maths (integ)
voltage v=25sin50πt is applied on a circuit. using integration, find mean and RMS over range t=0 to 10ms .
i got it to 25/2πt^2 cos(2π/9)+c but im not sure that im going the righ way.
If you are first of all looking for the integral of
v=25sin50πt
it should be
-1/(2π) cos(50πt) + c
the rest is physics, about which I know very little .
from there the mean is usually found by ...
1/(10-0) ( -1/(2π) cos (500π - (-1/2π cos(0) )
= 1/10 (-1/(2π) (1)
= -1/(20π)
I am using "the mean of a function" definition, don't know if it applies to this physics problem.
To find the mean and RMS (root mean square) of the voltage applied on the circuit using integration, we first need to calculate the definite integral of the given function over the specified range.
The given function is: v = 25sin(50πt)
To find the mean, we need to calculate the average value of the function over the given range. The formula for calculating the mean is:
mean = (1/(b-a)) ∫[a to b] f(t) dt
In this case, a = 0 and b = 10ms (which is equal to 0.01 seconds).
mean = (1/(0.01 - 0)) ∫[0 to 0.01] 25sin(50πt) dt
Now, let's calculate the definite integral of the function:
∫ 25sin(50πt) dt
To integrate this function, we can use the substitution method. Let u = 50πt and du = 50π dt. Rearranging, we get dt = du/(50π).
Substituting the values in the integral:
∫ 25sin(u) (du/(50π))
= (25/(50π)) ∫ sin(u) du
= -(25/(50π)) cos(u)
= -(25/(50π)) cos(50πt)
Now, plugging in the limits of integration:
mean = -(25/(50π)) (cos(50π(0.01)) - cos(50π(0)))
= -(25/(50π)) (cos(π) - cos(0))
= -(25/(50π)) (-1 - 1)
= 2/π ≈ 0.6366
Therefore, the mean value of the voltage over the range t = 0 to 10ms is approximately 0.6366.
Next, let's calculate the RMS value of the voltage. The formula for calculating RMS is:
RMS = sqrt((1/(b-a)) ∫[a to b] (f(t))^2 dt)
In this case, the function is v = 25sin(50πt), so we need to calculate:
RMS = sqrt((1/(0.01 - 0)) ∫[0 to 0.01] (25sin(50πt))^2 dt.
Now, let's calculate the definite integral:
∫ (25sin(50πt))^2 dt
= ∫ (625sin^2(50πt)) dt
= (625/2) ∫ (1 - cos(100πt)) dt
= (625/2) (t - (1/(100π)) sin(100πt))
Plugging in the limits of integration:
RMS = sqrt((1/(0.01 - 0)) (625/2) [(0.01) - (1/(100π)) sin(100π(0.01)) - (0) + (1/(100π)) sin(100π(0))])
RMS = sqrt((625/2) (0.01 - 0))
RMS = sqrt((625/2) (0.01))
RMS = sqrt(3.125)
RMS ≈ 1.76777
Therefore, the RMS value of the voltage over the range t = 0 to 10ms is approximately 1.76777.
To find the mean and RMS (Root Mean Square) of a voltage function using integration, you need to follow these steps:
1. Calculate the integral of the squared voltage function over the given time range.
2. Take the square root of the result from step 1 to find the RMS value.
3. Calculate the integral of the voltage function itself over the given time range.
4. Divide the result from step 3 by the total time to find the mean value.
Let me walk you through the process step by step:
Step 1: Calculate the integral of the squared voltage function:
In our case, the squared voltage function is (25sin(50πt))^2 = 625sin^2(50πt).
To find the integral of sin^2(50πt), you can use the trigonometric identity sin^2(x) = (1 - cos(2x))/2.
∫[0,10ms] 625sin^2(50πt) dt = ∫[0,10ms] 625(1 - cos(100πt))/2 dt
= [625t/2 - 625/200πsin(100πt)] between 0 and 10ms
= (625(10ms)/2 - 625/200πsin(100π(10ms))) - (625(0)/2 - 625/200πsin(100π(0)))
= 3125ms - 625/200πsin(1000π) - 0
Simplifying the expression, we find:
∫[0,10ms] 625sin^2(50πt) dt = 3125ms - 625/200πsin(1000π)
Step 2: Calculate the square root of the integral from step 1 to find the RMS value:
RMS = √(3125ms - 625/200πsin(1000π))
Step 3: Calculate the integral of the voltage function itself:
In our case, the voltage function is v = 25sin(50πt).
∫[0,10ms] 25sin(50πt) dt = -(25/50π)cos(50πt) between 0 and 10ms
= -(25/50π)cos(500π) + (25/50π)cos(0)
= -(25/50π)cos(500π) + (25/50π)
Step 4: Divide the integral from step 3 by the total time (10ms) to find the mean value:
Mean = (-(25/50π)cos(500π) + (25/50π)) / 10ms
Simplifying the expression, we find:
Mean = (-(25/50π)cos(500π) + (25/50π)) / 10ms
So, the final expressions for the mean and RMS values of the voltage function v = 25sin(50πt) over the range t = 0 to 10ms are:
Mean = (-(25/50π)cos(500π) + (25/50π)) / 10ms
RMS = √(3125ms - 625/200πsin(1000π))
Please note that you should double-check the calculations and trigonometric identities used in the process to be confident in the accuracy of the results.