A woman pushes a trunk on a railway platform which has a rough surface. She applies a force of \(100\) N over a distance of \(10\) m. Thereafter, she gets progressively tired and her applied force reduces linearly with distance to \(50\) N. The total distance through which the trunk has been moved is \(20\) m. The plot of force applied by the woman and the frictional force, which is \(50\) N versus displacement is given below. Work done by the two forces over \(20\) m are:

         

1. \(1750\) J and \(-1000\) J
2. \(1750\) J and \(1000\) J
3. \(-1750\) J and \(1000\) J
4. \(-1750\) J and \(-1000\) J

A block of mass \(m=1\) kg, moving on a horizontal surface with speed \(v_i=\mathrm{2~m/s}\) enters a rough patch ranging from \(\mathrm{x=0.10~m}\) to \(\mathrm{x=2.01~m}\). The retarding force \(F_r\) on the block in this range is inversely proportional to \(x\) over this range,
\(\begin{aligned} \mathrm{F}_{\mathrm{r}} & =\frac{-\mathrm{k}}{\mathrm{x}} \text { for } 0.1<\mathrm{x}<2.01 \mathrm{~m} \\ & =0 \quad \text { for } \mathrm{x}<0.1 \mathrm{~m} \text { and } \mathrm{x}>2.01 \mathrm{~m} \end{aligned}\)
where \(k=0.5~\mathrm{J}\). What is the final kinetic energy and speed \(v_f\) of the block as it crosses this patch?
1. 5 J and 1 m/s 
2. 1 J and 5 m/s
3. 0.5 J and 1 m/s
4. 0.05 J and 2 m/s 

A bob of mass m is suspended by a light string of length L. It is imparted a horizontal velocity ${\mathrm{v}}_0$ at the lowest point A such that it completes a semi-circular trajectory in the vertical plane with the string becoming slack only on reaching the topmost point, the horizontal velocity ${\mathrm{v}}_0$ is:

1.   $\sqrt{5\mathrm{gl}}$

2.   $\sqrt{4\mathrm{gl}}$

3.  $\sqrt{3\mathrm{gl}}$

4.   $\sqrt{\mathrm{gl}}$

A bob of mass m is suspended by a light string of length \(L.\) It is imparted a horizontal velocity \(v_0\) at the lowest point A such that it completes a semi-circular trajectory in the vertical plane with the string becoming slack only on reaching the topmost point, the ratio of the kinetic energies \(\frac{K_B}{K_C}\) ${\mathrm{}}_{}$at points \(\mathrm{B}\) and \(\mathrm{C}\) is:
${\mathrm{}}_{}$

                  
1. \(1:3\)
2. \(3:1\)
3. \(1:5\)
4. \(5:1\)

A bob of mass m is suspended by a light string of length L. It is imparted a horizontal velocity ${\mathrm{v}}_0$ at the lowest point A such that it completes a semi-circular trajectory in the vertical plane with the string becoming slack only on reaching the topmost point. If the connecting string is cut at point C, the trajectory of bob is:

1.   Straight line

2.   Circular 

3.   Projectile 

4.   Can't be determined

To simulate car accidents, auto manufacturers study the collisions of moving cars with mounted springs of different spring constants. Consider a typical simulation with a car of mass \(1000\) kg moving with a speed \(18.0\) km/h on a smooth road and colliding with a horizontally mounted spring of spring constant \(6.25\times 10^3~{\mathrm{N/m}}\). The maximum compression in the spring is:
1. \(3\) m
2. \(0.2\) m
3. \(1\) m
4. \(2\) m

To simulate car accidents, auto manufacturers study the collisions of moving cars with mounted springs of different spring constants. Consider a typical simulation with a car of mass 1000 kg moving with a speed of 18.0 km/h on a rough road and colliding with a horizontally mounted spring of spring constant $2.5\times {10}^3 \mathrm{N}/\mathrm{m}$. If the coefficient of friction between road and tyre of the car, µ, to be 0.375. Maximum compression of the spring is:

1.   3.5 m

2.   2.0 m

3.   1.5 m

4.   2.5 m

The values of energy required to break one bond in DNA \((10^{-20}~\mathrm{J})\) and the kinetic energy of an air molecule \((10^{-21}~\mathrm{J})\) in eV respectively are:

An elevator can carry a maximum load of \(1800\) kg (elevator + passengers) is moving up with a constant speed of \(2\) m/s. The frictional force opposing the motion is \(4000\) N. The minimum power delivered by the motor to the elevator is:
1. \(59000\) W
2. \(44000\) W
3. \(11000\) W
4. \(22000\) W

In a nuclear reactor, a neutron of high speed (typically ${10}^7$ m/s) must be slowed to ${10}^3$ m/s so that it can have a high probability of interacting with isotope ${}_{92}{}^{235}\mathrm{U}$ and causing it to fission. The material making up the light nuclei, usually heavy water $\left({\mathrm{D}}_2\mathrm{O}\right)$ or graphite, is called a moderator. Find the fraction of the kinetic energy of the neutron lost by it in an elastic collision with light nuclei like deuterium.

1.  $\frac{1}{9}$

2.  $\frac{8}{9}$

3.  $\frac{9}{8}$

4.  $\frac{1}{8}$