Newton's Laws of Motion

Q. Ball $1$ collides head on with another identical ball $2$, initially at rest. Velocity of ball $2$ after collision becomes two times that of ball $1$ after collision. The coefficient of restitution between the two balls is :

1. $e=\frac{1}{3}$
2. $e=\frac{1}{2}$
3. $e=\frac{1}{4}$
4. $e=\frac{2}{3}$

Q. In the figure shown, find the maximum amplitude of the system, so that there is no slipping between any of the blocks.

1. $\frac{2}{7}\text{m}$
2. $\frac{3}{4}\text{m}$
3. $\frac{10}{3}\text{m}$
4. $\frac{4}{9}\text{m}$

Q.

If force is proportional to the square of velocity, then the dimensions of the proportionality constant are

1. ${\text{[ML}}^{-1}\text{T]}$

2. $\left[{\mathrm{ML}}^{-1}{\mathrm{T}}^0\right]$

3. $[{\mathrm{MLT}}^0]$

4. ${\text{[ M}}^0{\mathrm{LT}}^{-1}]$

Q. From a fixed pulley the masses $2\text{kg}$, $1\text{kg}$ and $3\text{kg}$ are suspended as shown in the figure. Find the extension in spring if its stiffness is $k=100\text{N/m}$. Assume the acceleration of $1\text{kg}$ and $3\text{kg}$ to be same.
(Take $g=10{\text{m/s}}^2$)

1. $0.4\text{m}$
2. $0.2\text{m}$
3. $0.3\text{m}$
4. $0.5\text{m}$

Q. A man of mass $75\text{kg}$ is standing on a platform of mass $15\text{kg}$ kept on a smooth horizontal surface. The man starts moving on the platform with a velocity of $20\text{m/s}$ relative to the platform. Then, magnitude of the recoil velocity of the platform is

1. $12.33\text{m/s}$
2. $20\text{m/s}$
3. $25\text{m/s}$
4. $16.66\text{m/s}$

Q. Two balls of masses $m$ and $2m$ and momenta $4p$ and $2p$ (in the directions shown) collide as shown in figure.


During collision, the value of linear impulse between them is $J$. In terms of $J$ and $p$, which of the following is the correct condition for elastic collision.

1. $J=\frac{3}{20}p$
   
2. $J=p$
   
3. $J=\frac{20}{3}p$
   
4. $J=\frac{16}{21}p$
   

Q. A body of mass $10\text{kg}$ moving with velocity of $10\text{m/s}$ hits another body of mass $30\text{kg}$ moving with velocity $3\text{m/s}$ in same direction. The co-efficient of restitution is $\frac{1}{4}$. The velocity of centre of mass after collision will be

1. $20\text{m/s}$
2. $40\text{m/s}$
3. $\frac{19}{4}\text{m/s}$
4. $\frac{23}{4}\text{m/s}$

Q. Consider a two particle system with particles having masses $m_1$ and $m_2$. If the first particle is pushed towards the centre of mass through a distance $d$ by what distance should the second particle is moved, so as to keep the centre of mass at the same position?

1. $\frac{m_1}{m_1+m_2}d$
2. $\frac{m_2}{m_1}d$
3. $\frac{m_1}{m_2}d$
4. $d$

Q. The velocities of two particles $A$ and $B$ of same mass are $V_A=a\hat{i}$ and $V_B=b\hat{j}$, where $a$ and $b$ are constants. The acceleration of particle $A$ is $(2a\hat{i}+4b\hat{j})$ and acceleration of particle $B$ is $(a\hat{i}-b\hat{j})$ (in ${\text{m/s}}^2)$. The centre of mass of two particles will move in

1. Straight line
2. parabola
3. ellipse
4. circle

Q. For a spring block system as shown in figure, a time varying force $F=5t\text{N}$ is applied on mass $2\text{kg}$. After $10\text{seconds}$, velocity of $3\text{kg}$ mass is $30\text{m/s}$. Find the velocity of $2\text{kg}$ mass at this instant.

