91Ó°ÊÓ

A small disc of mass \(m\) slides down a smooth hill of height \(h\) without any initial velocity and lands upon a plank of mass \(M\) lying along the horizontal plane at the base of the hill. Due to friction between the disc and the plank the disc slows down till the two move together as a single piece with a certain common velocity. Find the work done by the force of friction in the process. Solution Since mass \(m\) falls through hcight \(h\) just before landing on the plank of mass \(M\), so velocity of at this moment is \(v-\sqrt{2 g h}\). Now, the frictional force betwcen \(M\) and \(m\) is the internal force the system so momentum of system remains conserved. I lence, \(m v-(M \mid m) v^{\prime}\) or \(v^{\prime}-\frac{m v}{M+m}\) or \(v^{\prime}-\frac{m \sqrt{2 g h}}{M+m}\) So, initial kinetic cnergy of the system \(K_{i}-\frac{1}{2} m v^{2} \quad\) or \(\quad K_{i}-m g h\) \(\ldots(2)\) and linal kinctic cnergy of the system \(K_{f}-\frac{1}{2}(M+m) v^{\prime 2}-\frac{1}{2}(M+m) \frac{m^{2} \times 2 g h}{(M+m)^{2}} \quad \Rightarrow K_{f}-\frac{m^{2} g h}{M+m} \quad \ldots(3)\) So, change of cnergy \(\Delta K-K_{f} K_{t} \Rightarrow \Delta K-\frac{m M g h}{M \mid m}\) Hence, work-done by friction \(W-\Delta K \quad \therefore W--\frac{m M g h}{M \mathrm{I} m}\)

Statement-1: \(\Lambda\) cyclist always bends inwards while negotiating a curve. Statement-2: By bending he lowers his centre of gravity.

A particle of mass \(m\) (starting at rest) moves vertically upwards from the surface of earth under an external force \(\vec{F}\) which varies with height \(z\) as \(\vec{F}-(2-\alpha z) m \vec{g}\), where \(\alpha\) is a positive constant. If \(H\) is the maximum height to which particle rises. Then (a) \(H-\frac{1}{\alpha}\) (b) Work done by \(\vec{F}\) during motion upto \(\frac{H}{2}\) is \(\frac{3 m g}{2 \alpha}\) (c) \(H-\frac{2}{\alpha}\) (d) Velocity of particle at \(\frac{H}{2}\) is \(\sqrt{\frac{g}{\alpha}}\)

A uniform ring, having radius \(a\) and mass \(m\) is to be rotated in the horizontal plane about its own axis with constant angular velocity \(\omega\), what would be the tension in the ring and nature of force? (a) \(\frac{m a \omega^{2}}{2 \pi}\) tensile (b) \(m a \omega^{2}\) tensile (c) \(\frac{m a \omega^{2}}{2}\) compressive (d) \(m a \omega^{2}\) compressive

The mean kinetic energy of a particle of mass \(m\) moving under a constant acceleration in any interval of time when initial and final vclocities are \(u_{1}\) and \(u_{2}\) (a) \(\frac{1}{2} m\left(u_{1}^{2} \mid u_{2}^{2}\right)\) (b) \(\frac{1}{2} m\left(u_{1}^{2} \quad u_{2}^{2}\right)\) (c) \(\frac{1}{6} m\left(u_{1}^{2}\left|u_{2}^{2}\right| u_{1} u_{2}\right)(\) d) Zero

A block of mass \(m\) slides along the track with coc\lceilicient o[ kinctic friction \(\mu .\) A man pulls the block through a rope which makes an angle \(\theta\) with the horizontal as shown in the figuro. The block moves with constant speed \(V\). Power delivered by the man is (a) \(T V\) (b) \(T V \cos \theta\) (c) \((T \cos \theta-\mu m g) V\) (d) zero