Question

A carousel is initially at rest, Ron is seating 3.0 m from the center. At an initial t=0, the carousel is given a constant angular acceleration of a=0.065 rad /s². At t=10.0 s, find the following quantities: (a) angular velocity of the carousel, (b) linear velocity of Ron, (c) his tangential acceleration, (d) his centripetal acceleration, and (e) his total acceleration.

Answer 1

(a) The angular velocity of the carousel can be found as follows:

$$\omega=\omega_0+\alpha t=0+0.065\ \dfrac{rad}{s^2}\times10\ s=0.65\ \dfrac{rad}{s}.$$

(b) Linear velocity of Ron can be found as follows:

$$v=\omega r=0.65\ \dfrac{rad}{s}\times3.0\ m=1.95\ \dfrac{m}{s}.$$

(c) Tangential acceleration of Ron can be found as follows:

$$a_t=\alpha r=0.065\ \dfrac{rad}{s^2}\times3.0\ m=0.195\ \dfrac{m}{s^2}.$$

(d) Centripetal acceleration of Ron can be found as follows:

$$a_c=\dfrac{v^2}{r}=\dfrac{(1.95\ \dfrac{m}{s})^2}{3.0}=1.27\ \dfrac{m}{s^2}.$$

(e) Total acceleration of Ron can be found as follows:

$$a=\sqrt{a_t^2+a_c^2}=\sqrt{(0.195\ \dfrac{m}{s^2})^2+(1.27\ \dfrac{m}{s^2})^2}=1.28\ \dfrac{m}{s^2}.$$