Friction and Vertical Circular Motion Problems

Q.

Three identical cars A, B and C are moving at the same speed on three bridges. The car A goes on a plane bridge, B on a bridge convex upward and C goes on a bridge concave upward. Let $F_A,F_{B}and{F}_C$ be the normal forces exerted by the cars on the bridges when they are at the middle of bridges.

1. $F_A$ is maximum of the three forces.

2. $F_B$ is maximum of the three forces.

3. $F_C$ is maximum of the three forces.

4. $F_A$ = $F_B$ = $F_C$

Q. A small spherical ball is suspended through a string of length $l$. The whole arrangement is placed in a vehicle which is moving with velocity $v$. Now, suddenly the vehicle stops and ball starts moving along a circular path. If tension in the string at the highest point is twice the weight of the ball then (assume that the ball completes the vertical circle)

1. $v=\sqrt{5gl}$
2. $v=\sqrt{7gl}$
3. velocity of the ball at highest point is $\sqrt{gl}$
4. velocity of the ball at the highest point is $\sqrt{3gl}$

Q. A particle of mass $m$ begins to slide down a fixed smooth sphere from the top as shown. What is its acceleration when it breaks off the sphere?

1. $2g/3$
2. $\sqrt{5}g/3$
3. $g$
4. $g/3$

Q. A body is allowed to slide on a frictionless track from rest position under gravity. The track ends into a circular loop of diameter $D$. What should be the minimum height of the body in terms of $D$ so that it may complete the loop successfully?

1. $\frac{4}{5}D$
2. $\frac{5}{4}D$
3. $1D$
4. $2D$