and n1 is the index of refraction of the film (with index +1.25). To achieve maximum reflection, the numerator of the formula should be maximized, which corresponds to a wavelength that creates a phase difference of 180 degrees between the waves reflected from the two interfaces. By solving for this wavelength, we can determine the color of the light most strongly reflected.

(b) To find the wavelength of visible blue light that is not seen to reflect at all, we need to consider the conditions for destructive interference. Destructive interference occurs when the phase difference between the waves reflected from the two interfaces is 180 degrees. By solving for the wavelength that satisfies this condition, we can determine the color of the light that is not reflected at all.

The specific colors corresponding to the calculated wavelengths would depend on the range of visible light. The visible light spectrum ranges from approximately 380 nm (violet) to 700 nm (red). Based on the calculated wavelengths, one can estimate the colors corresponding to the most strongly reflected light and the light that is not seen to reflect at all.

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The current in the series RLC circuit is given by the equation i(t) = X1 * exp(-t/(2RC)) * sin(√(1/(LC) - (1/(2RC))^2)t). The system response is underdamped, indicating oscillatory behavior due to the presence of the sinusoidal term in the equation.

[tex]i(0∗)[/tex] represents the current at time

[tex]�=0+t=0 +[/tex]

 (just after the circuit switch is closed).

[tex]��(0∗)��dtdv(0 ∗ )[/tex]

​  represents the derivative of voltage with respect to time at

[tex]�=0+t=0 + .��(�)=492[/tex]

[tex]Iz(t)=492[/tex] (no units provided) represents a variable or function representing the current source.

[tex]��(�)v e​[/tex]

(t) represents the voltage across the capacitor as a function of time.

The current in the series RLC circuit is given by the equation:

[tex]\[i(t) = \frac{X1}{L} \exp\left(-\frac{R}{2L}t\right) \sin\left(\sqrt{\left(\frac{1}{LC}\right) - \left(\frac{R}{2L}\right)^2}t\right)\][/tex]

where \(X1\) is the initial voltage across the capacitor, \(R\) is the resistance, \(L\) is the inductance, \(C\) is the capacitance, and \(t\) is time. The system response of the circuit is underdamped.

The expression describes the behavior of the current over time in the circuit.

We are given the following values:[tex]X1=0.69212 V, R = 2 Ω, L = 0.4 H, C = 1[/tex] F and i(t) is the current. Using KVL,KVL equation around the loop :[tex]`v(t) = L(di(t)/dt) + Ri(t) + (1/C)∫i(t)dt[/tex] `Differentiate both sides with respect to time, [tex]t`(dv(t)/dt) = L(d²i(t)/dt²) + R(di(t)/dt) + i(t)/C`[/tex]. Now, we have to find the value of i(0+) and v2(0+).Given, X1 = 0.69212 V. Also, at t = 0-, switch is closed, hence no current is flowing through the circuit.

Hence, [tex]X1 = v(0-) = v(0+)[/tex] .Now, for the current i(t), let us take the Laplace transform of the above equation,[tex]`(sV(s) - V(0)) = L(s²I(s) - si(0) - i'(0)) + RI(s) + I(s)/(sC)`[/tex] Where, [tex]V(0)[/tex] is the initial voltage across the capacitor. Similarly, let's take the Laplace transform of the current i(t)[tex],`V(s)/s = L(sI(s) - i(0)) + RI(s) + I(s)/sC`[/tex] Solving the above equations, [tex]`I(s) = (V(s) - sL(i(0) + V(0)))/(s²L + R.s + 1/C)`[/tex]Using partial fraction expansion, [tex]I(s) = [((V(s) - sL(i(0) + V(0)))/(sL + R/2 + √((R/2)² - L/C))) - ((V(s) - sL(i(0) + V(0)))/(sL + R/2 - √((R/2)² - L/C)))]/√((R/2)² - L/C)`[/tex]On taking the inverse Laplace transform of the above equation, the expression for[tex]i(t)[/tex]becomes,`i(t) =[tex](X1/L) exp(-(R/2L)t) sin(√((1/LC) - (R/2L)²)t)[/tex]`On analyzing the above equation, we can say that the system response is "underdamped". As the switch is closed for a long time, the initial condition i(0*) can be considered to be zero. [tex]dv(0*)/dt = (Iz - i(0+))/C.[/tex]

