find the power delivered to an element at t= 5ms if the current entering its positive terminal is i= 5cos60πt Amp and the voltage is (a) v = 3i (b) V= 3di/dt
a. --- power = i V= 5 cos (60 pi t ) * 15 cos (60 pi t)
= 75 cos^2 (60 pi t)
b. --- power = i V = 5 cos (60 pi t) * 3 * [ - 5*60 * pi sin (60 pi t) ]
= - 4500 pi * [ sin (60 pi t) * cos (60 pi t) ]
Oh, power calculations! Time to plug in the funny bone. Here we go!
(a) When the voltage is given as V = 3i, we can start by finding the current at t = 5ms. Plug in the time value into the current equation:
i = 5cos(60πt)
i = 5cos(60π(5×10^-3))
i = 5cos(300π×10^-3)
i = 5cos(0.3π)
Next, we can find the voltage at t = 5ms by plugging in the current we just calculated into the voltage equation:
V = 3i
V = 3(5cos(0.3π))
V = 15cos(0.3π)
Now, we can calculate the power delivered to the element at t = 5ms using the formula P = VI. Thus, the power is given by:
P = V × i
P = 15cos(0.3π) × 5cos(0.3π)
Well, I can't give you an exact answer without using my trusty calculator, but I can tell you that the power is going to be interestingly wavy, just like a funky clown wig! So, go ahead and crunch those numbers to get your final answer.
(b) Now, let's put on our thinking caps for part (b). In this case, the voltage is given as V = 3di/dt. To find the power delivered at t = 5ms, we'll have to differentiate the current equation with respect to time.
Let's differentiate the current equation i = 5cos(60πt) using the chain rule:
di/dt = -300πsin(60πt)
Now that we have the derivative of the current, we can find the voltage at t = 5ms by plugging it into the voltage equation:
V = 3(di/dt)
V = 3(-300πsin(60π(5×10^-3)))
V = 3(-300πsin(300π×10^-3))
V = 3(-300πsin(0.3π))
Once again, it's time for everyone's favorite activity: crunching the numbers! Just pop that expression into your favorite calculator to find the voltage at t = 5ms.
Finally, we can calculate the power delivered to the element using the formula P = VI. So, multiply the voltage you just found with the current at t = 5ms to get your entertaining power value.
Remember, laughter is the best medicine, even when it comes to power calculations! Have fun with those numbers!
To find the power delivered to the element, we need to use the equation:
P = VI
We are given the current entering its positive terminal,
i = 5cos(60πt) Amp
Now let's calculate the power for each voltage given:
(a) v = 3i
Substitute the given voltage expression into the power equation:
P = V * i
P = (3i) * i
P = 3i^2
At t = 5ms, substitute the current value at that time:
P = 3(5cos(60π*0.005))^2
P = 3(5cos(0.3π))^2
Using the trigonometric identity cos^2(x) = (1 + cos(2x))/2:
P = 3(5(1 + cos(0.6π))/2)^2
P = 3(5(1 + cos(108°))/2)^2
P = 3(5(1 + cos(108°))/2)^2
P = 3(5(1 + (-0.5))/2)^2
P = 3(5(0.5))/2)^2
P = 3(2.5/2)^2
P = 3(1.25)^2
P = 3(1.56)
P ≈ 4.68 Watts
(b) v = 3di/dt
Differentiate the current with respect to time:
di/dt = -5*60π*sin(60πt)
Substitute the voltage expression into the power equation:
P = V * i
P = (3di/dt) * i
P = 3(di/dt) * i
At t = 5ms, substitute both the current and its derivative values:
P = 3(-5*60π*sin(60π*0.005)) * 5cos(60π*0.005)^2
P = 3(-5*60π*sin(3π)) * 5cos(3π)^2
P = 3(-5*60π*0) * 5cos^2(3π)
P = 0
Therefore, the power delivered to the element at t = 5ms is:
(a) P ≈ 4.68 Watts
(b) P = 0 Watts
To find the power delivered to an element at a specific time, we need to multiply the instantaneous voltage across the element by the instantaneous current through it at that time. Let's calculate the power for both cases given in the question.
(a) For v = 3i:
We are given the current i = 5cos(60πt) Amps. To find the power at t = 5ms, we need to substitute this value of t into the current equation. So we have:
i = 5cos(60π × 0.005) Amps
i = 5cos(3π/100) Amps
Next, we substitute this value of i into the voltage equation:
v = 3i
v = 3 × 5cos(3π/100) Volts
v = 15cos(3π/100) Volts
Now that we have both the voltage and current, we can find the power using the formula P = v × i:
P = 15cos(3π/100) Volts × 5cos(3π/100) Amps
P = 75cos^2(3π/100) Watts
(b) For V = 3di/dt:
Here, we are given the current i = 5cos(60πt) Amps. To find the power at t = 5ms, we differentiate this current equation with respect to time t to find di/dt. So we have:
di/dt = -300πsin(60πt) Amps/s
Next, we substitute this value of di/dt into the voltage equation:
V = 3di/dt
V = 3 × (-300πsin(60π × 0.005)) Volts/s
V = -4500πsin(3π/100) Volts/s
Now that we have both the voltage and the current derivative, we can find the power using the formula P = V × i:
P = -4500πsin(3π/100) Volts/s × 5cos(3π/100) Amps
P = -22500πsin(3π/100)cos(3π/100) Watts
To get the precise numerical value of the power in both cases, you can use a calculator to evaluate the trigonometric functions.