Question

1. A turntable is initially rotating a 1.50 rad/s and accelerates at 2.50 rad/s for 5.0s.
a) Calculate the final angular velocity in rad/s
b) Compute the angular displacement during the 5.0 s in radians and revolutions

Answer 1

Answer on Question 49077, Physics, Mechanics | Kinematics | Dynamics

Question:

1. A turntable is initially rotating at 1.50 rad/s and accelerates at 2.50 rad/s for 5.0s.

a) Calculate the final angular velocity in rad/s

b) Compute the angular displacement during the 5.0 s in radians and revolutions

Solution:

a) By the definition of the angular acceleration we have:

$$\alpha = \frac {\Delta \omega}{\Delta t} = \frac {\left(\omega_ {f} - \omega_ {i}\right)}{t},$$

where $\omega_{i}$ is the initial angular velocity, $\omega_{f}$ is the final angular velocity and $t$ is the time. So, from the formula for angular acceleration we can find the final angular velocity:

$$\omega_ {f} = \omega_ {i} + \alpha t = 1.50 \frac {\mathrm{rad}}{\mathrm{s}} + 2.50 \frac {\mathrm{rad}}{\mathrm{s} ^ {2}} \cdot 5.0 \mathrm{s} = 14 \frac {\mathrm{rad}}{\mathrm{s}}.$$

b) By the definition of the angular displacement we have:

$$\theta = \frac {1}{2} \alpha t ^ {2} + \omega_ {i} t = 0.5 \cdot 2.50 \frac {\mathrm{rad}}{\mathrm{s} ^ {2}} \cdot (5.0 \mathrm{s}) ^ {2} + 1.50 \frac {\mathrm{rad}}{\mathrm{s}} \cdot 5.0 \mathrm{s} = 38.75 \mathrm{rad} = \frac {38.75 \mathrm{rad}}{2 \pi} = 6.2 \mathrm{rev}.$$

Answer:

a) The final angular velocity is $\omega_{f} = 14\frac{\mathrm{rad}}{\mathrm{s}}$.

b) The angular displacement is $\theta = 38.75\mathrm{rad}$ or $\theta = 6.2\mathrm{rev}$.

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