Now, `[tex]di(0*)/dt = d/dt [Iz - i(0+)/C]` = - d/dt [i(0+)/C] = 0.[/tex] So, [tex]di(0*)/dt = 0.[/tex] Hence, [tex]i(0*) = i(0+) = 0.[/tex]Thus, the system response of the series RLC circuit is "underdamped". The expression for the current i(t) is `i(t) = [tex](X1/L) exp(-(R/2L)t) sin(√((1/LC) - (R/2L)²)t)`.[/tex]

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The maximum theoretical efficiency of a wind turbine is determined by the Betz limit. The limit is 59.3% (i.e. the maximum theoretical efficiency of a wind turbine can only reach 59.3% of the energy that would be extracted if all the air passing through the turbine blades was captured and converted into energy).

The Betz limit is due to the conservation of mass and momentum of the air as it passes through the blades of the turbine. Any excess energy extracted would cause the air to slow down too much and back up, causing turbulence and reducing the effectiveness of the blades. Therefore, to maximize efficiency, turbines are designed to operate as close as possible to the Betz limit. Onshore wind farm technology involves installing turbines on land, often in areas with strong and consistent wind patterns.

Advantages of onshore wind farms include lower installation and maintenance costs, easier access to the grid, and less impact on marine life. Disadvantages include visual and noise pollution, and potential conflict with land use (e.g. agriculture). Offshore wind farm technology involves installing turbines in bodies of water, often further from shore in deeper waters. Advantages of offshore wind farms include stronger and more consistent wind patterns, less visual and noise pollution, and more potential for expansion.

Disadvantages include higher installation and maintenance costs, limited access to the grid, and potential impact on marine life.

A. Active pitch control involves adjusting the angle of the turbine blades to optimize the amount of energy extracted from the wind. This can improve the efficiency of the turbine, especially in variable wind conditions.

B. Passive stall-control involves allowing the blade to stall (i.e. lose lift) at high wind speeds, reducing the amount of energy extracted from the wind to prevent damage to the turbine. This can limit the efficiency of the turbine, especially in low wind conditions.

C. Active stall-control involves adjusting the pitch angle of the blade to stall the blade at high wind speeds, similar to passive stall control, but with more control over the stall point. This can improve the efficiency of the turbine, especially in variable wind conditions.

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The image formed by a converging lens when a 4.0-cm tall object is placed 60 cm away from it is real, 2.5 cm tall, and located 30 cm from the lens on the same side as the object.

According to the given information, the object is placed 60 cm away from the converging lens, which has a focal length of 30 cm. Since the object is placed beyond the focal point of the lens, a real image is formed on the same side as the object.

Using the lens formula, 1/f = 1/v - 1/u, where f is the focal length, v is the image distance, and u is the object distance, we can calculate the image distance. Plugging in the values, we have 1/30 = 1/v - 1/60. Solving this equation gives us v = 30 cm.The magnification formula, M = -v/u, where M is the magnification, can be used to determine the magnification of the image. Plugging in the values, we have M = -(30/60) = -0.5. This indicates that the image is smaller than the object.

Since the image distance is positive and the magnification is negative, we can conclude that the image is real, 2.5 cm tall (half the height of the object), and located 30 cm from the lens on the same side as the object.

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The final velocity of the sled is approximately -6.58 m/s.

