SHM in Vertical Spring Block

Q.

In the given figure, a mass $M$ is attached to a horizontal spring which is fixed on one side to a rigid support. The spring constant of the spring is $k$. The mass oscillates on a frictionless surface with time period $T$ and amplitude $A$. When the mass is in an equilibrium position, as shown in the figure, another mass $m$ is gently fixed upon it. The new amplitude of oscillation will be:

1. $A\sqrt{\frac{M}{M+m}}$

2. $A\sqrt{\frac{M}{M-m}}$

3. $A\sqrt{\frac{M-m}{M}}$

4. $A\sqrt{\frac{M+m}{M}}$

Q.

Two bodies A and B of equal mass are suspended from two separate massless springs of spring constant $k_{1}and{k}_2$ respectively. If the bodies oscillate vertically such that their maximum velocities are equal, the ratio of the amplitude of A to that of B is

1. $\frac{k_1}{k_2}$

2. $\sqrt{\frac{k_1}{k_2}}$

3. $\frac{k_2}{k_1}$

4. $\sqrt{\frac{k_2}{k_1}}$

Q. Two masses $m$ and $2m$ are suspended together by a massless spring of force constant $K$. When the masses are in equilibrium, mass $m$ is removed without disturbing the system. Then, the amplitude of vibration of the system is:

1. $\frac{2mg}{K}$
2. $\frac{mg}{K}$
3. $\frac{3mg}{K}$
4. $\frac{mg}{2K}$

Q. In the system shown in the figure, the string, springs and pulley are weightless. The force constants of the two springs are $k_1=k$ and $k_2=2k$. Block of mass $M$ is pulled vertically down from its equilibrium position and released. Calculate the angular frequency of oscillation.
[Assume the top surface of the block (represented by line $AB$) always remains horizontal]

1. $\omega =\sqrt{\frac{4k}{3M}}$
2. $\omega =\sqrt{\frac{8k}{3M}}$
3. $\omega =\sqrt{\frac{2k}{3M}}$
4. $\omega =\sqrt{\frac{k}{3M}}$

Q. A heavy metal sphere is suspended from a spring. When pulled down a little and released, it oscillates up and down with a frequency f. If it is taken to the moon (where the acceleration due to gravity is one – sixth that on earth) the frequency of vertical oscillation will

1. Remain unchanged
2. become 6f
3. become $\frac{f}{6}$
4. become $\sqrt{6}f$

Q. The coefficient of friction between the blocks of mass $m$ and $2m$ is $\mu =2\tan \theta$. There is no friction between the mass $2m$ and inclined plane. The maximum amplitude of the two block system for which there is no relative motion between both the blocks is:

1. $\frac{mg\sin \theta }{K}$
2. $\frac{3mg\sin \theta }{K}$
3. $g\sin \theta \sqrt{\frac{K}{m}}$
4. $\frac{2mg\sin \theta }{K}$

Q. A horizontal platform with an object placed on it is executing SHM in the vertical direction. The amplitude of oscillation is 2.5 cm. what must be the least period of these oscillations so that the object is not detached from the platform? Take g = 10 $m{s}^{-2}$

1. $0.1\pi sec$
2. $0.5\pi sec$
3. $\pi$
4. $2\pi sec$

Q. A body of mass $m$ hangs from a smooth fixed pulley $P_1$ by an inextensible string fitted with the springs of stiffness constants $k_1$ and $k_3$ The string passes over a smooth light pulley $P_2$ , which is connected with another ideal spring of stiffness constant $k_2$. Find the period of small oscillations of the body.

1. $T=2\pi {\left(m(\frac{2}{k_1}+\frac{4}{k_2}+\frac{2}{k_3})\right)}^{\frac{1}{2}}$
2. $T=2\pi {\left(m(\frac{1}{k_1}+\frac{4}{k_2}+\frac{1}{k_3})\right)}^{\frac{1}{2}}$
3. $T=2\pi {\left(m(\frac{2}{k_1}+\frac{2}{k_2}+\frac{2}{k_3})\right)}^{\frac{1}{2}}$
4. $T=2\pi {\left(m(\frac{1}{k_1}+\frac{1}{k_2}+\frac{1}{k_3})\right)}^{\frac{1}{2}}$

Q.

An ideal spring with spring-constant k is hung from the ceiling and a block of mass M is attached to its lower end. The mass is released suddenly with the spring initially unstretched. Then the maximum extension in the spring is

1. $\frac{4Mg}{k}$

2. $\frac{2Mg}{k}$

3. $\frac{Mg}{k}$

4. $\frac{Mg}{2k}$

Q. In the system shown in the figure, the string, springs and pulley are weightless. The force constants of the two springs are $k_1=k$ and $k_2=2k$. Block of mass $M$ is pulled vertically down from its equilibrium position and released. Calculate the angular frequency of oscillation.
[Assume the top surface of the block (represented by line $AB$) always remains horizontal]

1. $\omega =\sqrt{\frac{4k}{3M}}$
2. $\omega =\sqrt{\frac{8k}{3M}}$
3. $\omega =\sqrt{\frac{2k}{3M}}$
4. $\omega =\sqrt{\frac{k}{3M}}$

Q. Find the time period of the spring mass system shown in the figure below.

1. $T=2\pi \sqrt{\frac{m}{2K}}$
2. $T=2\pi \sqrt{\frac{m}{K}}$
3. $T=2\pi \sqrt{\frac{2m}{K}}$
4. $T=2\pi \sqrt{\frac{3m}{2K}}$

Q.

A block is resting on a piston which is moving vertically with SHM of period 1 s. What is the minimum amplitude to be given to it so that the block and piston can separate?

1. 0.2 m

2. 0.25 m

3. 0.3 m

4. 0.35 m

Q. When a body of mass $1.0\text{kg}$ is suspended from a certain light spring hanging vertically, its length increases by $5\text{cm}$. If by replacing the body of mass $1.0\text{kg}$ and suspending $2.0\text{kg}$ block to the spring, the block is pulled through $10\text{cm}$ from its new equilibrium position and released, the maximum velocity achieved by the block is

1. $0.5\text{m/s}$
2. $1\text{m/s}$
3. $2\text{m/s}$
4. $4\text{m/s}$

Q. Two springs of equal lengths and equal cross – sectional areas are made of material whose Young’s modulii are in the ratio of 2 : 3. They are suspended and loaded with the same mass. When stretched and released, they will oscillate with time periods in the ratio of

1. $\sqrt{3}:\sqrt{2}$
2. 3: 2
3. $3\sqrt{3}:2\sqrt{2}$
4. 9 : 4

Q.

Youngs modulus of a perfectly rigid body is

1. zero

2. unity

3. finite and low

4. infinite

Q.

A spring is loaded with two blocks $m_1$ and $m_2$ where $m_1$ is rigidly fixed with the spring and $m_2$ is just kept on the block $m_1$ as shown in the figure. The maximum energy of oscillation that is possible for the system having the block $m_2$ in contact with $m_1$ is

1. $\frac{m_1^2{g}^2}{2k}$

2. $\frac{m_2^2{g}^2}{2k}$

3. $\frac{(m_1+m_2{)}^2{g}^2}{2k}$

4. $\frac{(m_1+m_2{)}^2{g}^2}{k}$

Q. Two ideal springs of different lengths have been suspended from points $A$ and $B$ such that their free ends $C$ and $D$ are at the same horizontal level. A massless rod $PQ$ is attached to the ends of the springs. A block of mass $m$ is attached to the rod at point $R$. The rod remains horizontal in equilibrium. Now, the block is pulled down and released. It performs vertical oscillations with time period $T=2\pi \sqrt{\frac{m}{3k}}$ where $k$ is the force constant of the longer spring.


Find the ratio of lengths $RC$ and $RD$.

1. $3:1$
2. $3:2$
3. $2:1$
4. $4:3$

Q. A mass M attached to a light spring oscillates with a period of 2 seconds. If the mass is increased by 2 kg the period increases by 1 second. What is the value of M?

1. 1.0 kg
2. 1.6 kg
3. 2.0 kg
4. 2.4 kg

Q.
A spring block system with mass M and spring constant k is suspended vertically and left to oscillate. It has a natural frequency fo f1. Another spring block system with same mass and spring constant is kept on a horizontal smooth surface.Its natural frequency is observed to be f2.
What do you think will be the relation between f1 and f2.

1. f1>f2
2. f1<f2
3. f1=f2
4. Data Insufficient

Q.

A block of mass m moves with a speed v towards the right block in equilibrium with a spring. If the surface is frictionless and collistions are elastic, the frequency of collisions between the masses will be :

1. $\frac{v}{2L}+\frac{1}{\pi }\sqrt{\frac{K}{m}}$

2. 2 $[\frac{v}{2L}+\frac{1}{\pi }\sqrt{\frac{K}{m}}]$

3. $\frac{2}{[\frac{2L}{v}+\frac{1}{\pi }\sqrt{\frac{K}{m}}]}$

4. $\frac{4}{[\frac{2L}{v}+\frac{1}{\pi }\sqrt{\frac{K}{m}}]}$

Q.

One end of a spring of force constant k is fixed to a vertical wall and the other to a block of mass m resting on a smooth horizontal surface. There is another wall at a distance $x_0$ from the block. The spring is then compressed by $2x_0$ and released. The time taken to strike the wall is

1. $\frac{1}{6}\pi \sqrt{\frac{k}{m}}$

2. $\sqrt{\frac{k}{m}}$

3. $\frac{2\pi }{3}\sqrt{\frac{m}{k}}$

4. $\frac{\pi }{4}\sqrt{\frac{k}{m}}$

Q. A block of mass $m$ suspended from a spring executes vertical SHM of time period $T$ as shown in the figure. The amplitude of the SHM is $A$ and the spring is never in compressed state during the oscillation. The magnitude of minimum force exerted by the spring on the block is

1. $mg-\frac{4{\pi }^2}{T^2}mA$
2. $mg+\frac{4{\pi }^2}{T^2}mA$
3. $mg-\frac{{\pi }^2}{T^2}mA$
4. $mg+\frac{{\pi }^2}{T^2}mA$

