Constrained Motion : Intuitive Approach

Q. The sum of two forces $\overrightarrow{P}$ and $\overrightarrow{Q}$ is $\overrightarrow{R}$ such that $|\overrightarrow{R}|=|\overrightarrow{P}|.$ The angle $\theta$ (in degrees) that the resultant of $2\overrightarrow{P}$ and $\overrightarrow{Q}$ will make with $\overrightarrow{Q}$ is _____.

Q. Two forces P and Q, of magnitude $2F\text{and}3F$ respectively are at an angle $\theta$ with each other. If the force $Q$ is doubled, then their resultant also gets doubled. Then the angle $\theta$ is

1. ${120}^{\circ }$
2. ${60}^{\circ }$
3. ${90}^{\circ }$
4. ${30}^{\circ }$

Q. Three blocks $\text{a, b and c}$ of masses $10kg$, $10kg$ and $20kg$ respectively are arranged as shown in figure. All the surfaces are frictionless and string is inextensible. A constant force $F=20N$ is applied on block $\text{a}$ as shown. Pulley and string are light. Part of the string connecting both pulleys is vertical and part of the strings connecting pulleys with masses a and b are horizontal. Choose the correct option(s) for the given system.

1. Acceleration of block b is $0.5m/s^2$
2. Acceleration of block $\text{b}$ is $1m/s^2$
3. Tension in the string is $10N$
4. Acceleration of block $\text{c}$ is $0.5m/s^2$

Q.

If $\mathrm{s}={\mathrm{ae}}^{\mathrm{t}}+(\mathrm{b}/{\mathrm{e}}^{\mathrm{t}})$is the distance in seconds described by a particle in $\mathrm{t}$ seconds, then the particles acceleration at the time $\mathrm{t}$ is

1. proportional to t

2. Proportional to s

3. s

4. Constant

Q. A ball of mass \(m\) is dropped from height \(h\) on a light platform fixed at the top of vertical spring as shown in the figure. The maximum compression of spring is \(x\). What is the spring constant ?

Q. Both springs, string and pulley shown in the figure are light. Initially when both the springs were in their natural state, the system was released from rest. The maximum displacement of the block having mass $m$ is:

1. $\frac{5mg}{k}$
2. $\frac{3mg}{k}$
3. $\frac{10mg}{k}$
4. $\frac{4mg}{k}$

Q. If pulleys shown in the diagram are smooth and massless and $a_1$ & $a_2$ are acceleration of blocks of mass $4\text{kg}$ and $8\text{kg}$ respectively. Then find $a_2$.

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

Q. Velocities of blocks $A,B$ and pulley $P_2$ are shown in the figure given below, then

1. Velocity of pulley $P_1$ will be $5\text{m/s}$ downwards.
2. Velocity of block $C$ will be $25\text{m/s}$ upwards.
3. Velocity of pulley $P_1$ will be $5\text{m/s}$ upwards.
4. Velocity of block $C$ will be $15\text{m/s}$ downwards.

Q.

A block of mass $m$ is connected to a massless pulley and massless spring of stiffness$k.$ The pulley is frictionless, and the string is massless. Initially, the block is not hanging, it is supported by some unknown support. Also, the spring is not stretched when the block is released. When the block is released, the spring is maximum stretched. Find the ratio of tension in the rope and weight of the block (give integer value)

Q. In the figure shown the velocity of lift is 2 m/s while string is winding on the motor shaft with velocity 2 m/s and block A is moving downwards with velocity of 2 m/s, the find out the velocity of block B.

1. $2\frac{m}{s}$
2. $4\frac{m}{s}$
3. $5\frac{m}{s}$
4. $6\frac{m}{s}$

Q. In the arrangement shown in the figure, mass of the block $A$ is $4$ times that of block $B$. The height $h=20\text{cm}$. At a certain instant, the block $A$ is released and the system is set in motion. What is the maximum height (in $\text{cm})$ that the body $B$ will go up? Assume enough space allowed for $B$ and $A$ (i.e $A$ and $B$ are of small size) and take $g=10{\text{m/s}}^2$

1. $40\text{cm}$
2. $60\text{cm}$
3. $20\text{cm}$
4. $0\text{cm}$

Q. In the figure shown, the pulleys and strings are massless. The acceleration of the block of mass 4m just after the system is released from rest is $(\theta =si{n}^{-1}\frac{3}{5})$

1. $\frac{2g}{5}$ upwards
2. $\frac{2g}{5}$ downwards
3. $\frac{5g}{11}$ downwards.
4. $\frac{5g}{11}$ upwards

Q. A racing car has uniform acceleration of $4m{s}^{-2}$. What distance will it cover in $10s$ after start?

1. $200m$
2. $80m$
3. $400m$
4. $40m$

Q. The lower end of a $4\text{m}$ long uniform rod $AB$ is pulled with constant speed $v=4\text{m/s}$. The speed of the centre of mass of the rod when $\theta ={60}^{\circ }$ will be

1. $\frac{4}{\sqrt{3}}\text{m/s}$
2. $2\sqrt{3}\text{m/s}$
3. $4\text{m/s}$
4. $4\sqrt{3}\text{m/s}$

Q. Find the velocity of the hanging block if the velocities of the free ends of the rope are as indicated in the figure. Take the upward velocity to be positive.

