A block of mass \(m\) is in contact with the cart \((C)\) as shown in the figure. 
                    
The coefficient of static friction between the block and the cart is \(\mu.\) The acceleration \(a\) of the cart that will prevent the block from falling satisfies:
1. \(a > \dfrac{mg}{\mu}\)
2. \(a > \dfrac{g}{\mu m}\)
3. \(a \ge \dfrac{g}{\mu}\)
4. \(a < \dfrac{g}{\mu}\)

The mass of a ${}_3{}^7\mathrm{Li}$ nucleus is \(0.042~\text{u}\) less than the sum of the masses of all its nucleons. The binding energy per nucleon of the ${}_3{}^7\mathrm{Li}$ nucleus is near:
1. \(4.6~\text{MeV}\)
2. \(5.6~\text{MeV}\)
3. \(3.9~\text{MeV}\)
4. \(23~\text{MeV}\)

A circular disk of a moment of inertia \(\mathrm{I_t}\) is rotating in a horizontal plane, about its symmetric axis, with a constant angular speed \(\omega_i.\) Another disk of a moment of inertia \(\mathrm{I_b}\) is dropped coaxially onto the rotating disk. Initially, the second disk has zero angular speed. Eventually, both the disks rotate with a constant angular speed \(\omega_f.\) The energy lost by the initially rotating disc due to friction is:
1. \( \frac{1}{2} \frac{\mathrm{I}_{\mathrm{b}}^2}{\left(\mathrm{I}_{\mathrm{t}}+\mathrm{I}_{\mathrm{b}}\right)} \omega_{\mathrm{i}}^2\)

2. \( \frac{1}{2} \frac{\mathrm{I}_{\mathrm{t}}^2}{\left(\mathrm{I}_{\mathrm{t}}+\mathrm{I}_{\mathrm{b}}\right)} \omega_{\mathrm{i}}^2\)

3. \( \frac{1}{2} \frac{\mathrm{I}_{\mathrm{b}}-\mathrm{I}_{\mathrm{t}}}{\left(\mathrm{I}_{\mathrm{t}}+\mathrm{I}_{\mathrm{b}}\right)} \omega_{\mathrm{i}}^2 \)

4. \( \frac{1}{2} \frac{\mathrm{I}_{\mathrm{b}} \mathrm{I}_{\mathrm{t}}}{\left(\mathrm{I}_{\mathrm{t}}+\mathrm{I}_{\mathrm{b}}\right)} \omega_{\mathrm{i}}^2 \)

Which one of the following statements is false?

The displacement of a particle along the x-axis is given by, x = asin²$\mathrm{\omega }$t. The motion of the particle corresponds to:

The radii of circular orbits of two satellites A and B of the earth are \(4R\) and \(R\) respectively. If the speed of satellite A is \(3v,\) then the speed of satellite B will be:
1. \(3v/4\)
2. \(6v\)
3. \(12v\)
4. \(3v/2\)

A beam of cathode rays is subjected to cross Electric (E) and magnetic fields(B). The fields are adjusted such that the beam is not deflected. The specific charge of the cathode rays is given by:

1. $\frac{B^2}{2V{E}^2}$

2. $\frac{2V{B}^2}{E^2}$

3. $\frac{\mathrm{2V}{E}^2}{B^2}$

4. $\frac{E^2}{2V{B}^2}$

(where V is the potential difference between cathode and anode)

A ball is dropped from a high-rise platform at \(t=0\) starting from rest. After \(6\) seconds, another ball is thrown downwards from the same platform with speed \(v\). The two balls meet after \(18\) seconds. What is the value of \(v\)?

A ray of light travelling in a transparent medium of refractive index \(\mu\) falls on a surface separating the medium from the air at an angle of incidence of \(45^{\circ}\). For which of the following value of \(\mu\)$$, the ray can undergo total internal reflection?
1. \(\mu = 1.33\)
2. \(\mu =1.40\)
3. \(\mu=1.50\)
4. \(\mu = 1.25\)