A smooth ring of mass m can slide on a fixed horizontal rod.A string tied to the ring passes over a fixed pulley and carries a mass M(<m).At an instant the angle between the rod and the string is θ.Show that if the ring slides with a speed v,the block descends with a speed v cosθ.With what acceleration will the ring start to move if the system is released from rest at θ=30*?

I take z axis downwards,x axis rightwards.

The force equations:T cosθ=m D²x
Mg-T=M D²z

Now,we are to find the constraint equation.

I got this:(using the length conservation)

√[x²+c²]+z=L...............c is a const
Differentiating twice w.r.t. t we get: Dz=-Dx (cosθ)


What is annoying me is the (-)ve sign before the ansswer.

Somehow this is not correct.Because,in the next part we require this result:

D²z=-D²x cosθ+0(initially Dx=0)

This gives an error in the final answer.

Can anyone please help?

however,i got it.I did a mistake:


T cosθ=m D²x

in my sign convention,it should be -D²x

Hi neelakash,
I'm glad you sorted it out. Keeping track of axis directions can be a right pain at times.
Just one thing:
I take z axis downwards,x axis rightwards.
"x-axis rightwards" doesn't tell us anything unless we know whether the ring is to the left or the right of the pulley.
This is a mistake I often make. It's just so easy to assume that people picture the apparatus with the same orientation as you did.
The best way to avoid it is with a labelled diagram, of course, but that's hard to do on an online discussion forum (maybe an add-on to vBulletin... anyone?)

May be ...I do not know actually.
Some places have optiob to upload images.But here we cannot....
ring is to the left of the pullet.