Spring Block's Time Period

Q. Find the time period of small oscillations for the block of mass $m$ as shown in the figure. [All springs are identical]

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

Q. A $10\text{kg}$ metal block is attached to a spring of spring constant $1000{\text{Nm}}^{-1}$. The block is displaced from equilibrium position by $10\text{cm}$ and released. The maximum acceleration of the block is

1. $10{\text{ms}}^{-2}$
2. $100{\text{ms}}^{-2}$
3. $200{\text{ms}}^{-2}$
4. $0.1{\text{ms}}^{-2}$

Q. A mass $m$ is suspended from a spring of spring constant $K$ and just touches another identical spring fixed to the floor as shown in the figure. The time period of small oscillations is

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

Q. A copper sphere attached to the bottom of a vertical spring is oscillating with time period $10\text{s}$. If the copper sphere is immersed in a fluid (assume the viscosity of the fluid is negligible) of specific gravity $\frac{1}{4}$ of that of the copper, then time period of the oscillation is

1. $5\text{s}$
2. $10\text{s}$
3. $2.5\text{s}$
4. $20\text{s}$

Q. The potential energy of a particle of mass $1\text{kg}$ moving along the $X$-axis is given by $U=4(1+\cos 2x)\text{J},$ where $x$ is in meters. Find the time period of small oscillation (in second).

1. $2\pi$
2. $\pi$
3. $\frac{\pi }{2}$
4. $\sqrt{2}x$

Q. All the surfaces shown in the figure are frictionless. The mass of the inclined wedge is $M$, that of the block is $m$ and spring constant of the spring is $k$. Initially, the block and wedge are at rest. The spring is stretched by a small length and then released. Find the time period of oscillation of the system.

1. $2\pi \sqrt{\frac{M}{k}}$
2. $2\pi \sqrt{\frac{M}{k}}$
3. $2\pi \sqrt{\frac{m+M}{2k}}$
4. $2\pi \sqrt{\frac{Mm}{k(m+M)}}$

Q. Molten - wax of mass $m$ drops on a block of mass $M$, which is oscillating on a frictionless table as shown.
Select the incorrect option.

1. If the collision takes place at extreme position, amplitude does not change
2. If the collision takes place at mean position, amplitude decreases
3. If the collision takes place at mean position, time period decreases
4. If the collision takes place at extreme position, time period increases

Q. A block of mass $10\text{kg}$ attached to a horizontal spring with force constant $250\text{N/m}$ is moving with simple harmonic motion having amplitude $10\text{m}$. At the instant when the block passes through its equilibrium position, a lump of putty with mass $5\text{kg}$ is dropped vertically on the block from a very small height and sticks to it. Find the new amplitude and period?

1. $8.1\text{m},1.54\text{s}$
2. $10\text{m},1.54\text{s}$
3. $8.1\text{m},1.25\text{s}$
4. $12.2\text{m},1.54\text{s}$

Q. A particle of mass 200g executes simple harmonic motion. A restoring force is provided by a spring of spring constant 80N/m. Find the time period.

1. 0.31 s
2. 0.52 s
3. 0.1 s
4. 0.9 s

Q. In a gravity free space, a particle is connected with four springs as shown below. If the particle is pulled down gently and left, then find the period of oscillation of the particle.
$(\text{Given}:\theta ={45}^{\circ },\beta ={30}^{\circ })$

1. $T=2\pi \sqrt{\frac{m}{4k}}$
2. $T=2\pi \sqrt{\frac{m}{6k}}$
3. $T=2\pi \sqrt{\frac{3m}{4k}}$
4. $T=2\pi \sqrt{\frac{m}{2k}}$

Q.

A tray of mass $M=10kg$ is supported on two identical springs, each of spring constant k, as shown in the figure. When the tray is depressed a little and released, it executes simple harmonic motion of period 1.5 s. When a block of mass m is placed on the tray, the period of oscillation becomes 3.0 s. The value of m is

1. 10 kg

2. 20 kg

3. 30 kg

4. 40 kg

Q. A mass $m$ is suspended from a spring of spring constant $K$ and just touches another identical spring fixed to the floor as shown in the figure. The time period of small oscillations is

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

Q.

