Two similar springs \(P\) and \(Q\) have spring constants \(k_P\) and \(k_Q\), such that \(k_P>k_Q\). They are stretched, first by the same amount (case a), then by the same force (case b). The work done by the springs \(W_P\) and \(W_Q\) are related as, in case (a) and case (b), respectively:
| 1. | \(W_P=W_Q;~W_P>W_Q\) |
| 2. | \(W_P=W_Q;~W_P=W_Q\) |
| 3. | \(W_P>W_Q;~W_P<W_Q\) |
| 4. | \(W_P<W_Q;~W_P<W_Q\) |
| 1. | \(475\) J | 2. | \(450\) J |
| 3. | \(275\) J | 4. | \(250\) J |
| 1. | \( \sqrt{\frac{m k}{2}} t^{-1 / 2} \) | 2. | \( \sqrt{m k} t^{-1 / 2} \) |
| 3. | \( \sqrt{2 m k} t^{-1 / 2} \) | 4. | \( \frac{1}{2} \sqrt{m k} t^{-1 / 2}\) |
Two particles of masses \(m_1\) and \(m_2\) move with initial velocities \(u_1\) and \(u_2\) respectively. On collision, one of the particles gets excited to a higher level, after absorbing energy \(E\). If the final velocities of particles are \(v_1\) and \(v_2\), then we must have:
| 1. | \(m_1^2u_1+m_2^2u_2-E = m_1^2v_1+m_2^2v_2\) |
| 2. | \(\frac{1}{2}m_1u_1^2+\frac{1}{2}m_2u_2^2= \frac{1}{2}m_1v_1^2+\frac{1}{2}m_2v_2^2\) |
| 3. | \(\frac{1}{2}m_1u_1^2+\frac{1}{2}m_2u_2^2-E= \frac{1}{2}m_1v_1^2+\frac{1}{2}m_2v_2^2\) |
| 4. | \(\frac{1}{2}m_1^2u_1^2+\frac{1}{2}m_2^2u_2^2+E = \frac{1}{2}m_1^2v_1^2+\frac{1}{2}m_2^2v_2^2\) |
| 1. | \(wx \over d\) | 2. | \(wd \over x\) |
| 3. | \(w(d-x) \over x\) | 4. | \(w(d-x) \over d\) |
A mass \(m\) moves in a circle on a smooth horizontal plane with velocity \(v_0\) at a radius \(R_0.\) The mass is attached to a string that passes through a smooth hole in the plane, as shown in the figure.
The tension in the string is increased gradually and finally, \(m\) moves in a circle of radius \(\frac{R_0}{2}.\) The final value of the kinetic energy is:
| 1. | \( m v_0^2 \) | 2. | \( \dfrac{1}{4} m v_0^2 \) |
| 3. | \( 2 m v_0^2 \) | 4. | \( \dfrac{1}{2} m v_0^2\) |
Three identical spherical shells, each of mass \(m\) and radius \(r\) are placed as shown in the figure. Consider an axis \(XX',\) which is touching two shells and passing through the diameter of the third shell. The moment of inertia of the system consisting of these three spherical shells about the \(XX'\) axis is:
| 1. | \(\dfrac{11}{5}mr^2\) | 2. | \(3mr^2\) |
| 3. | \(\dfrac{16}{5}mr^2\) | 4. | \(4mr^2\) |
Kepler's third law states that the square of the period of revolution (\(T\)) of a planet around the sun, is proportional to the third power of average distance \(r\) between the sun and planet i.e. \(T^2 = Kr^3\), here \(K\) is constant. If the masses of the sun and planet are \(M\) and \(m\) respectively, then as per Newton's law of gravitation, the force of attraction between them is \(F = \frac{GMm}{r^2},\) here \(G\) is the gravitational constant. The relation between \(G\) and \(K\) is described as:
1. \(GK = 4\pi^2\)
2. \(GMK = 4\pi^2\)
3. \(K =G\)
4. \(K = \frac{1}{G}\)
Two spherical bodies of masses \(M\) and \(5M\) and radii \(R\) and \(2R\) are released in free space with initial separation between their centres equal to \(12R.\) If they attract each other due to gravitational force only, then the distance covered by the smaller body before the collision is:
| 1. | \(2.5R\) | 2. | \(4.5R\) |
| 3. | \(7.5R\) | 4. | \(1.5R\) |
On observing light from three different stars \(P,\) \(Q,\) and \(R,\) it was found that the intensity of the violet colour is maximum in the spectrum of \(P,\) the intensity of the green colour is maximum in the spectrum of \(R\) and the intensity of the red colour is maximum in the spectrum of \(Q.\) If \(T_P,\) \(T_Q,\) and \(T_R\) are the respective absolute temperatures of \(P,\) \(Q,\) and \(R,\) then it can be concluded from the above observations that:
1. \(T_P>T_Q>T_R\)
2. \(T_P>T_R>T_Q\)
3. \(T_P<T_R<T_Q\)
4. \(T_P<T_Q<T_R\)
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