Question

What is the crankshaft's angular acceleration at t = 1 s?

Answer 1

The text of the problem is incomplete. Complete version:
"The following figure hows the angular velocity graph of the crankshaft in a car. (image: http://d2vlcm61l7u1fs.cloudfront.net/media%2F473%2F473f5b4d-b7d0-47de-bf54-0d19985ab4cf%2FphplfrYOY..... )

a) What is the crankshaft's angular acceleration at t=1s?

b) What is the crankshaft's angular acceleration at t=3s?

c) What is the crankshaft's angular acceleration at t=5s?

Solution

The angular acceleration is the derivative of the angular velocity with respect to time:

$\alpha = \frac{\Delta \omega}{\Delta t}$

This means that, in the graph $\Delta \omega/\Delta t$, the angular acceleraion is the slope of the curve. To solve the exercise we can simply calculate the slope of the curve in the three different points.

a) t=1 s

The initial angular velocity of this part of the motion is 50 rad/s, the final velocity is 250 rad/s, the final time is 2 s and the initial time is 0 s. The angular acceleration is

$\alpha = \frac{50 rad/s - 250 rad/s}{2 s-0 s}=-100 rad/s^{2}$

where the negative sign means it is a deceleration.

b) t= 3 s

The initial angular velocity of this part of the motion is 50 rad/s, the final velocity is still 50 rad/s, the final time is 4 s and the initial time is 2 s. The angular acceleration is

$\alpha = \frac{50 rad/s - 50 rad/s}{4 s-2 s}=0 rad/s^{2}$

The angular acceleration in this part is zero, because the angular velocity of the object is not changing.

c) t= 5 s

The initial angular velocity of this part of the motion is 50 rad/s, the final velocity is 250 rad/s, the final time is 7 s and the initial time is 4 s. The angular acceleration is

$\alpha = \frac{200 rad/s - 50 rad/s}{7 s-4 s}=50 rad/s^{2}$

where the positive sign means it is an acceleration.