find the mean and RMS of v=25sin50πt
over the range t=0 to t=20ms using integration.
i have integrated to: v=-cos(50πt)/2π+c
but i don't know how to get the mean and rms.
rms = root of the mean of the square.
v = 25 sin 50πt
we want the square:
v^2 = 625 sin^2(50πt)
the mean is ∫[0,20] v^2 dv / 20
= 1/20 ∫[0,20] 625 sin^2(50πt) dt
= 312.5
so, the root mean square is
17.68
the mean is just
1/20 ∫[0,20] 25 sin(50πt) dt = 0
To find the mean and RMS of the given function v = 25sin(50πt) over the range t = 0 to t = 20ms using integration, you need to follow these steps:
1. Find the definite integral of the function v with respect to time (t) over the given range (0 to 20ms). Let's denote this result as V(t):
V(t) = ∫(v) dt
2. Evaluate the definite integral V(t) using the antiderivative you have found:
V(t) = ∫[-cos(50πt)/(2π)] dt
3. Solve for the upper and lower limits of integration:
V(20ms) - V(0)
4. Calculate the mean (average value) of the function by dividing the result of step 3 by the range of integration (20ms):
Mean = [V(20ms) - V(0)] / (20ms)
5. Calculate the RMS (root mean square) value of the function using the formula:
RMS = sqrt([V(20ms) - V(0)] / (20ms))
Here's a step-by-step breakdown of the calculation:
1. Integrate v = -cos(50πt)/(2π) with respect to t to find V(t):
V(t) = [-sin(50πt)/(100π^2)] + C
2. Evaluate the definite integral V(t) at t = 20ms and t = 0:
V(20ms) = [-sin(50π(20ms))/(100π^2)] + C
V(0) = [-sin(50π(0))/(100π^2)] + C
3. Subtract the values of V(0) from V(20ms):
[V(20ms) - V(0)] = [-sin(50π(20ms))/(100π^2)] - [-sin(50π(0))/(100π^2)]
4. Divide the result of step 3 by the range of integration (20ms) to find the mean:
Mean = [-sin(50π(20ms))/(100π^2) + sin(50π(0))/(100π^2)] / (20ms)
5. Calculate the RMS using the formula:
RMS = sqrt([-sin(50π(20ms))/(100π^2) + sin(50π(0))/(100π^2)] / (20ms))
Now you can evaluate these equations to find the mean and RMS of the given function over the specified range.