When a rubber band is stretched by a distance \(x,\) it exerts a restoring force given by; \(F=(ax+bx^2),\) where \(a\) and \(b\) are constants. What is the total work done in stretching the rubber band from its natural (unstretched) length to a length \(L\text{?}\)

1.	\(\dfrac{1}{2}(aL^2+bL^3)\)	2.	\(\left ( \dfrac{aL^2}{2}+\dfrac{bL^3}{3} \right ) \)
3.	\(\dfrac{1}{2}\left(\dfrac{aL^2}{2}+\dfrac{bL^3}{3}\right) \)	4.	\((aL^2+bL^3)\)

A particle of mass \(m\) moving in the \(x\)-direction with speed \(2v\) is hit by another particle of mass \(2m\) moving in the \(y\)-direction with speed \(v.\) If the collision is perfectly inelastic, the percentage loss in the energy during the collision is close to:
1. \(44\%\)
2. \(50\%\)
3. \(56\%\)
4. \(62\%\)

A person trying to lose weight by burning fat lifts a mass of \(10~\text{kg}\) upto a height of \(1~\text{m}\) \(1000\) times. Assume that the potential energy lost each time he lowers the mass is dissipated. How much fat will he use up considering the work done only when the weight is lifted up? Fat supplies \(3.8\times 10^7~\text{J}\) of energy per kg which is converted to mechanical energy with a \(20\%\) efficiency rate. Take \(g= 9.8~\text{ms}^{-2}\):
1. \(2.45\times 10^{-3}~\text{kg}\)
2. \(6.45\times 10^{-3}~\text{kg}\)
3. \(9.89\times 10^{-3}~\text{kg}\)
4. \(12.89\times 10^{-3}~\text{kg}\)

A point particle of mass \(m\), moves along the uniformly rough track \(PQR\) as shown in the figure. The coefficient of friction, between the particle and the rough track equals \(\mu\). The particle is released, from rest, from the point \(P\) and it comes to rest at a point \(R\). The energies, lost by the ball, over the parts, \(PQ\) and \(QR\), of the track, are equal to each other, and no energy is lost when the particle changes direction from \(PQ\) to \(QR\).
The values of the coefficient of friction \(\mu\) and the distance \(x~(=QR)\), are, respectively close to:

             
1. \(0.2~\text{and}~6.5~\text{m}\)
2. \(0.2~\text{and}~3.5~\text{m}\)
3. \(0.29~\text{and}~3.5~\text{m}\)
4. \(0.29~\text{and}~6.5~\text{m}\)

A time dependent force \(F = 6t \) acts on a particle of mass \(1~\text{kg}. \) If the particle starts from rest, the work done by the force during the first \(1~\text s\) will be:
1. \(4.5~\text{J} \)
2. \(22~\text{J} \)
3. \(9~\text{J} \)
4. \(18~\text{J} \)

A person pushes a box on a rough horizontal plateform surface. He applies a force of \(200~\text{N}\) over a distance of \(15~\text{m}\). Thereafter, he gets progressively tired and his applied force reduces linearly with distance to \(100~\text{N}\). The total distance through which the box has been moved is \(30~\text{m}\). What is the work done by the person during the total movement of the box?
1. \(5690~\text{J}\)
2. \(3280~\text{J}\)
3. \(2780~\text{J}\)
4. \(5250~\text{J}\)

A small bob, attached to one end of a thin string of length \(1~\text m,\) is performing vertical circular motion. The ratio of the maximum tension to the minimum tension in the string is \(5:1.\) What is the bob's velocity at the highest point?
(take \(g=10~\text{m/s}^2\))
1. \(2~\text{m/s}\)
2. \(5~\text{m/s}\)
3. \(6~\text{m/s}\)
4. \(9~\text{m/s}\)
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Two bodies of the same mass are moving with the same speed, but in different directions in a plane. They have a completely inelastic collision and move together thereafter with a final speed which is half of their initial speed. The angle between the initial velocities of the two bodies (in degree) is:

1. 30^o

2. 60^o

3^. 90^o

4^. 120^o

If the potential energy between two molecules is given by \(U = -\dfrac{A}{r^6}+ \dfrac{B}{r^{12}}, \) then the potential energy at equilibrium separation between molecules is:

Two solids \(A\) and \(B\) of mass \(1\) kg and \(2\) kg respectively are moving with equal linear momentum. The ratio of their kinetic energies \((KE)_A:(KE)_B\) will be:
1. \(1:2\)
2. \(2:1\)
3. \(1:4\)
4. \(4:1\)