Question

An astronaut is being tested in a centrifuge. the centrifuge has a radius of 10 m and, in starting, rotates according to θ = 0.30t2 , where t in seconds gives θ in radians. when t = 5.0 s, what are the astronaut’s angular velocity?

Answer 1

The astronaut's angular velocity is 2.1 radians per second.

Step-by-step explanation:

To find the astronaut's angular velocity, we need to first calculate the angular displacement, which is the change in angle over time. The formula for angular displacement is given by θ = ω t + (1/2) α t^2, where θ is the angle, ω is the initial angular velocity, α is the angular acceleration, and t is the time.

In this case, we are given that the formula for θ is θ = 0.30t^2, so we can differentiate it twice with respect to time to find the angular acceleration, which is the second derivative of the angle function. Taking the derivative of θ with respect to t, we get ω = dθ/dt = 0.60t. Taking the derivative of ω with respect to t, we get α = dω/dt = 0.60.

Now we can substitute the values into the first formula to find the angular displacement. When t = 5.0 s, the angular displacement is given by θ = (0.60)(5.0) + (1/2)(0.60)(5.0)^2 = 3.0 + 7.5 = 10.5 radians.

The angular velocity is the rate at which the angle changes over time. So, we divide the angular displacement by the time taken: ω = θ/t = 10.5/5.0 = 2.1 radians per second.