The potential energy function for a particle executing linear simple harmonic motion is given by \(V(x)=\frac{kx^2}{2}\), where \(k\) is the force constant of the oscillator. For \(k=0.5~\text{N/m},\) the graph of \(V(x)\) versus \(x\) is shown in the figure. A particle of total energy \(1~\text J\) moving under this potential must turn back when it reaches:

1. \({x}=\pm 1~\text m\) 
2. \({x}=\pm 2~\text m\) 
3. \({x}=\pm 3~\text m\) 
4. \({x}=\pm 4~\text m\)

Which one of the following statement is wrong?

An electron and a proton are detected in a cosmic ray experiment, the first with kinetic energy \(10~\text{keV},\) and the second with \(100~\text{keV}.\) The ratio of their speeds is:
(take \(m_e=9.31\times 10^{-31}~\text{kg},\) \(m_p=1.67\times 10^{-27}~\text{kg},\) \(1~\text{eV}=1.60\times 10^{-19}~\text{kg}\) )
1. \(13.5: 1\)
2. \(1: 1\)
3. \(1: 13.5\)
4. \(2: 1\)

A raindrop of radius \(2~\text{mm}\) falls from a height of \(500~\text{m}\) above the ground. It falls with decreasing acceleration (due to viscous resistance of the air) until, at half its original height, it attains its maximum (terminal) speed, and moves with uniform speed thereafter. What is the work done by the gravitational force on the drop in the first and second half of its journey?

The bob of a pendulum is released from a horizontal position. If the length of the pendulum is \(1.5~\text m,\) what is the speed with which the bob arrives at the lowermost point, given that it dissipates \(5\%\) of its initial energy against air resistance?
1. \(2.5~\text{m/s}\)
2. \(3.9~\text{m/s}\)
3. \(4.7~\text{m/s}\)
4. \(5.3~\text{m/s}\)

A body of mass 0.5 kg travels in a straight line with velocity $\mathrm{v}={\mathrm{ax}}^{3/\mathrm{2 }}$where $\mathrm{a}=5 {\mathrm{m}}^{-1/2} {\mathrm{s}}^{-1}$. What is the work done by the net force during its displacement from x = 0 to x = 2 m?

1. 50 J

2. 45 J

3. 68 J

4. 90 J

A person trying to lose weight (dieter) lifts a \(10~\text{kg}\) mass, one thousand times, to a height of \(0.5~\text m\) each time. Assume that the potential energy lost each time she lowers the mass is dissipated. How much work does she do against the gravitational force?
1. \(29,000~\text J\)
2. \(49,000~\text J\)
3. \(21,000~\text J\)
4. \(18,000~\text J\)

A 1 kg block situated on a rough incline is connected to a spring of spring constant $100 N m^{-1}$ as shown in the figure. The block is released from rest with the spring in the unstretched position. The block moves 10 cm down the incline before coming to rest. The coefficient of friction between the block and the incline is:

(Assume that the spring has a negligible mass and the pulley is frictionless.)

1. 0.03
2. 0.14
3. 0.11
4. 0.08

A bolt of mass 0.3 kg falls from the ceiling of an elevator moving down at a uniform speed of 7 m/s. It hits the floor of the elevator (length of the elevator = 3 m) and does not rebound. What is the heat produced by the impact?

1.  8.82 J

2.  7.65 J

3.  7.01 J

4.  7.98 J