1. $80\text{m/s}$
2. $\frac{80}{3}\text{m/s}$
3. $20\text{m/s}$
4. $40\text{m/s}$

Q. Two blocks are attached to the pulley as shown in the arrangement.

If the string is inextensible and all surfaces are frictionless, find out the magnitude acceleration of COM of the system. (Take $g=10m/s^2$)

1. $0.4{\text{m/s}}^2$
2. $0.16{\text{m/s}}^2$
3. $4{\text{m/s}}^2$
4. $1.6{\text{m/s}}^2$

Q. In the adjoining figure $m_1=4m_2$. The pulleys are smooth and light. At time t = 0, the system is at rest. If the system is released and if the acceleration of mass m₁ is a, then the acceleration of $m_2$ will

1. 
2. 
3. 
4. 

Q. Assuming all the surfaces to be frictionless, the acceleration of block $C$ shown in the figure is

1. $5{\text{ms}}^{-2}$
2. $7{\text{ms}}^{-2}$
3. $3.5{\text{ms}}^{-2}$
4. $4{\text{ms}}^{-2}$

Q. For the arrangement shown in the figure, $m_1=6\text{kg}$ and $m_2=4\text{kg}$. If the string is light and inextensible and the pulley and surfaces are frictionless, find the magnitude of the acceleration of the center of mass.
(Take $g=10m/s^2$)

1. $2.88{\text{m/s}}^2$
2. $6{\text{m/s}}^2$
3. $4{\text{m/s}}^2$
4. $10{\text{m/s}}^2$

Q. The force on a particle of mass $10\text{g}$ is $(10\hat{i}+5\hat{j})$. If it starts from rest from origin, what would be its position at time $\text{t}=5\text{s}?$

1. $(6250\hat{i}+12500\hat{j})\text{m}$
2. $(-12500\hat{i}+6250\hat{j})\text{m}$
3. $(12500\hat{i}-6250\hat{j})\text{m}$
4. $(12500\hat{i}+6250\hat{j})\text{m}$

Q. A pulley mass arrangement is shown is the figure. A force of $50\text{N}$ is applied on the $2\text{kg}$ block. Find the acceleration of COM of the system.

1. $1.6{\text{m/s}}^2$
2. $2{\text{m/s}}^2$
3. $10{\text{m/s}}^2$
4. $0{\text{m/s}}^2$

Q. In the figure shown below, $m_A=2\text{kg}$ and $m_B=3\text{kg}$. A force $\text{F}=40\text{N}$ is applied on $2\text{kg}$ block as shown. Find the acceleration of COM of the system. (Neglect friction everywhere and assume the string to be inextensible)

1. $1.2{\text{m/s}}^2$
2. $1{\text{m/s}}^2$
3. $2.4{\text{m/s}}^2$
4. $2{\text{m/s}}^2$

Q. For the arrangement shown in the figure, $m_1=6\text{kg}$ and $m_2=4\text{kg}$. If the string is light and inextensible and the pulley and surfaces are frictionless, find the magnitude of the acceleration of the center of mass.
(Take $g=10m/s^2$)

1. $4{\text{m/s}}^2$
2. $10{\text{m/s}}^2$
3. $6{\text{m/s}}^2$
4. $2.88{\text{m/s}}^2$

Q. In the arrangement shown in the figure, the mass of wedge $A$ and that of the block $B$ are $3\text{m}$ and $\text{m}$ respectively. Friction exists between $A$ and $B$ only. The mass of the block $C$ is $m$. The force $F=19.5\text{m}g$ is applied on the block $C$ as shown in the figure. The minimum coefficient of friction $\left(\mu \right)$ between $A$ and $B$ so that $B$ remains stationary with respect to wedge $A$ will be $\frac{1}{x}$ then, find the value of $x$

