Angular Velocity of a centrifuge

[BJN153]

An astronaut is being tested in a centrifuge. The centrifuge has a radius of 10.5 m and, in starting, rotates according to Θ(theta) = 0.13t^2, where t in seconds gives Θ in radians. When t = 5.0 s, what are the astronaut's angular velocity?
What is the astronauts linear speed?
What is the astronauts tangential acceleration (magnitude only)?
What is the astronauts radial acceleration (magnitude only)?

 

[whozum]

By deriving:

$$\theta(t) = .13t^2$$ <-- Position Function
$$\omega(t) = .26t$$ <-- Velocity Function
$$\alpha(t) = .26$$ <-- Acceleration Function

Find the angular accelerations and speeds, then think of a relationship between the angular accel/speed and linear accel/speed using the given informaiton.

 

[charng]

I would like to first clarify that angular velocity is defined as the rate of change of angular displacement with respect to time. It is measured in radians per second (rad/s).

Using the given equation for Θ, we can find the angular displacement of the centrifuge at t = 5.0 s by substituting the value in the equation:

Θ = 0.13(5.0)^2 = 3.25 radians

Therefore, at t = 5.0 s, the astronaut's angular velocity can be calculated as the derivative of the angular displacement with respect to time:

ω = dΘ/dt = 0.13(2)(5.0) = 1.3 rad/s

The astronaut's linear speed can be calculated using the formula v = ωr, where r is the radius of the centrifuge.

v = (1.3 rad/s)(10.5 m) = 13.65 m/s

The astronaut's tangential acceleration can be calculated using the formula a = ω^2r.

a = (1.3 rad/s)^2(10.5 m) = 17.01 m/s^2

The astronaut's radial acceleration can be calculated using the formula a = ω^2r, with the direction being towards the center of the centrifuge.

a = (1.3 rad/s)^2(10.5 m) = 17.01 m/s^2

In conclusion, at t = 5.0 s, the astronaut's angular velocity is 1.3 rad/s, linear speed is 13.65 m/s, tangential acceleration is 17.01 m/s^2, and radial acceleration is 17.01 m/s^2 towards the center of the centrifuge. These values are important in understanding the effects of centrifugal force on the astronaut's body during the test.