Power Question: Power Calculation for Particle Moving in Circular Path

[prateek_34gem]

Power Question !Need Help

Homework Statement

A particle of mass 'm' is moving in a Circular path of constant radius 'r' such that it's centripetal acceleration is given by a_c= k²rt² where k is constant. The power delivered to the particle by the force acting on it is :-

A) 2(pi)mk²r²t
B) mk²r²t
C) (mk⁴r²t⁵)/3
D) zero

Homework Equations

The Attempt at a Solution

Centripetal force will be f=ma_c so f=mk²rt².

Now, Centripetal acceleration = v²/r.
so v= krt
now,
power = force*distance/time _{for which work is done}
(mk²rt²*vt)/t

from this i am getting power as mk³r²t³.

 

[Doc Al]

What direction does the force act? What direction is the displacement of the particle?

 

[prateek_34gem]

Force is centripetal so i think it is acting towards centre. And it is moving in circular track.
Do i have to take the component of force acting along displacement.

 

[prateek_34gem]

sorry i think force is acting away from centre

 

[Doc Al]

Force is centripetal so i think it is acting towards centre.

Good!

And it is moving in circular track.
Do i have to take the component of force acting along displacement.

Absolutely, in order to find the work done on the particle by that force.

sorry i think force is acting away from centre

No, you were right the first time.

 

[Redbelly98]

Now, Centripetal acceleration = v²/r.
so v= krt

Yes.

now,
power = force*distance/time _{for which work is done}
(mk²rt²*vt)/t

As Doc Al said, you need to use the force component along the particle's displacement.

Is there another equation for power that you are familiar with?

 

[prateek_34gem]

F.scosX/t ??

 

[prateek_34gem]

ohk , I got the answer.
Here is how i got it :

Let us assume that the particle moves a quarter. so its displacement will be
2^1/2r.
Now since the force acts in the direction towards centre.
so taking the component of displacement along force :
2^1/2rcos45.
now using the same formula:
Power = Force*displacement /time

(mk²r²t²*2^1/2rcos45)/t
=mk²r²t²*r/t
=mk²r²t

so, (B) is correct option.

Thank you guys.
PF rocks!

 

[Doc Al]

I just realized that I misread the question.

The power delivered to the particle by the force acting on it is

I was thinking that it said "Find the power delivered by the centripetal force", which is an entirely different question. The force acting on this particle is not simply centripetal.

Nonetheless, as already stated: Find the component of the force in the direction of the particle's velocity. Or find the energy of the particle as a function of time.