Question

A mass of 0.34 kg is fixed to the end of a 1.4 m long string that is fixed at the other end. Initially at rest, he mass is made to rotate around the fixed end with an angular acceleration of 3.31 rad/s. What centripetal force must act on the mass after 8 s so that it continues to move in a circular path

Answer 1

At time
`t` seconds, the mass has angular speed

`\omega = \left(3.31(\rm rad)/(\mathrm s^2)\right) t`

and hence linear speed

`v = (1.4\,\mathrm m) \omega = (1.4\,\mathrm m) \left(3.31(\rm rad)/(\mathrm s^2)\right) t`

After 8 s, its linear speed is

`v = (1.4\,\mathrm m) \left(3.31(\rm rad)/(\mathrm s^2)\right) (8\,\mathrm s) = 37.072 (\rm m)/(\rm s) \approx 37 (\rm m)/(\rm s)`

and has centripetal acceleration with magnitude

`a = (v^2)/(1.4\,\rm m) \approx 981.667(\rm m)/(\mathrm s^2) \approx 980 (\rm m)/(\mathrm s^2)`

To maintain this linear speed, by Newton's second law the required centripetal force should have magnitude

`F = (0.34\,\mathrm{kg}) a \approx 333.767\,\mathrm N \approx \boxed{330 \,\mathrm N}`