A thin circular ring of mass M and radius R is rotating in a horizontal plane about an axis vertical to its plane with a constant angular velocity ω. If two objects each of mass m are attached gently to the opposite ends of the diameter of the ring, the ring will then rotate with an angular velocity:

A black body at \(227^{\circ}~\mathrm{C}\) radiates heat at the rate of \(7~ \mathrm{cal-cm^{-2}s^{-1}}\).  At a temperature of \(727^{\circ}~\mathrm{C}\), the rate of heat radiated in the same units will be:
1. \(60\)
2. \(50\)
3. \(112\)
4. \(80\)

Two bodies of mass \(1\) kg and \(3\) kg have position vectors \(\hat{i}+2\hat{j}+\hat{k}\) and \(-3\hat{i}-2\hat{j}+\hat{k}\) respectively. The centre of mass of this system has a position vector:
1. \(-2\hat{i}+2\hat{k}\)
2. \(-2\hat{i}-\hat{j}+\hat{k}\)
3. \(2\hat{i}-\hat{j}-2\hat{k}\)
4. \(-\hat{i}+\hat{j}+\hat{k}\)

The internal energy change in a system that has absorbed \(2\) kcal of heat and done \(500\) J of work is:
1. \(8900\) J
2. \(6400\) J
3. \(5400\) J
4. \(7900\) J

An engine pumps water continuously through a hose. Water leaves the hose with a velocity \(v\) and \(m\) is the mass per unit length of the water jet. What is the rate at which kinetic energy is imparted to water?
1. \(\dfrac{1}{2}mv^3\)
2. \(mv^3\)
3. \(\dfrac{1}{2}mv^2\)
4. \(\dfrac{1}{2}m^2v^2\)

The driver of a car travelling at a speed of 30 m/s towards a hill sounds a horn of frequency 600 Hz. If the velocity of sound in air is 330 m/s, the frequency of reflected sound as heard by the driver is:

1. 550 Hz

2. 555.5 Hz

3. 720 Hz

4. 500 Hz

Three capacitors each of capacitance \(C\) and of breakdown voltage \(V\) are joined in series. The capacitance and breakdown voltage of the combination will be:
1. $\frac{C}{3},$ $\frac{V}{3}$

2. $3C,$ $\frac{V}{3}$

3. $\frac{C}{3},$ $3V$

4. \(3C,~3V\)