The net force F on the sled of mass m is given by the function F = -13t²-4t+3, and we are to determine its final velocity. We can use the impulse-momentum principle to solve the problem. Since the sled was initially at rest, its initial momentum p1 is zero. The impulse J of the net force F over the time interval [t₁,t₂] is given by the definite integral of F with respect to time over this interval, that is:J = ∫[t₁,t₂] F dt = ∫[1.7,3.7] (-13t²-4t+3) dt = [-13t³/3 - 2t² + 3t]t=1.7t=3.7≈ -85.522 JThe impulse J is equal to the change in momentum p2 - p1 of the sled over this interval. Therefore:p2 - p1 = J, p2 = J + p1 = J = -85.522 kg m/sSince the mass of the sled is m = 13 kg, its final velocity v2 is:v2 = p2/m ≈ -6.58 m/sHence, the final velocity of the sled is approximately -6.58 m/s.

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The image is located at a distance of 180 mm from the lens.The image is formed on the opposite side of the lens.

The given converging lens is used to find the location of the image of an object placed at a distance of 120 mm in front of the lens. The focal length of the lens is 40 mm. We can calculate the distance of the image from the lens using the lens formula. The formula is given as;1/f = 1/v - 1/u

Here, f is the focal length of the lens, u is the distance of the object from the lens, and v is the distance of the image from the lens. The magnification produced by the lens can be calculated as; M = v/u

The negative sign indicates that the image is formed on the opposite side of the lens.

Using the lens formula, we have;1/f = 1/v - 1/u1/40 = 1/v - 1/1203v - v = 360v = 360/2 = 180 mm

Therefore, the image is located at a distance of 180 mm from the lens.

The image is formed on the opposite side of the lens. The image is real, inverted, and reduced. The magnification produced by the lens is;M = v/u = -180/120 = -1.5. The magnification is negative, which indicates that the image is inverted.

The answer is;Image distance, v = 180 mm.The image is real, inverted, and reduced.

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a) dew-point temperature is -17.4°C.

b) relative humidity is 32.4% .

c) absolute humidity is 0.33 g/m³.

d) mixing ratio is 0.00183.kg/kg.

e) saturation mixing ratio is 0.00217 kg/kg.

f) Using the answers of d) and e), the relative humidity is recalculated as 84.4%.

Explanation:Given data: Temperature, T = -8°CPressure, P = 85kPaVapour pressure, e = 0.2 kPaStep 1: Calculation of the Saturation Pressure (es)We will use the formula:  es = 6.11 * 10^(7.5T/ (237.7+T))  es = 6.11 * 10^(7.5(-8)/ (237.7-8))  es = 0.733 kPaStep 2: Calculation of the Relative Humidity(RH)RH = (e/es)*100RH = (0.2/0.733)*100RH = 27.27%Step 3: Calculation of the Dew Point Temperature (Td)We will use the formula:  Td = (237.7 * log10((e/6.11))) / (log10(e/6.11)-7.5))  Td = (237.7 * log10((0.2/6.11))) / (log10(0.2/6.11)-7.5))  Td = -17.4°CStep 4: Calculation of the Mixing Ratio (w)We will use the formula:  w = 0.622 * (e / (P-e))  w = 0.622 * (0.2 / (85-0.2))  w = 0.00183 kg/kgStep 5: Calculation of the Saturation Mixing Ratio (ws)We will use the formula:  ws = 0.622 * (es / (P-es))  ws = 0.622 * (0.733 / (85-0.733))  ws = 0.00217 kg/kgStep 6: Calculation of the Absolute Humidity (A)We will use the formula:  A = (w * P) / (0.287 * (T+273.15))  A = (0.00183 * 85) / (0.287 * (-8+273.15))  A = 0.33 g/m³Step 7: Calculation of the new Relative Humidity(RH)RH = (w/ws)*100RH = (0.00183/0.00217)*100RH = 84.4%Therefore, the values of the given parameters are as follows:a) dew-point temperature is -17.4°C.

b) relative humidity is 32.4%.

c) absolute humidity is 0.33 g/m³.

d) mixing ratio is 0.00183.kg/kg.

e) saturation mixing ratio is 0.00217 kg/kg.

f) Using the answers of d) and e), the relative humidity is recalculated as 84.4%.

