A body constrained to move along the \({z}\)-axis of a coordinate system is subjected to constant force given by \(\vec{F}=-\hat{i}+2 \hat{j}+3 \hat{k}\) where \(\hat{i},\hat{j} \) and \(\hat{k}\) are unit vectors along the \({x}\)-axis, \({y}\)-axis and \({z}\)-axis of the system respectively. The work done by this force in moving the body a distance of \(4~\text m\) along the \({z}\)-axis will be:
1. \(15~\text J\) 
2. \(14~\text J\) 
3. \(13~\text J\) 
4. \(12~\text J\) 

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 \(v = ax^{3 / 2 }\) where \(a = 5 m^{- 1 / 2} s^{- 1}\). 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 bolt of mass \(0.3~\text{kg}\) falls from the ceiling of an elevator moving down at a uniform speed of \(7~\text{m/s}.\) It hits the floor of the elevator (length of the elevator = \(3\text{ m}\)) and does not rebound. What is the heat produced by the impact?

1.  \(8.82~\text{J}\)

2.  \(7.65~\text{J}\)

3.  \(7.01~\text{J}\)

4.  \(7.98~\text{J}\)

A body of mass 'm' is released from the top of a fixed rough inclined plane as shown in the figure. If the frictional force has magnitude F, then the body will reach the bottom with a velocity: $(\mathrm{L}=\sqrt{2}\mathrm{h})$

The diagram represents a particle's potential energy curve in a field. The particle will be in equilibrium at which position(s):

                   
1. \(B\) and \(D\)
2. \(A\) and \(C\)
3. \(A,B\) and \(C\)
4. \(A,B,C\) and \(D\)

In the diagram shown, force \(F\) acts on the free end of the string. If the weight \(W\) moves up slowly by distance \(h,\) then work done on the weight by the string holding it will be: (pulley and string are ideal)

1. \(Fh\)
2. \(2Fh\)
3. \(\dfrac{Fh}{2}\)
4. \(4Fh\)

The potential energy of a system increases if work is done:

1. by the system against a conservative force.

2. by the system against a non-conservative force.

3. upon the system by a conservative force.

4. upon the system by a non-conservative force.

The potential energy of a \(1 ~\text{kg}\) particle free to move along the \(x\text-\)axis is given by \(U(x)=\left(\frac {x^4}{ 4}-\frac {x^2}{ 2}\right)~\text J.\) The total mechanical energy of the particle is \(2~\text J.\) Then the maximum speed (in \(\text{ms}^{-1}\)) will be:
1. \(\dfrac{3}{\sqrt{2}} \)
2. \(\sqrt{2}\)
3. \(\dfrac{1}{\sqrt{2}}\)
4.  \(2\)

The power supplied to a particle of mass 2 kg varies with time as $\mathrm{P}=\frac{3{\mathrm{t}}^2}{2}$ Watt, where t is in seconds. If the velocity of a particle at t = 0 is v = 0, then the velocity of the particle at t = 2 s will be: