What is the velocity of a rod released inside a cart?

[Titan97]

Homework Statement

[/B]
A rod of mass m is attached on the ceiling inside a cart of mass 6m kept at rest. Length of rod is L. Find the velocity of rod with respect to cart when the rod becomes vertical after releasing it. The rod is free to rotate about its point of suspension.

Homework Equations

Conservation of energy and momentum

The Attempt at a Solution

With respect to earth, let velocity of center of mass of rod be ##v_1## and velocity of cart be ##v_2##.
Using COE,
$$mg\frac{l}{2}=\frac{1}{2}mv_1^2+\frac{1}{2}(6m)v_2^2$$
Since centre of mass remains at rest, ##mv_1=6mv_2##
$$V_1=\sqrt{\frac{7}{6}gl}$$
So ##v_{\text{rel}}=\frac{v}{6}+v##
But I am not getting the correct answer.

 

[TSny]

Is one end of the rod attached to the ceiling of the cart so that the rod is free to rotate about this end?
Did you account for all of the KE of the rod?

Also, the problem statement doesn't specify which point of the rod is used when finding the "velocity of rod with respect to cart".

 

[Titan97]

@TSny question edited. They have used concept of reduced mass to solve it. But wrong in my method?

 

[TSny]

You have left out some of the KE of the rod.

 

[Titan97]

Oh. Is it the rotational kinetic energy?

 

[TSny]

Oh. Is it the rotational kinetic energy?

Yes.

 

[conscience]

Hello , I get ##\frac{\sqrt{21gl}}{5}## as the answer .Is this what I should be getting ?

 

[TSny]

Hello , I get ##\frac{\sqrt{21gl}}{5}## as the answer .Is this what I should be getting ?

Yes, I believe that is the correct speed of the center of the rod relative to the cart when the rod is vertical.

 

[conscience]

Thanks