Question

The angular velocity of a process control motor is (16 - 1/2t²) rad/s, where t is in seconds, through what angle does the motor turn between t= 0s and the instant at which it reverses its direction?

Answer 1

To find the total angle turned by the process control motor, we must first determine the time when the angular velocity equals zero by solving the equation (16 - ½t²) = 0. Then, we integrate the angular velocity over the time interval from 0 to the time of reversal to find the angular displacement.

Step-by-step explanation:

Calculating the Angle Turned by a Process Control Motor

To determine through what angle the process control motor turns between t=0s and the instant it reverses direction, we need to figure out the time at which the angular velocity becomes zero. The angular velocity of the motor is given by the function ω(t) = (16 - ½t²) rad/s. We first find the time when ω(t) = 0, by setting the angular velocity equation to zero and solving for t.

Setting ω(t) = 0 gives us 0 = (16 - ½t²), which upon solving gives t = √(32). Then, we calculate the angle turned Δθ by integrating the angular velocity function ω(t) with respect to time from t=0 to t=√(32). The integral of ω(t) gives the angular displacement.

Therefore, we have Δθ = ∫ ω(t) dt = ∫ (16 - ½t²) dt from 0 to √(32). We can calculate this integral to get the total angle through which the motor turns before it comes to a stop and reverses direction.