Q. Two springs of equal lengths and equal cross – sectional areas are made of material whose Young’s modulii are in the ratio of 2 : 3. They are suspended and loaded with the same mass. When stretched and released, they will oscillate with time periods in the ratio of

1. $\sqrt{3}:\sqrt{2}$
2. 3: 2
3. $3\sqrt{3}:2\sqrt{2}$
4. 9 : 4

Q. A spring has a natural length of 50 cm and a force constant of $2.0\times {10}^{3}N{m}^{-1}$ . A body of mass 10 kg is suspended from it and the spring is stretched. If the body is pulled down further stretching the spring to a length of 58 cm and released, it executes simple harmonic motion. What is the net force on the body when it is at its lowermost position of its oscillation? Take $g=10m{s}^{-1}$

1. 20 N
2. 40 N
3. 60 N
4. 80 N

Q.

A spring is loaded with two blocks $m_1$ and $m_2$ where $m_1$ is rigidly fixed with the spring and $m_2$ is just kept on the block $m_1$ as shown in the figure. The maximum energy of oscillation that is possible for the system having the block $m_2$ in contact with $m_1$ is

1. $\frac{m_1^2{g}^2}{2k}$

2. $\frac{m_2^2{g}^2}{2k}$

3. $\frac{(m_1+m_2{)}^2{g}^2}{2k}$

4. $\frac{(m_1+m_2{)}^2{g}^2}{k}$

Q. A spring of negligible mass having a force constant k extends by an amount y when a mass m is hung from it. The mass is pulled down a little and released. The system begins to execute simple harmonic motion of amplitude A and angular frequency $\omega$. The total energy of the mass - spring system will be

1. $\frac{1}{2}m{A}^2{\omega }^2$
2. $\frac{1}{2}m{A}^2{\omega }^2+\frac{1}{2}k{y}^2$
3. $\frac{1}{2}k{y}^2$
4. $\frac{1}{2}m{A}^2{\omega }^2-\frac{1}{2}k{y}^2$

Q. If the mass of the pulleys shown in figure is very very small and the cord is inextensible, the angular frequency of oscillation of the system is

1. $\sqrt{\frac{k_a+k_b}{m}}$
2. $\sqrt{\frac{k_a{k}_b}{(k_a+k_b)m}}$
3. $\sqrt{\frac{k_a{k}_b}{4m(k_a+k_b)}}$
4. $\sqrt{\frac{4k_a{k}_b}{(k_a+k_b)m}}$

Q. A box $B$ of mass $M$ hangs from an ideal spring of force constant $k$. A small particle $A$ also of mass $M$ is stuck to the ceiling of the box and the system is in equilibrium. The particle gets detached from the ceiling and falls to strike the floor of the box. It takes time $t$ for the particle to hit the floor after it gets detached from the ceiling. The particle on hitting the floor sticks to it and the system thereafter oscillates with a time period of $T$. Find the height $H$ of the box if it is given that $t=\frac{T}{6}$. Assume that the floor and ceiling of the box always remain horizontal.

1. $\frac{Mg}{k}[1+\frac{{\pi }^2}{3}]$
2. $\frac{Mg}{k}[1+\frac{{\pi }^2}{9}]$
3. $\frac{Mg}{2k}[1+\frac{{\pi }^2}{3}]$
4. $\frac{Mg}{2k}[1+\frac{{\pi }^2}{9}]$

Q. In the figure shown, a block $\text{A}$ of mass $m$ is rigidly attached to a light spring of stiffness $k$ and suspended from a fixed support. Another block $\text{B}$ of same mass is just placed on it and the blocks are in equilibrium. Suddenly, the block $\text{B}$ is removed. Choose the correct option(s) afterward.

1. Block $\text{A}$ will start SHM.
2. Amplitude of oscillation of the block $\text{A}$ is $\frac{mg}{k}$.
3. Maximum speed acquired by the block $\text{A}$ is $\sqrt{\frac{m{g}^2}{k}}$.
4. Period of oscillation is $2\pi \sqrt{\frac{m}{2k}}$.

Q. A box $B$ of mass $M$ hangs from an ideal spring of force constant $k$. A small particle $A$ also of mass $M$ is stuck to the ceiling of the box and the system is in equilibrium. The particle gets detached from the ceiling and falls to strike the floor of the box. It takes time $t$ for the particle to hit the floor after it gets detached from the ceiling. The particle on hitting the floor sticks to it and the system thereafter oscillates with a time period of $T$. Find the height $H$ of the box if it is given that $t=\frac{T}{6}$. Assume that the floor and ceiling of the box always remain horizontal.

1. $\frac{Mg}{k}[1+\frac{{\pi }^2}{3}]$
2. $\frac{Mg}{k}[1+\frac{{\pi }^2}{9}]$
3. $\frac{Mg}{2k}[1+\frac{{\pi }^2}{3}]$
4. $\frac{Mg}{2k}[1+\frac{{\pi }^2}{9}]$