Spring Block's Time Period

Q. A particle of mass $m$ is attached with four identical springs, each of length $l.$ Initially, each spring has tension $F_0$. Neglecting gravity, the time period for small oscillations of the particle along a line perpendicular to the plane of the figure is given by $T=2\pi \sqrt{\frac{ml}{y{F}_0}}$. Find the value of $y$.

Q.

A device is designed in such a way that it displays green light whenever its speed is increasing and a blue light when the speed is decreasing. If the device is performing SHM with a time period of 12s and it starts from the extreme position, when is the device green?

1. at t = 1 seconds

2. at t = 3.2 seconds

3. at t = 7 seconds

4. at t = 11 seconds

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. A particle is executing S.H.M. After 2 seconds of crossing the equilibrium position it is at a distance of $\frac{\sqrt{3}}{2}$ of its amplitude. Find its time period?

1. 3 s
2. 4 s
3. 12 s
4. 2 s

Q.

A mass M is suspended from a spring of negligible mass. The spring is pulled a little and then released so that the mass executes SHM of time period T. If the mass is increased by m, the time period becomes$\frac{5T}{3}$ . Then the ratio of $\frac{m}{M}$ is:

1. $\frac{3}{5}$
   

2. $\frac{25}{9}$
   

3. $\frac{16}{9}$
   

4. $\frac{5}{3}$

Q.

A block P of mass m is placed on a frictionless horizontal surface. Another block Q of same mass is kept on P and connected to the wall with the help of a spring of spring constant k as shown in the figure. ${\mu }_s$ is the coefficient of friction between P and Q. The blocks move together performing SHM of amplitude A. The maximum value of the friction force between P and Q is:

1. $\frac{kA}{2}$
   

2. ${\mu }_{s}mg$

3. kA

4. Zero
   

Q.

A horizontal platform is executing simple harmonic motion in the vertical direction with a frequency $v$. A block of mass $m$ is placed on the platform. What is the maximum amplitude of the platform so that the block is not detached from it?

1. $\frac{g}{4{\pi }^2{v}^2}$

2. $\frac{g}{2{\pi }^2{v}^2}$

3. $\frac{mg}{2{\pi }^2{v}^2}$

4. $\frac{mg}{4{\pi }^2{v}^2}$

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. 40 kg

2. 20 kg

3. 30 kg

4. 10 kg

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. $\frac{f}{2}$

3. 2f
   

4. 4f
   

Q.

Two springs A and B have force constants $k_1$ and $k_2$ respectively. The ratio of work done on A to that done on B when they are stretched by the same force is

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

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

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

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