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Inessa05 [86]
3 years ago
8

A 217 Ω resistor, a 0.875 H inductor, and a 6.75 μF capacitor are connected in series across a voltage source that has voltage a

mplitude 30.0 V and an angular frequency of 220 rad/s . What is v at t= 22.0 ms ? What is vR at t= 22.0 ms ? What is vL at t= 22.0 ms ? What is vC at t= 22.0 ms ? What is vR? What is vL? What is vC?
Physics
1 answer:
Nataly [62]3 years ago
4 0

For an AC circuit:

I = V/Z

V = AC source voltage, I = total AC current, Z = total impedance

Note: We will be dealing with impedances which take on complex values where j is the square root of -1. All phasor angles are given in radians.

For a resistor R, inductor L, and capacitor C, their impedances are given by:

Z_{R} = R

R = resistance

Z_{L} = jωL

ω = voltage source angular frequency, L = inductance

Z_{C} = -j/(ωC)

ω = voltage source angular frequency, C = capacitance

Given values:

R = 217Ω, L = 0.875H, C = 6.75×10⁻⁶F, ω = 220rad/s

Plug in and calculate the impedances:

Z_{R} = 217Ω

Z_{L} = j(220)(0.875) = j192.5Ω

Z_{C} = -j/(220×6.75×10⁻⁶) = -j673.4Ω

Add up the impedances to get the total impedance Z, then convert Z to polar form:

Z = Z_{R} + Z_{L} + Z_{C}

Z = 217 + j192.5 - j673.4

Z = (217-j480.9)Ω

Z = (527.6∠-1.147)Ω

Back to I = V/Z

Given values:

V = (30.0∠0+220t)V (assume 0 initial phase, and t = time)

Z = (527.6∠-1.147)Ω (from previous computation)

Plug in and solve for I:

I = (30.0∠0+220t)/(527.6∠-1.147)

I = (0.0569∠1.147+220t)A

To get the voltages of each individual component, we'll just multiply I and each of their impedances:

v_{R} = I×Z_{R}

v_{L} = I×Z_{L}

v_{C} = I×Z_{C}

Given values:

I = (0.0569∠1.147+220t)A

Z_{R} = 217Ω = (217∠0)Ω

Z_{L} = j192.5Ω = (192.5∠π/2)Ω

Z_{C} = -j673.4Ω = (673.4∠-π/2)Ω

Plug in and calculate each component's voltage:

v_{R} = (0.0569∠1.147+220t)(217∠0) = (12.35∠1.147+220t)V

v_{L} = (0.0569∠1.147+220t)(192.5∠π/2) = (10.95∠2.718+220t)V

v_{C} = (0.0569∠1.147+220t)(673.4∠-π/2) = (38.32∠-0.4238+220t)V

Now we have the total and individual voltages as functions of time:

V = (30.0∠0+220t)V

v_{R} = (12.35∠1.147+220t)V

v_{L} = (10.95∠2.718+220t)V

v_{C} = (38.32∠-0.4238+220t)V

Plug in t = 22.0×10⁻³s into these values and take the real component (amplitude multiplied by the cosine of the phase) to determine the real voltage values at this point in time:

V = 30.0cos(0+220(22.0×10⁻³)) = 3.82V

v_{R} = 12.35cos(1.147+220(22.0×10⁻³)) = 11.8V

v_{L} = 10.95cos(2.718+220(22.0×10⁻³)) = 3.19V

v_{C} = 38.32cos(-0.4238+220(22.0×10⁻³)) = -11.2V

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Block 1 (the rightmost block) has

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∑ <em>F</em> = <em>N₁</em> - <em>m₁g</em> = 0

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Block 2 (middle) has much the same information:

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The same goes for block 3 (leftmost):

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∑ <em>F</em> = <em>T₂</em> - <em>f₃</em> = <em>m₃a</em>

• net vert. force

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We have <em>m₁</em> = <em>m₂</em> = <em>m₃</em> = 0.8 kg, so I'll just replace each with <em>m</em>. It follows that each normal force has the same magnitude, <em>N₁</em> = <em>N₂</em> = <em>N₃</em> = <em>mg</em>. And as a consequence of that, each frictional force has the same magnitude, <em>f₁</em> = <em>f₂</em> = <em>f₃</em> = 0.4<em>mg.</em>

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[2] …<em>T₁</em> - <em>T₂</em> - 0.4<em>mg</em> = <em>ma</em>

[3] … <em>T₂</em> - 0.4<em>mg</em> = <em>ma</em>

<em />

Adding [1], [2] and [3] together eliminates the tension forces, and we get

57.3 N - 1.2<em>mg</em> = 3<em>ma</em>

<em />

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