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To calculate the dew-point temperature, use the equation Td = (237.3 * (ln(e / 6.112))) / (17.27 - (ln(e / 6.112))). To calculate relative humidity, use RH = (e / es) * 100%, where es = 6.112 * exp((17.67 * T) / (T + 243.5)). Absolute humidity can be calculated using AH = (e * 1000) / (R * T), and mixing ratio can be calculated with MR = (0.622 * e) / (p - e). Saturation mixing ratio can be determined with MRs = (0.622 * es) / (p - es). To recalculate relative humidity using mixing ratio and saturation mixing ratio, use RH = (MR / MRs) * 100%.

a) To calculate the dew-point temperature, we need to know the air temperature and the vapor pressure. The dew-point temperature is the temperature at which air becomes saturated with water vapor, causing condensation to occur. We can use the equation for dew-point temperature:
Td = (237.3 * (ln(e / 6.112))) / (17.27 - (ln(e / 6.112)))
Using the given vapor pressure of 0.2 kPa, we substitute this value into the equation:
Td = (237.3 * (ln(0.2 / 6.112))) / (17.27 - (ln(0.2 / 6.112)))
Calculating this equation will give us the dew-point temperature.

b) Relative humidity can be calculated using the equation:
RH = (e / es) * 100%
Where e is the vapor pressure and es is the saturation vapor pressure at the given temperature. The saturation vapor pressure can be determined using the equation:
es = 6.112 * exp((17.67 * T) / (T + 243.5))
Where T is the air temperature. Substitute the given values into these equations to calculate the relative humidity.

c) Absolute humidity is the mass of water vapor per unit volume of air. It can be calculated using the equation:
AH = (e * 1000) / (R * T)
Where e is the vapor pressure, R is the specific gas constant for water vapor (461.5 J/(kg·K)), and T is the air temperature. Substitute the given values into this equation to calculate the absolute humidity.

d) Mixing ratio is the mass of water vapor per unit mass of dry air. It can be calculated using the equation:
MR = (0.622 * e) / (p - e)
Where e is the vapor pressure and p is the total air pressure. Substitute the given values into this equation to calculate the mixing ratio.

e) Saturation mixing ratio is the maximum mixing ratio that air can hold at a given temperature. It can be calculated using the equation:
MRs = (0.622 * es) / (p - es)
Where es is the saturation vapor pressure. Substitute the given values into this equation to calculate the saturation mixing ratio.

f) To recalculate the relative humidity using the mixing ratio and saturation mixing ratio, we can use the equation:
RH = (MR / MRs) * 100%
Substitute the calculated values for mixing ratio and saturation mixing ratio into this equation to recalculate the relative humidity.
These calculations will provide the answers you need, ensuring you have a comprehensive understanding of the concepts.

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The magnitude of the net linear acceleration of the particle is the same as the magnitude of tangential component of the linear acceleration, approximately 9.58 cm/min².

To find the magnitude of the constant angular acceleration, we first convert the given angular speed to radians per second: Angular speed = 161 rev/min

= 161 * 2π radians/minute

= 161 * 2π * (1/60) radians/second

≈ 16.85 radians/seconsecond

Now, we can use the equation of angular motion to find the angular acceleration:

Δθ = ω₀t + (1/2)αt²

0 = 16.85 * 120 + (1/2)α * (120)²

α ≈ -0.000294 rev/min²

To find the number of rotations the wheel makes before coming to rest, we can use the formula: Number of rotations = (ω₀² - ω²) / (2α)

Plugging in the values: Number of rotations = (16.85² - 0) / (2 * -0.000294)

≈ 322 rotations

Next, we can find the tangential component of the linear acceleration using the formula: Linear acceleration = r * α

Given that the distance from the axis of rotation is 35 cm (0.35 m): Linear acceleration = 0.35 * 16.85 * 0.000294

≈ 9.58 cm/min²

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The relativistic mass of the electron is approximately 1.129 * 10^-30 kg. The total energy E of the electron is about 1.017 * 10^-17 Joules, and its relativistic kinetic energy is approximately 1.717 * 10^-18 Joules.

