Converting angual velocity to linear velocity

[crybllrd]

Homework Statement

An astronaut is being tested in a centrifuge. The centrifuge has a radius of 10m and, in starting, rotates $$\theta$$=0.30t$$^{2}$$, there t is in seconds and $$\theta$$ in radians. When t=5.0s, what are the magnitudes of the astronaut's (a) angular velocity, (b) linear velocity, (c) tangential acceleration, and (d) radial acceleration?

Homework Equations

Kinematics equations

The Attempt at a Solution

For (a), I used $$\theta$$=.30(5)$$^{2}$$=7.5rad, plugging in 5.0s for t. Then knowing initial angular velocity = 0, time = 5s, and now $$\theta$$=7.5rad, I used a kinematics equation to determine final angular velocity to be 3rad/s.

Part (b) is where I'm stuck. I was thinking to convert 3 rad/s to degrees, then divide by 360 degrees to get 4.7 rotations a second, but not sure where to go from there (or if I'm on the right track).

 

[rock.freak667]

Well you can formulate the expression for v using v=rω and you know that ω=dθ/dt.

 

[crybllrd]

Thanks for the fast reply.
Where does that leave me in units? Seems like it would be (rad*m)/s. Is there a way I can divide out the radians?

 

[rock.freak667]

Thanks for the fast reply.
Where does that leave me in units? Seems like it would be (rad*m)/s. Is there a way I can divide out the radians?

a radian is the ratio of arc length to radius so essentially, you can leave out the radian with no consequence.

 

[rwisz]

To convert angular velocity to linear velocity, we can use the formula v = rω, where v is the linear velocity, r is the radius, and ω is the angular velocity. In this case, r = 10m and ω = 3 rad/s, so the linear velocity would be 30 m/s. This means that the astronaut is moving at a speed of 30 m/s along the circumference of the centrifuge.

For (c) and (d), we can use the formulas a_t = rα and a_r = ω^2r, where a_t is the tangential acceleration, a_r is the radial acceleration, and α is the angular acceleration. Plugging in the given values, we get a_t = 10 m/s^2 and a_r = 90 m/s^2.

Overall, the astronaut is experiencing a tangential acceleration of 10 m/s^2 and a radial acceleration of 90 m/s^2 while in the centrifuge. These values are important for understanding the forces acting on the astronaut's body and the effects of the centrifugal force.