A mass m slips along the wall of a semispherical surface of radius R. The velocity at the bottom of the surface is [ MP PMT 1993]

(1) $\sqrt{Rg}$

(2) $\sqrt{2Rg}$

(3) $2\sqrt{\pi Rg}$

(4) $\sqrt{\pi Rg}$

Three different objects of mass $m_1,$ $m_2$ and m₃ are allowed to fall from rest and from the same point ‘O’ along three different frictionless paths. The speeds of the three objects, on reaching the ground, will be in the ratio of:

1. $m_1:m_2:m_3$

2. $m_1:2m_2:3m_3$

3. 1 : 1 : 1

4. $\frac{1}{m_1}:\frac{1}{m_2}:\frac{1}{m_3}$ 

When a body moves with a constant speed along a circle 

(1) No work is done on it

(2) No acceleration is produced in the body

(3) No force acts on the body

(4) Its velocity remains constant

A sphere of mass m is tied to end of a string of length l and rotated through the other end along a horizontal circular path with speed v. The work done by centripetal force in full horizontal circle is 

(1) 0

(2) $\left(\frac{{\displaystyle m{v}^2}}{{\displaystyle l}}\right)\hspace{0.17em}.\hspace{0.17em}2\pi l$

(3) $\text{mg\hspace{0.17em}.\hspace{0.17em}2πl}$

(4) $\left(\frac{{\displaystyle m{v}^2}}{{\displaystyle l}}\right)\hspace{0.17em}.\hspace{0.17em}\left(l\right)$

In a circus stuntman rides a motorbike in a circular track of radius R in the vertical plane. The minimum speed at highest point of track will be 

(1) $\sqrt{2gR}$

(2) 2gR

(3) $\sqrt{3gR}$

(4) $\sqrt{gR}$

A stone of mass 1 kg tied to a light inextensible string of length $L=\frac{10}{3}m$ is whirling in a circular path of radius L in a vertical plane. If the ratio of the maximum tension in the string to the minimum tension in the string is 4 and if g is taken to be 10 m/sec², the speed of the stone at the highest point of the circle is 

(1) 20 m/sec

(2) $10\sqrt{3}m/\sec$

(3) $5\sqrt{2}m/\sec$

(4) 10 m/sec

A stone tied to a string of length L is whirled in a vertical circle with the other end of the string at the centre. At a certain instant of time, the stone is at its lowest position and has a speed u. The magnitude of the change in its velocity as it reaches a position where the string is horizontal is:

(1) $\sqrt{u^2-2gL}$

(2) $\sqrt{2gL}$

(3) $\sqrt{u^2-gl}$

(4) $\sqrt{2(u^2-gL)}$

A 100 g iron ball having velocity 10 m/s collides with a wall at an angle 30° and rebounds with the same angle. If the period of contact between the ball and wall is 0.1 second, then the force experienced by the wall is 

(1) 10 N

(2) 100 N

(3) 1.0 N

(4) 0.1 N

A man fires a bullet of mass 200 g at a speed of 5 m/s. The gun is of one kg mass. By what velocity the gun rebounds backwards ?

(1) 0.1 m/s

(2) 10 m/s

(3) 1 m/s

(4) 0.01 m/s

A body of mass M at rest explodes into three pieces, two of which of mass M/4 each are thrown off in perpendicular directions with velocities of 3 m/s and 4 m/s respectively. The third piece will be thrown off with a velocity of 

(1) 1.5 m/s

(2) 2.0 m/s

(3) 2.5 m/s

(4) 3.0 m/s