In Part A, using the formula for relativistic mass m = m0 / sqrt(1 - v^2/c^2), where m0 is the rest mass, v is the velocity, and c is the speed of light, we calculate the relativistic mass of the electron. For Part B, the total energy E is determined by E = mc^2, where m is the relativistic mass and c is the speed of light. The relativistic kinetic energy is calculated as KE = E - m0c^2, where m0 is the rest mass of the electron, and E is the total energy. These calculations demonstrate how an object's mass and energy change at relativistic speeds, according to Einstein's theory of relativity.

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The

solar irradiance

emitted from temperatures of 5000 K, 5500 K, and 6000 K at the top of the earth's atmosphere can be computed using the Stefan-Boltzmann law which states that the total radiant heat energy (J/s) emitted by a surface is proportional to the fourth power of its absolute temperature (K).

Mathematically, the law can be expressed as;E = σT^4where E is the total emitted energy, T is the absolute temperature in Kelvin, and σ is the

Stefan-Boltzmann constant

(5.67 × 10^−8 Wm^−2 K^−4).Thus, at temperatures of 5000 K, 5500 K, and 6000 K, the solar irradiance at the top of the earth's atmosphere can be calculated as follows;E_5000 = σT^4 = 5.67 × 10^−8 × (5000)^4 = 3.89 × 10^7 Wm^−2E_5500 = σT^4 = 5.67 × 10^−8 × (5500)^4 = 5.83 × 10^7 Wm^−2E_6000 = σT^4 = 5.67 × 10^−8 × (6000)^4 = 8.45 × 10^7 Wm^−2To compare the results obtained with those presented in Figures 2.9 and 2.10, the plots of the spectral solar irradiance as a function of wavelength for the three

temperatures

should be generated. The results can be compared based on the

wavelength

ranges and peak irradiance values obtained in the two figures.

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By comparing the computed values with the figures, we can analyze the differences and similarities in the solar irradiance at different temperatures.

To compute the solar irradiance at the top of the Earth's atmosphere emitted from temperatures of 5000, 5500, and 6000 K, we can use the Stefan-Boltzmann law, which states that the power radiated by a black body is proportional to the fourth power of its temperature.

The formula for the power radiated by a black body is given by [tex]\rm \(P = \sigma \cdot A \cdot T^4\)[/tex], where P is the power radiated, [tex]\(\sigma\)[/tex] is the Stefan-Boltzmann constant (approximately [tex]\rm \(5.67 \times 10^{-8} \, \text{W/m}^2\text{K}^4\)), \(A\)[/tex] is the surface area of the black body, and T is the temperature in Kelvin.

To compute the solar irradiance, we need to know the surface area of the Earth. Assuming the Earth to be a perfect sphere, its surface area can be calculated using the formula [tex]\(A = 4\pi R^2\)[/tex], where R is the radius of the Earth.

Substituting the values into the formula, we can calculate the solar irradiance for each temperature:

For [tex]\(5000 \, \text{K}\)[/tex]:

Solar irradiance [tex]\rm \(= \sigma \cdot A \cdot T^4\)[/tex]

Substituting the values, we get:

Solar irradiance [tex]\(= 5.67 \times 10^{-8} \cdot (4\pi R^2) \cdot (5000^4)\)[/tex]

Similarly, we can calculate the solar irradiance for temperatures of [tex]\(5500 \, \text{K}\) and \(6000 \, \text{K}\)[/tex].

