Pulley Problem

Q.

A solid cylinder of mass 'm' and radius 'R' rests on a plank of mass '2m' lying on a smooth horizontal surface. String connecting the cylinder to the plank is passing over a massless pulley mounted on a movable light block B, and the friction between the cylinder and the plank is sufficient to prevent slipping. If the block B is pulled with a constant force 'F', find the acceleration of the cylinder and that of the plank.

1. $\frac{3F}{7m},\frac{2F}{7m}$

2. $\frac{2F}{7m},\frac{3F}{7m}$

3. $\frac{4F}{7m},\frac{3F}{7m}$

4. $\frac{3F}{7m},\frac{4F}{7m}$

Q. A block of mass 2 kg hangs from the rim of a wheel of radius 0.5 m. On releasing 2 kg from rest the block falls through 5 m height in 2 s. The moment of inertia of the wheel will be

1. $1kg-m^2$
2. $3.2kg-m^2$
3. $2.5kg-m^2$
4. $1.5kg-m^2$

Q. A circular pulley of mass ${}^{\prime }{M}^{\prime }$ and radius ${}^{\prime }{R}^{\prime }$, is hinged at the centre. A long string is wound over this disc and two bodies of mass $m_1$ and $m_2$ are attached at it's free ends. Now the bodies are released. Find the accceleration of each body.

1. $\frac{2(m_1-m_2)g}{M+(m_1+m_2)}$
2. $\frac{(m_1-m_2)g}{M+2(m_1+m_2)}$
3. $\frac{2(m_1-m_2)g}{M+2(m_1+m_2)}$
4. $\frac{(m_1-m_2)g}{M+(m_1+m_2)}$

Q. A mass m hangs with the help of a string wrapped around a pulley on a frictionless bearing. The pulley has mass m and radius R. Assuming pulley to be a perfect uniform circular disc, the acceleration of the mass m, if the string does not slip on the pulley, is

1. 3g/2
2. g
3. 2g/3
4. g/3

Q. In the system shown in figure, masses of the blocks are such that when the sytem is released, the acceleration of pulley $P_1$ is upwards and the acceleration of block 1 is $a_1$ upwards. It is found that the acceleration of block 3 is same as that of 1 both in magnitude and direction.
Given $a_1>a>\frac{a_1}{2}$, match the following:


$\begin{array}{cccc}& \text{Column I}& & \text{Column II}\\ \text{i.}& \text{Acceleration of 2}& \text{a.}& 2a+a_1\\ \text{ii.}& \text{Acceleration of 4}& \text{b.}& 2a-a_1\\ \text{iii.}& \text{Acceleration of 2 with respect to 3}& \text{c.}& \text{Upwards}\\ \text{iv.}& \text{Acceleration of 2 with respect to 4}& \text{d.}& \text{Downwards}\end{array}$

1. i - b, c; ii - a; iii - d; iv - c
2. i - b; ii - a, d; iii - d; iv - c
3. i - b, c; ii - a, d; iii - d; iv - c
4. i - b, c; ii - a; iii - c; iv - c