Two springs of force constants $K_1$ and $K_2$ are connected to a mass m as shown. The frequency of oscillation of the mass is f. If both $K_1$ and $K_2$ are made four times their original values, the frequency of oscillation becomes:

1. $\frac{f}{4}$
   

2. 4f
   

3. 2f
   

4. $\frac{f}{2}$

Q. Two identical masses are connected to identical springs of spring constant $K$ as shown in the figure. Find the time period of small oscillations.

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

Q. A particle of mass $200\text{g}$ executes simple harmonic motion. The restoring force is provided by a spring of spring constant $80{\text{Nm}}^{-1}$. What would be the time period?

1. $0.93\text{s}$
2. $0.62\text{s}$
3. $0.31\text{s}$
4. None of these

Q. A mass of $0.2\text{kg}$ is attached to the lower end of a massless spring of force constant $200{\text{Nm}}^{-1}$, the upper end of which is fixed to a rigid support. Which of the following statements is true?

1. In equilibrium, the spring will be stretched by $1\text{cm}$
2. If the body is raised till the spring attains its natural length and then released, it will go down by $2\text{cm}$ before moving upwards
3. The frequency of oscillation will be nearly $50\text{Hz}$
4. All of the above

Q. A body at the end of a spring executes S.H.M. with a period $t_1,$ while the corresponding period for another spring is $t_2.$ If the period of oscillation with the two springs in series is $T$, then

1. $T=t_1+t_2$
2. $T^2=t_1^2+t_2^2$
3. $\frac{1}{T}=\frac{1}{t_1}+\frac{1}{t_2}$
4. $\frac{1}{T^2}=\frac{1}{t_1^2}+\frac{1}{t_2^2}$

Q. In spring-block system match the following columns.

$\begin{array}{cc}\text{Column I}& \text{Column II}\\ \text{(A) If}k\text{(the spring constant) is made 4 times}& \text{(p) Angular speed will become 16 times}\\ \text{(B) If}m\text{(the mass of block) is made 4 times}& \text{(q) Potential energy will become 4 times}\\ \text{(C) If}k\text{and}m\text{both are made 4-times}& \text{(r) Kinetic energy will remain unchanged}\\ & \text{(s) None}\end{array}$

1. $A\to s,B\to q,C\to q$
2. $A\to q,B\to r,C\to q$
3. $A\to q,B\to s,C\to r$
4. $A\to r,B\to p,C\to s$

Q. A body of mass $m$ is attached to the lower end of a spring whose upper end is fixed. The spring has negligible mass. When the mass $m$ is slightly pulled down and released, it oscillates with a time period of $3s$. When the mass $m$ is increased by $1kg$, the time period of oscillation becomes $5s$. The value of $m$ in $kg$ is

1. $\frac{9}{16}$
2. $\frac{3}{4}$
3. $\frac{4}{3}$
4. $\frac{16}{9}$

Q. A body of mass $m$ is attached to the lower end of a spring whose upper end is fixed. The spring has negligible mass. When the mass $m$ is slightly pulled down and released, it oscillates with a time period of $3\text{s}$. When the mass $m$ is increased by $1\text{kg}$, the time period of oscillations becomes $5\text{s}$. The value of $m$ in $\text{kg}$ is

1. $\frac{3}{4}$
2. $\frac{4}{3}$
3. $\frac{16}{9}$
4. $\frac{9}{16}$

Q. A spring having a spring constant $1200{\text{Nm}}^{-1}$ is mounted on a horizontal table as shown. A mass of $3.0\text{kg}$ is attached to the free end of the spring. The mass is then pulled sideways to a distance of $2\text{cm}$ and released. Determine
(i) The frequency of oscillations.
(ii) The maximum acceleration of the mass, and
(iii) the maximum speed of the mass?

