Question

The angular velocity of a gear is controlled according to ω = 79 − 5t2 where ω, in radians per second, is positive in the clockwise sense and where t is the time in seconds. Find the net angular displacement Δθ (poisitive in the clockwise sense) from the time t = 0 to t = 4 s. Also find the total number of revolutions N through which the gear turns during the 4 seconds.

Answer 1

Answer:

$\theta=209.33\ rad=11993.92^{\circ}$ is the angular displacement in 4 seconds.

$N\approx33.32\ revolutions$

Explanation:

Given:

time of observation, $t=4\ s$

angular velocity as a function of time, $\omega=79-5t^2$

where:

$\omega=$ angular velocity

$t=$ time taken

Now for the angular displacement we integrate the expression of angular velocity:

$\theta=\int^{4}_{0} (79-5t^2)dt$

$\theta=[79t-\frac{5}{3} t^3]^{4}_{0}$

$\theta=209.33\ rad=11993.92^{\circ}$ is the angular displacement in 4 seconds.

Now the no. of revolutions made during this time;

$N=\frac{\theta^{\circ}}{360^{\circ}}$

$N=\frac{11993.92^{\circ}}{360^{\circ}}$

$N\approx33.32\ revolutions$