A body is thrown vertically up with a certain initial velocity. The potential and the kinetic energy of the body are equal at a point P in its path. If the same body is thrown with double the velocity upwards, the ratio of the potential and the kinetic energies of the body when it crosses at the same point will be: 

1. 1:1

2. 1:4

3. 1:7

4. 1:8

A body is displaced from (0,0) to (1m,1m) along the path x=y by a force $F=\left(x^2\hat{j}+y\hat{i}\right)N$. The work done by this force will be :

1. $\frac{4}{3}J$

2. $\frac{5}{6}J$

3. $\frac{3}{2}J$

4. $\frac{7}{5}J$

In the figure shown, the potential energy U of a particle is plotted against its position 'x' from the origin. Then which of the following statement is correct?

1. ${\mathrm{x}}_1$ is in stable equilibrium

2. ${\mathrm{x}}_2$ is in stable equilibrium

3. ${\mathrm{x}}_3$ is in stable equilibrium

4. none of these

A weightless rod of length 2l carries two equal mass 'm', one tied at lower end A and the other at the middle of the rod at B. The rod can rotate in a vertical plane about a fixed horizontal axis passing through C. The rod is released from rest in the horizontal position. The speed of the mass B at the instant rod becomes vertical is:

1. \(\sqrt{\frac{3 g l}{5}} \)

2. \(\sqrt{\frac{4 g l}{5}} \)

3. \(\sqrt{\frac{6 g l}{5}} \)

4. \(\sqrt{\frac{7 g l}{5}} \)

Potential energy \((U)\) related to coordinates is given by; \(U=3(x+y).\) Work done by the conservative force when the particle is going from \((0,0), (2,3)\) is:
1. \(15\) J
2. \(-15\) J
3. \(12\) J
4. \(10\) J

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:

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

A ball is suspended by a thread of length l. What minimum horizontal velocity has to be imparted to the ball for it to reach the height of the suspension:

1. gl

2. 2 gl

3. $\sqrt{gl}$

4. $\sqrt{2gl}$

A body of mass m hangs at one end of a string of length l, the other end of which is fixed. It is given a horizontal velocity so that the string would just reach where it makes an angle of 60° with the vertical. The tension in the string at mean position is 

1. 2 mg

2. mg

3. 3 mg

4. $\sqrt{3}mg$

The kinetic energy k of a particle moving along a circle of radius R depends on the distance covered s as k = as² where a is a constant. The force acting on the particle is 

(1) $2a\frac{s^2}{R}$

(2) $2as{\left(1+\frac{{\displaystyle s^2}}{{\displaystyle R^2}}\right)}^{1/2}$

(3) 2 as

(4) $2a\frac{R^2}{s}$