Q. Two blocks of masses $5\text{kg}$ and $2\text{kg}$ are placed at rest on a frictionless surface and connected by a spring. An external hit gives a velocity of $21\text{m/s}$ to the heavier block towards the lighter one. Velocities of both blocks (heavier and lighter one) in the centre of mass frame, just after the kick will be respectively:
(Consider direction of motion of heavier block as $+ve$ direction)

1. $6\text{m/s},10\text{m/s}$
2. $15\text{m/s},15\text{m/s}$
3. $6\text{m/s},-15\text{m/s}$
4. $3\text{m/s},5\text{m/s}$

Q. A particle of mass $m$ is moving with a constant acceleration $2a$ towards another particle of mass $4m$ as shown in figure. Mass $4m$ is also moving with acceleration $a$ towards the particle of mass $m$. Find the acceleration of $COM$ of the system.

1. $a/5$ (towards left)
2. $2a/5$ (towards right)
3. $2a/5$ (towards left)
4. $a/5$ (towards right)

Q. A uniform chain of length $L$ and mass $M$ overhangs with two third of its length on the table. The coefficient of friction between table and chain is $\mu$. The work done by the friction during the time, the chain slips off the table is

1. $-\frac{2\mu MgL}{9}$
2. $-\frac{\mu MgL}{9}$
3. $-\frac{\mu MgL}{18}$
4. $-\mu MgL$

Q. Each of the two blocks shown in the figure has a mass \(m\). The coefficient of friction for all surfaces in contact is \(\mu\). A horizontal force \(P\) is applied to move the bottom block. The value of \(P\), for which acceleration of block \(A\) is same in both cases is

Q. A magnet of mass $2\text{kg}$ is thrown with a velocity of $5\text{m/s}$ towards a moving metal block of unknown mass. Both magnet and block are moving along a smooth horizontal surface. After the collision, the magnet sticks to the metal block. If the initial momentum of system is $48\text{kg.m/s}$ and the initial velocity of $\text{COM}$ of system is $3\text{m/s}$, then what is the magnitude of initial velocity of the metal block?

1. $2.7\text{m/s}$
2. $3.92\text{m/s}$
3. $1.92\text{m/s}$
4. $4.92\text{m/s}$

Q. A body of mass $m_1$ collides head on elastically with a stationary body of mass $m_2$. If velocities of $m_1$ before and after the collision are $v$ and $–v/3$ respectively then the value of $m_1/m_2$ is

1. $2$
2. $1$
3. $4$
4. $0.5$

Q. A sphere of mass $m$ moving with a constant velocity hits another stationary sphere of the same mass. If $e$ is the coefficient of restitution, the ratio of speed of the first sphere to the speed of the second sphere, after the head on collision will be,

1. $\left(\frac{1-e}{1+e}\right)$
2. $\left(\frac{1+e}{1-e}\right)$
3. $\left(\frac{e+1}{e-1}\right)$
4. $\left(\frac{e-1}{e+1}\right)$

Q. The apparent weight of a person inside a lift is $W_1$ when lift moves up with certain acceleration and is $W_2$ when lift moves down with same acceleration. The weight of person when lift moves up with constant speed is

1. $\frac{W_1-W_2}{2}$
2. ${2W}_1$
3. ${2W}_2$
4. $\frac{W_1+W_2}{2}$

Q. In the system shown, the acceleration of the wedge of mass $5$M is?(there is no friction anywhere)

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

Q. A small roller coaster starts at point $A$ with a speed $u$ on a curved track as shown in the figure.
The friction between the roller coaster and the track is negligible and its always remains in contact with the track. The speed of roller coaster at point $D$ on the track will be:

1. $(u^2+2gh{)}^{1/2}$
2. $(u^2+gh{)}^{1/2}$
3. $(u^2+4gh{)}^{1/2}$
4. $u$

Q. Two masses 5 kg and 3 kg are suspended from the ends of an inextensible light string passing over a frictionless pulley. When the masses are released, force exerted by the string on the pulley is :

1. $8N$
2. $2N$
3. $5N$
4. $65N$