Question

A car is traveling along a road, and its engine is turning over with an angular velocity of +230 rad/s. The driver steps on the accelerator, and in a time of 8.0 s the angular velocity increases to +270 rad/s. (a) What would have been the angular displacement of the engine if its angular velocity had remained constant at the initial value of +230 rad/s during the entire 8.0-s interval? (b) What would have been the angular displacement if the angular velocity had been equal to its final value of +270 rad/s during the entire 8.0-s interval? (c) Determine the actual value of the angular displacement during the 8.0-s interval.

Answer 1

Solution :

Given :

Initial angular velocity (_i) = 230 rad/s

Final angular velocity (_f) = 270 rad/s

time interval (t) = 8 sec

Part (a) Solution :

If its angular velocity had remained constant at the initial value of +230 rad/s during the entire 8.0-s interval :

Angular displacement () = _i t = (230 rad/s)(8 s) = 1840 rad

Part (b) Solution :

If the angular velocity had been equal to its final value of +270 rad/s during the entire 8.0-s interval:

Angular displacement () = _f t = (270 rad/s)(8 s) = 2160 rad

Part (c) Solution :

Angular acceleration () = ( _f - _i) / t = {(270 rad/s) - (230 rad/s)} / (8 sec) = 5 rad/s²

So, Actual angular displacement () = _i t + (1/2) t²

Therefore : = (230 rad/s)(8 s) + (0.5)(5 rad/s²)(8 s)² = 2000 rad