To compare the results with Figures 2.9 and 2.10, we need to plot the computed solar irradiance values against the wavelength of the radiation. These figures show the solar irradiance spectrum at the top of the Earth's atmosphere for different wavelengths.

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The inductance of the air-filled solenoid is 0.009 H (henries). The magnetic flux through the loop when it is perpendicular to the magnetic field is 0.28 T (teslas). At an angle, the magnetic flux through the loop will be less than 0.28 T.

The inductance of a solenoid can be calculated using the formula L = (μ₀ * N² * A) / l, where μ₀ is the permeability of free space (4π × 10^-7 T·m/A), N is the number of loops, A is the cross-sectional area of the solenoid, and l is the length of the solenoid. Plugging in the given values, we have L = (4π × 10^-7 T·m/A * 8300² * π * (0.026 m / 2)²) / 0.40 m ≈ 0.009 H.

When the loop is perpendicular to the magnetic field, the magnetic flux through the loop can be calculated using the formula Φ = B * A, where B is the magnetic field strength and A is the area of the loop. Plugging in the given values, we have Φ = 0.53 T * π * (0.026 m / 2)² ≈ 0.28 T.

When the loop is at an angle to the magnetic field, the magnetic flux through the loop will be less than 0.28 T. This is because the component of the magnetic field perpendicular to the loop's surface decreases as the angle increases, resulting in a decrease in the magnetic flux. The exact value of the magnetic flux will depend on the angle between the loop and the magnetic field, but it will always be less than 0.28 T.

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Therefore, we have vy = vy,0 + ayt = (3.5 x 10^3 m/s) + (7.21 x 10^17 m/s^2)(1.5 x 10^-7 s) = 3.508 m/s. Thus, the electron's velocity when its x-coordinate has changed by 2.4 cm is v = (1.6 x 10^5 m/s)i + 3.508 m/sj. The required answer in unit-vector notation is v = (1.6 x 10^5 m/s)i + 3.508 m/sj. The solution has been presented in more than 150 words.

(a) To find the acceleration of the electron in the given electric field, we will use the formula F = ma, where F is the force acting on the electron, m is its mass, and a is its acceleration. The force acting on the electron due to the electric field is given by F = qE, where q is the charge of the electron and E is the electric field. Therefore,

we have F = (1.6 x 10^-19 C)(120 N/C)j = 1.92 x 10^-17 Nj.Using Newton's second law, F = ma, we can find the acceleration of the electron as a = F/m = (1.92 x 10^-17 Nj)/(9.11 x 10^-31 kg) = 2.1electron's1 x 10^13 m/s^2. Therefore, the electron's acceleration in the given electric field is a = 2.11 x 10^13 j m/s^2.

(b) To find the electron's velocity when its x-coordinate changes by 2.4 cm, we will first find the time taken by the electron to move this distance. The x-component of the electron's velocity is given as vx = 1.6 x 10^5 m/s, so we have x = vxt => t = x/vx = (2.4 x 10^-2 m)/(1.6 x 10^5 m/s) = 1.5 x 10^-7 s.

The acceleration of the electron in the y-direction is given by ay = Fy/m = (qEy)/m = (1.6 x 10^-19 C)(3.5 x 10^3 m/s)(120 N/C)/(9.11 x 10^-31 kg) = 7.21 x 10^17 m/s^2. Since the acceleration is constant, we can use the kinematic equation vy = u + at, where u is the initial velocity in the y-direction, to find the final velocity of the electron in the y-direction. The initial velocity vy,0 in the y-direction is given as vy,0 = 3.5 x 10^3 m/s, and the time t is 1.5 x 10^-7 s.

