A carriage with the mass M 20 kg has a vertical post attached. At the top end of

Question

A carriage with the mass M 20 kg has a vertical post attached. At the top end of the post (0.75m above the cart, point P in Figure 3) fasten a thin rope (rope and pole mass are neglected). IN the end of the line (with the length R 0.5 m) is one point mass attached (mass n kg). The cart and The point mass has the velocity V when they are collide with another carriage with the mass m 10 kg. After the collision continues the carts together as a body. Calculate minimum speed V required to The point mass n will complete a vertical circular motion around the point P (dashed path in Figure 3). Neglect friction and assume that m >> n

Explanation / Answer

We have from conservation of momentum for the 2 colliding carts:

MV=(M+m)V' where V' is the common velocity of the 2 carts after collision.

Then V'=MV/(M+m).

We also have for the motion of the center of mass of the first cart and the pendulum attached to it:

P=MV+nv=(M+m+n) V' where V' is the velocity of the center of mass of the system after collision and v is the velocity of the point mass at the bottom of its circular path.

On simplifying the above equation we have v=MV/((m+n)*n).

For the mass n to complete vertical motion around the point P we must have:

mv2/R>mg.

After substituting the above value of v in the above relation we will have the answer.

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