At time t =0, the current to the dc motor is reversed, resulting in an angular displacement of the motor shaft given by è = (198 rad/s)t – (24 rad/s2)t2 – (2 rad/s3)t3 (15 marks)
a. At what time is the angular velocity of the motor shaft zero?
b. Calculate the angular acceleration at the instant that the motor shaft has zero angular velocity.
c. Through how many revolutions does the motor shaft turn between the time when the current is reversed and the instant when the angular velocity is zero?
d. How fast was the motor shaft rotating at t=0, when the current was reversed?
e. Calculate the average angular velocity for the time period from t = 0 to the time calculated in part (a).
To solve the problem, we need to follow a step-by-step approach. Let's go through each part of the question and explain how to find the answers.
a. At what time is the angular velocity of the motor shaft zero?
To find the time when the angular velocity of the motor shaft is zero, we need to find the time at which the derivative of the angular displacement equation is zero. The derivative of the equation would give us the angular velocity. So, we differentiate the given angular displacement equation with respect to time:
ω = dè/dt = (198 rad/s) - 2(24 rad/s^2)t - 3(2 rad/s^3)t^2
To find the time when the angular velocity is zero, we set ω = 0 and solve the equation:
0 = (198 rad/s) - 2(24 rad/s^2)t - 3(2 rad/s^3)t^2
We can solve this quadratic equation to find the value of t when the angular velocity is zero.
b. Calculate the angular acceleration at the instant that the motor shaft has zero angular velocity.
To calculate the angular acceleration at the instant when the motor shaft has zero angular velocity, we need to find the derivative of the angular velocity equation with respect to time. The angular acceleration is the second derivative of the angular displacement equation. So, we differentiate the angular velocity equation:
α = dω/dt = -2(24 rad/s^2) - 6(2 rad/s^3)t
We substitute the time t when the angular velocity is zero into the angular acceleration equation to find the value of α.
c. Through how many revolutions does the motor shaft turn between the time when the current is reversed and the instant when the angular velocity is zero?
To find the number of revolutions, we need to integrate the angular velocity equation from the time when the current is reversed to the time when the angular velocity is zero. The integral of angular velocity with respect to time gives us the total angular displacement.
Revolutions = ∫(ω dt)
We evaluate the integral over the given time interval to find the number of revolutions.
d. How fast was the motor shaft rotating at t=0 when the current was reversed?
To find the initial angular velocity when t = 0, we substitute t = 0 into the angular velocity equation:
ω(0) = (198 rad/s) - 2(24 rad/s^2)(0) - 3(2 rad/s^3)(0)^2
Solving this equation gives us the initial angular velocity when the current is reversed.
e. Calculate the average angular velocity for the time period from t = 0 to the time calculated in part (a).
To calculate the average angular velocity, we find the total change in angular displacement from t = 0 to the time calculated in part (a). Then we divide this change in angular displacement by the time interval.
Average angular velocity = Δè / Δt
We calculate the difference in angular displacement between the given time points and divide it by the time difference to find the average angular velocity.