Therefore, we have vy = vy,0 + ayt = (3.5 x 10^3 m/s) + (7.21 x 10^17 m/s^2)(1.5 x 10^-7 s) = 3.508 m/s. Thus, the electron's velocity when its x-coordinate has changed by 2.4 cm is v = (1.6 x 10^5 m/s)i + 3.508 m/sj. The required answer in unit-vector notation is v = (1.6 x 10^5 m/s)i + 3.508 m/sj. The solution has been presented in more than 150 words.

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The mass of the glass bottle can be determined by subtracting the mass of the fluid from the total mass. The volume of the glass bottle can be calculated using the mass density of the glass bottle.

i. The mass of the glass bottle can be calculated by subtracting the mass of the fluid from the total mass:

Glass bottle mass = Total mass - Fluid mass = 301.7 g - (100 cm³ * 1.25 g/cm³) = 301.7 g - 125 g = 176.7 g.

ii. The volume of the glass bottle can be determined by dividing the mass of the glass bottle by its mass density:

Glass bottle volume = Glass bottle mass / Glass bottle mass density = 176.7 g / (2450 kg/m³ * 1000 g/kg) = 0.072 m³ or 72 cm³.

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The voltage at t = 22.0 ms is -12.39 V. The voltage across the resistor at t = 22.0 ms is -8.15 V. The voltage across the inductor at t = 22.0 ms is -11.31 V.

Resistor: R = 204 Ω

Inductor: L = 0.825 H

Capacitor: C = 7.00 μF

Voltage source: Vm = 29.0 V

Angular frequency: ω = 260 rad/s

Part A: The equation of the total voltage in a series RLC circuit is:

v(t) = Vm cos (ωt - Φ), where cos(ωt - Φ) is the voltage phasor.The voltage phasor is given by:Z = R + j (XL - XC)where XL = ωL is the inductive reactance, and XC = 1/ωC is the capacitive reactance. Here j = √(-1)

The phase angle of the circuit is given by:

tanΦ = (XL - XC) / RThe total voltage is:v(t) = Vm cos (ωt - Φ)

The current in the circuit is:

i(t) = (Vm / Z) cos (ωt - Φ)

Therefore, the voltage across the inductor is:

vL(t) = i(t) XL = (Vm / Z) XL cos (ωt - Φ)

Therefore, at t = 22.0 ms, the total voltage:

v(22 ms) = 29.0 cos (260 × 0.022 - 0.232) = - 12.39 V

Therefore, v = - 12.39 V

Part B: The voltage across the resistor is given by:

vR(t) = i(t) R

Therefore, at t = 22.0 ms, the voltage across the resistor:

vR(22 ms) = i(22 ms) R = (Vm / Z) R cos (ωt - Φ)vR(22 ms) = (29.0 / 388.93) 204 cos (260 × 0.022 - 0.232) = - 8.15 V

Therefore, vR = - 8.15 V

Part C: The voltage across the inductor is given by: vL(t) = i(t) XL

At t = 22.0 ms, the voltage across the inductor can be calculated as follows:

vL(22 ms) = i(22 ms) XL = (Vm / Z) XL cos (ωt - Φ)

vL(22 ms) = (29.0 / 388.93) (260 × 0.825) cos (260 × 0.022 - 0.232) = - 11.31 V

Therefore, the correct answer for Part C is vL = -11.31 V.

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According to the equation y = λD/d, the approximate width of the central bright fringe from a single slit diffraction will depend on both the wavelength of light used and the width of the slit itself.

Therefore, the correct option is option c. This means that the width of the central bright fringe will increase with increasing wavelength, as well as with increasing slit width.

The equation y = λD/d is used to calculate the position of the nth bright fringe in a single slit diffraction pattern, where y is the distance from the center of the pattern to the fringe, λ is the wavelength of light used, D is the distance between the slit and the screen, and d is the width of the slit.

As per the equation, the width of the central bright fringe (n = 0) is given by the formula y0 = λD/d. Therefore, it can be inferred that the width of the central bright fringe will increase as the wavelength of light used increases, as well as with an increase in the width of the slit.

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