1. $3.18\text{Hz},8{\text{m/s}}^2,0.40\text{m/s}$
2. $1.57\text{Hz},16{\text{m/s}}^2,0.80\text{m/s}$
3. $3.18\text{Hz},32{\text{m/s}}^2,1.6\text{m/s}$.
4. $3.18\text{Hz},16{\text{m/s}}^2,0.80\text{m/s}$

Q. A block of mass $20\text{kg}$ attached to a horizontal spring with force constant $250\text{N/m}$ is moving with simple harmonic motion having amplitude $10\text{m}$. At the instant when the block is at extreme position, a lump of putty with mass $5\text{kg}$ is dropped vertically on the block from a very small height and sticks to it. Find the new amplitude and time period.

1. $1.78\text{m},2\text{s}$
2. $8.9\text{m},2\text{s}$
3. $11.18\text{m},2\text{s}$
4. $16\text{m},4\text{s}$

Q. Find the time period of small oscillations for the block of mass $m$ as shown in the figure. [All springs are identical]

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

Q. A spring block system (mass = $m$, spring constant $k$) is placed on a smooth inclined plane ($\theta$= angle of inclination). The plane is accelerated horizontally with a such that the block does not loose contact with the plane. The time period of small oscillation of the block is

1. $2\pi \sqrt{\frac{m}{k}}$
2. $2\pi \sqrt{\frac{msin\theta }{k}}$
3. $2\pi \sqrt{\frac{mg}{ka}}$
4. none of the above

Q.

A mass attached to a spring executes SHM

Match the following conditions.

(i) $v_x>0,a_x>0\left(I\right)x>0$

(ii) $v_x<0,a_x>0$

(iii) $v_x>0,a_x<0\left(II\right)x<0$

(iv) $v_x<0,a_x<0$

1. (i) - II; (ii) - I; (iii) - I; (iv) - I

2. (i) - II; (ii) - II; (iii) - I; (iv) - I

3. (i) - I; (ii) - II; (iii) - I; (iv) - I

4. (i) - I; (ii) - I; (iii) - II; (iv) - II

Q.

The mass M shown in the figure oscillates in simple harmonic motion with amplitude A. The amplitude of the point P is

1. $\frac{k_{1}A}{k_2}$
   

2. $\frac{k_{2}A}{k_1}$
   

3. $\frac{k_{1}A}{k_1+k_2}$
   

4. $\frac{k_{2}A}{k_1+k_2}$

Q. In a gravity free space, a particle is connected with four springs as shown below. If the particle is pulled down gently and left, then find the period of oscillation of the particle.
$(\text{Given}:\theta ={45}^{\circ },\beta ={30}^{\circ })$

1. $T=2\pi \sqrt{\frac{m}{4k}}$
2. $T=2\pi \sqrt{\frac{m}{6k}}$
3. $T=2\pi \sqrt{\frac{3m}{4k}}$
4. $T=2\pi \sqrt{\frac{m}{2k}}$

Q. The potential energy of a particle of mass $1\text{kg}$ moving along the $X$-axis is given by $U=4(1+\cos 2x)\text{J},$ where $x$ is in meters. Find the time period of small oscillation (in second).

1. $2\pi$
2. $\pi$
3. $\frac{\pi }{2}$
4. $\sqrt{2}x$

Q. A body of mass $m$ is attached to the lower end of a spring whose upper end is fixed. The spring has negligible mass. When the mass $m$ is slightly pulled down and released, it oscillates with a time period of $3\text{s}$. When the mass $m$ is increased by $1\text{kg}$, the time period of oscillations becomes $5\text{s}$. The value of $m$ in $\text{kg}$ is

1. $\frac{3}{4}$
2. $\frac{4}{3}$
3. $\frac{16}{9}$
4. $\frac{9}{16}$

Q. Some springs are combined in series and parallel arrangement as shown in the figure and a mass $m$ is suspended from them. The ratio of their frequencies will be

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