Question

The angular velocity of an object is given by the following equation: ω(t)=(5rads3)t2\omega\left(t\right)=\left(5\frac{rad}{s^3}\right)t^2ω(t)=(5s3rad​)t2 What is the angular displacement of the object (in rad) between t = 2 s and t = 4 s?

Answer 1

Answer:

The angular displacement of the object between $t = 2\,s$ and $t = 4\,s$ is 20 radians.

Explanation:

The angular velocity of the object ($\omega$), in radians per second, is given by the following expression:

$\omega(t) = 5\cdot t^{2}$ (1)

Where $t$ is the time, measured in seconds.

The change in the angular displacement ($\Delta \theta$), in radians, is found by means of the following definite integral:

$\Delta \theta = \int\limits^{4}_{2} {5\cdot t^{2}} \, dt$ (2)

Then we proceed to integrate on the function in time:

$\Delta \theta = \frac{5}{3}\cdot (4^{2}-2^{2})$

$\Delta \theta = 20\,rad$

The angular displacement of the object between $t = 2\,s$ and $t = 4\,s$ is 20 radians.