The potential energy of a particle of mass \(1\) kg free to move along the \(X\text-\)axis is given by \(U(x) = (3x^2-4x+6)~\text{J}\). The force acting on the particle at \(x=0\) will be:
1. \(2\hat i~\text{N}\)
2. \(-4\hat i~\text{N}\)
3. \(5\hat i~\text{N}\)
4. \(4\hat i~\text{N}\)

The potential energy of a particle of mass m varies as $U=a{x}^2+by.$ The magnitude of the acceleration of the particle at (0, 3) is (symbols have their usual meaning)

1.  $\sqrt{\frac{\mathrm{b}}{\mathrm{m}}}$

2.  $\sqrt{\frac{3\mathrm{b}}{\mathrm{m}}}$

3.  $\frac{\mathrm{b}}{\mathrm{m}}$

4.  Zero

A ball of mass 100 g at rest is thrown on a horizontal surface as shown in the figure. The acceleration-displacement graph for the motion is given. The maximum height attained by the ball on the smooth inclined plane is

(1)  10 m

(2)  4 m

(3)  2 m

(4)  1 m

A body moving with speed 10 m/s is stopped by applying constant braking power in 5 seconds. If the speed of the body is 30 m/s, then time in which the body can be stopped by applying the same retarding power is

(1)  10 s

(2)  15 s

(3)  30 s

(4)  45 s

A small ball is given a velocity $\sqrt{2gl}$ on the smooth horizontal floor which leads to a smooth vertical circular path. The ball will Smooth

(1)  complete the loop.

(2)  reach only up to point P.

(3)  reach to point Q.

(4)  leave the contact somewhere between P and Q.

A vertically compressed spring of constant k releases a ball of mass m as shown in the figure. The maximum height attained by the ball from the compressed point is (x is the initial compression in the spring)

1.  $\frac{{\mathrm{kx}}^2}{2\mathrm{mg}}$

2.  $\frac{{\mathrm{kx}}^2}{\mathrm{mg}}$

3.  $\frac{{\mathrm{kx}}^2}{4\mathrm{mg}}$

4.  $\frac{2{\mathrm{kx}}^2}{\mathrm{mg}}$

Work done the spring force in a time interval maybe

(1)  Positive

(2)  Negative

(3)  Zero

(4)  All of these

 A ball of mass m at rest starts moving from point A. The irregular surface is frictionless. The speed of the ball at the point C on the track is

1.  $\sqrt{\frac{4\mathrm{gH}}{3}}$

2.  $\sqrt{\frac{2\mathrm{gH}}{3}}$

3.  $\sqrt{\mathrm{gH}}$

4.  Zero

A ball is allowed to fall from a height of 10 m. If there is a 40% loss of energy due to impact, then after one impact ball will go up by

(1)  10 m

(2)  8 m

(3)  4 m

(4)  6 m