a sphere rotates about an axis passing through its center. the angular velocity, \\(\\omega\\), of the…

a sphere rotates about an axis passing through its center. the angular velocity, \\(\\omega\\), of the sphere in rad/s is given by the following equations, where \\(t\\) is the time in seconds.\\(\\omega = 4, 0 \\leq t \\leq 1\\)\\(\\omega = \\frac{4}{t^2}, t > 1\\)what is the angular displacement of the sphere at time \\(t = 2\\) seconds?a 1 radianb 2 radiansc 4 radiansd 6 radians

a sphere rotates about an axis passing through its center. the angular velocity, \\(\\omega\\), of the sphere in rad/s is given by the following equations, where \\(t\\) is the time in seconds.\\(\\omega = 4, 0 \\leq t \\leq 1\\)\\(\\omega = \\frac{4}{t^2}, t > 1\\)what is the angular displacement of the sphere at time \\(t = 2\\) seconds?a 1 radianb 2 radiansc 4 radiansd 6 radians

Answer

Explanation:

Step1: Calculate displacement from ( t = 0 ) to ( t = 1 )

Angular velocity ( \omega = 4 ) for ( 0 \leq t \leq 1 ). Angular displacement ( \theta_1 ) is the integral of ( \omega ) with respect to ( t ) over this interval. Since ( \omega ) is constant, ( \theta_1=\omega\times(1 - 0)=4\times1 = 4 ) radians.

Step2: Calculate displacement from ( t = 1 ) to ( t = 2 )

For ( t>1 ), ( \omega=\frac{4}{t^2} ). The angular displacement ( \theta_2 ) from ( t = 1 ) to ( t = 2 ) is the integral of ( \frac{4}{t^2} ) with respect to ( t ) from 1 to 2. The integral of ( \frac{4}{t^2}=4t^{-2} ) is ( -4t^{-1}+C ). Evaluating from 1 to 2: ( \left(-\frac{4}{2}\right)-\left(-\frac{4}{1}\right)=-2 + 4=2 ) radians.

Step3: Total angular displacement

Total angular displacement ( \theta=\theta_1+\theta_2=4 + 2=6 ) radians.

Answer:

D. 6 radians