Power given to particle by centripetal forces

[Ghost Repeater]

Homework Statement

A particle of mass m is moving in a circular path of constant radius r such that its centripetal acceleration a varies with time t as a = k²rt², where k is a constant. Show that the power delivered to the particle by the forces acting on it is mk⁴r²t⁵/3. [/B]

I have solved this problem, but I am still confused about it conceptually. If a particle's centripetal acceleration is changing with time, how can it continue to travel on a circular path of constant radius?

 

[jbriggs444]

If a particle's centripetal acceleration is changing with time, how can it continue to travel on a circular path of constant radius?

It can speed up and slow down. I see no language in the problem statement restricting its tangential acceleration.

Clearly, knowing the centripetal acceleration as a function of time gives tangential velocity as a function of time and, accordingly, the tangential acceleration as a function of time.

 

[gneill]

It may be moving along a circular track of some sort (think bead on a wire), or perhaps it's fixed to the end of a light rod that pivots about the center. Or, perhaps it is kept on trajectory by tiny rockets

 

[Ghost Repeater]

Ok, so centripetal and tangential vectors are independent of one another, analogous to the way that horizontal and vertical components of motion are independent in projectile problems?

But in that case how can a centripetal acceleration change the tangential velocity?

 

[jbriggs444]

Ok, so centripetal and tangential vectors are independent of one another, analogous to the way that horizontal and vertical components of motion are independent in projectile problems?

But in that case how can a centripetal acceleration change the tangential velocity?

There is no reason for it to need to. We are told how the centripetal acceleration changes with time. We are told that the object follows a circular path. We are not told that this causes the tangential acceleration. We merely infer the tangential acceleration because it is required to make the givens of the problem possible.

 

[CWatters]

+1

An example would be the pendulum in a clock. As it swings back and forth the velocity and centripetal acceleration changes but the radius remains constant.