1. $+\frac{3}{2}m/s$
2. $-\frac{3}{2}m/s$
3. $+\frac{1}{2}m/s$
4. $-\frac{1}{2}m/s$

Q. An electric dipole of moment p is placed at the origin along the x-axis. The electric field at a point p , whose position vector makes an angle $\theta$ with the x-axis, will make an angle ..... with the x-axis, where
$tan\alpha =\frac{1}{2}tan\theta$

1. $\alpha$
   
2. $\theta +\alpha$
   
3. $\theta +2\alpha$
4. $\theta$
   

Q. A ball is thrown from the ground to clear a wall $3\text{m}$ high at a distance of $6\text{m}$ and falls $18\text{m}$ away from the wall, the angle of projection of ball with horizontal is

1. ${\tan }^{-1}\left(\frac{3}{2}\right)$
2. ${\tan }^{-1}\left(\frac{2}{3}\right)$
3. ${\tan }^{-1}\left(\frac{1}{2}\right)$
4. ${\tan }^{-1}\left(\frac{3}{4}\right)$

Q. A uniform ring is connected to a light axle with light spokes so as to form a wheel. The wheel is placed on a horizontal surface with its plane vertical and a constant horizontal force $F$ is applied to the axle. Surface AB is rough and surface right of B is smooth. The wheel does not slip when it moves from A to B. The wheel moves from A to B in time $T$.

1. Rotational KE remains constant right of B.
2. Energy is dissipated from A to B.
3. Linear acceleration right of B is more than linear acceleration left of B.
4. The wheel undergoes constant non-zero acceleration to the right of B.

Q. An arrangement of pulley, string and blocks is shown in figure. Given $m_1=4\text{kg},m_2=2\text{kg}$ and $m_3=4\text{kg}$. All surfaces are frictionless, string and pulleys are massless. Find the acceleration of block $m_1$. Assume $g=10{\text{m/s}}^2$.

1. $\frac{36}{11}{\text{m/s}}^2$
2. $\frac{40}{11}{\text{m/s}}^2$
3. $\frac{40}{7}{\text{m/s}}^2$
4. $\frac{36}{7}{\text{m/s}}^2$

Q.

Find acceleration of 3kg block? Given string and pulleys are massless. Pulley and wedge is frictionless and wedge is fixed.

1. 2m/

2. 10m/

3. 4m/

4. 3 m/

Q. A block A of mass $5\text{kg}$ is placed on a table and connected to another block B of mass $3\text{kg}$ with a light string passing over a frictionless pulley. The coefficient of friction between the table the block A is $0.4$. The block B is suddenly dropped and the string breaks after $3$ seconds. What is the total distance covered by block A?

1. $5.625\text{m}$
2. $1.75\text{m}$
3. $7.375\text{m}$
4. $6.5\text{m}$

Q. In the arrangement shown, end $A$ of light inextensible string is pulled up with constant velocity $v$. The velocity of block $B$ is

1. $\frac{v}{2}$
2. $\frac{v}{3}$
3. $v$
4. $3v$

Q. A system is shown in the figure. A man standing on the block is pulling the rope. If the man pulls $2m$ of the rope in each second, then the velocity of the block will be
[Assume that the block does not rotate and rope is inextensible.]

1. $3m/s$
2. $\frac{1}{2}m/s$
3. $1m/s$
4. $2m/s$

Q. Find out the velocity of block $E$ for the arrangement as shown in figure. Consider all pulleys to be light and frictionless. All surfaces are smooth and the string is massless and inextensible.

1. $\frac{21}{2}\text{m/s}$ (upwards)
2. $\frac{31}{2}\text{m/s}$ (downwards)
3. $\frac{31}{2}\text{m/s}$ (upwards)
4. $\frac{21}{2}\text{m/s}$ (downwards)

Q. If block B moves downward with an acceleration of $3m/s^2,$ find the acceleration of block A.

Q.

A block of mass 2 kg slides down the face of a smooth ${45}^{\circ }$ wedge of mass 9kg as shown in figure. The wedge is placed on a frictionless horizontal surface. Determine the acceleration of the wedge.

1. none of these

2. 1 m/.

3. m/.

4. 2 m/.

Q. The figure shows mass $m$ with velocity $U$. Ring is restricted to move on smooth fixed horizontal rod. The velocity of ring at that moment is :

1. $2U$
2. $\frac{2U}{\sqrt{3}}$
3. $\frac{U}{2}$
4. $\frac{\sqrt{3}U}{2}$

Q. Velocity of block B in the given arrangement is $300mm/sec$ towards right. Then velocity of A will be

1. $200mm/sec$
2. $100mm/sec$
3. $450mm/sec$
4. $150mm/sec$

Q. A block of mass $1\text{kg}$ is situated on a rough inclined plane and is connected to a spring of spring constant $100{\text{Nm}}^{-1}$ as shown in figure. The block is released from rest when the spring is unstretched. The block moves through a distance of $10\text{cm}$ down the plane before coming to rest. Assuming that spring and pulley are massless, and the pulley is frictionless. Find the coefficient of friction between the block and the inclined surface.
(Take $g=10{\text{ms}}^{-2}$)

1. $\frac{1}{6}$
2. $\frac{1}{8}$
3. $\frac{1}{5}$
4. $\frac{3}{5}$

Q. Derive following equations for a uniformly accelerated motion where the symbols have their usual meanings: $v=u+at$