At t=0 the current to a dc electric motor is reversed, resulting in an angular displacement of the motor shaft given by
theta(t)= 258t - 20.4t^2 - 1.47t^3
How many revolutions does the motor shaft turn through between the time when the current is reversed and the instant when the angular velocity is zero?
I figured it out! I wasn't putting the t at omega=0 in the first equation in properly before converting from radians to revolutions!
To find the number of revolutions the motor shaft turns through between the time when the current is reversed and the instant when the angular velocity is zero, we need to calculate the integral of the angular velocity with respect to time and convert the result to revolutions.
The angular velocity is the derivative of the angular displacement:
ω(t) = dθ(t)/dt
To find the time when the angular velocity is zero, we need to solve the equation:
ω(t) = 0
Differentiating the expression for angular displacement, we can find the angular velocity:
ω(t) = d/dt (258t - 20.4t^2 - 1.47t^3)
= 258 - 40.8t - 4.41t^2
Setting the angular velocity to zero:
258 - 40.8t - 4.41t^2 = 0
This is a quadratic equation that we can solve to find the values of t when the angular velocity is zero.
Once we have these values of t, we can integrate the angular velocity from the time when the current is reversed (t=0) to the time when the angular velocity is zero, to find the total angular displacement:
∫[0 to t0] (258 - 40.8t - 4.41t^2) dt
Where t0 is the time when the angular velocity is zero.
Finally, to convert the angular displacement to revolutions, we need to divide it by 2π, since one revolution corresponds to an angle of 2π radians.
So, the steps to find the number of revolutions the motor shaft turns through are:
1. Solve the equation 258 - 40.8t - 4.41t^2 = 0 to find the values of t when the angular velocity is zero.
2. Evaluate the integral ∫[0 to t0] (258 - 40.8t - 4.41t^2) dt to find the total angular displacement.
3. Divide the total angular displacement by 2π to convert it to revolutions.