Springs in Parallel and Series

Q.

The periodic time of a body executing SHM is $4\mathrm{s}$. After how much interval from time $\mathrm{t}=0$, its displacement will be half of its amplitude?

Q. A trolley of mass m is connected to two identical springs, each of force constant k, as shown in figure. The trolley is displaced from its equilibrium position by a distance x and released. The trolley executes simple harmonic motion of period T. After some time it comes to rest due to friction. The total energy dissipated as heat is (assume the damping force to be weak)

1. $\frac{1}{2}k{x}^2$
2. $k{x}^2$
3. $\frac{2{\pi }^{2}m{x}^2}{r^2}$
4. $\frac{m{x}^2}{r^2}$

Q. A uniform spring has a certain mass suspended from it and its period for vertical oscillations is $T_1$ . The spring is now cut into two equal halves and the same mass is suspended from one of the halves. The period of vertical oscillations now is $T_2$ . The ratio $\frac{T_2}{T_1}$ is

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

Q. A spring of force constant k is cut into lengths of ratio $1:2:3.$ They are connected in series and the new force constant is $k^{\prime }$. Then they are connected in parallel and force constant is $k^{\prime \prime }$. The ratio $k^{\prime }:k^{\prime \prime }$ is

1. $1:19$
2. $1:11$
3. $1:14$
4. $1:16$

Q. One end of a long metallic wire of length $L$ is tied to the ceiling. The other end is tied to a massless spring of spring constant $k$. A mass $m$ hangs freely from the free end of the spring. The area of cross-section and Young's modulus of the wire are $A$ and $Y$ respectively. If the mass is slightly pulled down and released, it will oscillate with a time period $T$ equal to
[Assume that , within the elastic limit, wire behaves as a spring]

1. $2\pi \left(\frac{m}{k}\right)$
2. $2\pi \sqrt{\frac{(YA+kl)m}{YAk}}$
3. $2\pi \frac{mYA}{kL}$
4. $2\pi \frac{mL}{YA}$

Q. Figure shows a smooth horizontal table in $‘x-y^{\prime }$ plane between two identical fixed walls. Two springs are connected to the small ball. The length of the springs in the free state is $l$. The ball is shifted slightly from the equilibrium position in two different ways, once along the axis $OX$ and second along the y-axis and it begins to perform vibrations. The time period for these motions is (Assume vibrations to be small)

1. motion along x-axis is simple harmonic
2. motion along y-axis is simple harmonic
3. motion along y-axis is not simple harmonic
4. $T_x=2\pi \sqrt{\frac{m}{2k}}$

Q. A spring of force constant k is cut into two equal halves. The force constant of each half is

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

Q.

A particle of mass m is attached to three springs A, B and C of equal force constants k as shown in figure. if the particle is pushed slightly against the spring C and released, find the time period of oscillation.

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

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

4. None of these

Q. The springs in fig. $A$ and $B$ are identical but the length of spring $A$ is three times the length of each spring in $B$. The ratio of period $T_A/T_B$ is

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

Q.

Three S.H.M's of equal amplitude A and equal time period in the same direction combine. The difference in phase between each pair is ${60}^{\circ }$ ahead of the other. The amplitude of the resultant oscillation is :

1. a

2. 2a

3. 0

4. 4a

Q.

A simple harmonic motion of a spring-block system has an amplitude A & time period T. Consider x=0 for equilibrium position. The time required by the block to travel from $x=Atox=\frac{A}{2}$ is -

1. $\frac{T}{6}$

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

3. $\frac{T}{3}$

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

Q. Four massless springs whose force constants are $2k,2k,k$ and $2k$, respectively, are attached to a mass $M$ kept on a frictionless horizontal plane (as shown in figure). If the other ends of the springs are fixed and the mass $M$ is displaced , then the frequency of oscillation of the system is

1. $\frac{1}{2\pi }\sqrt{\frac{k}{4M}}$
2. $\frac{1}{2\pi }\sqrt{\frac{4k}{M}}$
3. $\frac{1}{2\pi }\sqrt{\frac{k}{7M}}$
4. $\frac{1}{2\pi }\sqrt{\frac{7k}{M}}$

Q. If a spring of stiffness ${}^{\prime }{K}^{\prime }$ is cut into two parts ${}^{\prime }{A}^{\prime }$ and ${}^{\prime }{B}^{\prime }$ of length $l_A=l_B=2:3$, then the stiffness of spring ${}^{\prime }{A}^{\prime }$ is given by:

1. $\frac{3K}{2}$
2. $\frac{2K}{5}$
3. $K$
4. $\frac{5K}{2}$

Q. In the given figure, the block is displaced slightly and released. Then, the time period of oscillation is:

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

Q.

A simple harmonic motion of a spring-block system has an amplitude A & time period T. Consider x=0 for equilibrium position. The time required by the block to travel from $x=Atox=\frac{A}{2}$ is -

1. $\frac{T}{6}$

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

3. $\frac{T}{3}$

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

Q. A block $M$ shown in the figure oscillates in a 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. The resultant spring constant of two identical springs attached in series is lower than that of an individual spring.

1. True
2. False

Q.

Two sine waves travel in the same direction in a medium. The amplitude of each wave is A and the phase difference between the two waves is${120}^0$. The resultant amplitude will be

1. A

2. 2A

3. 4A

4. $\sqrt{2A}$

Q. The resultant spring constant of two identical springs attached in series is lower than that of an individual spring.

1. True
2. False

Q.

A particle of mass m is attached to three springs A, B and C of equal force constants k as shown in figure. if the particle is pushed slightly against the spring C and released, find the time period of oscillation.

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

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

4. None of these

Q. Four massless springs whose force constants are $2k,2k,k$ and $2k$, respectively, are attached to a mass $M$ kept on a frictionless horizontal plane (as shown in figure). If the other ends of the springs are fixed and the mass $M$ is displaced , then the frequency of oscillation of the system is

1. $\frac{1}{2\pi }\sqrt{\frac{k}{4M}}$
2. $\frac{1}{2\pi }\sqrt{\frac{4k}{M}}$
3. $\frac{1}{2\pi }\sqrt{\frac{k}{7M}}$
4. $\frac{1}{2\pi }\sqrt{\frac{7k}{M}}$

Q.

A particle is subjected to two simple harmonic motions, one along the X-axis and the other on a line making an angle of ${45}^{\circ }$ with the X-axis. The two motions are given by

$x=x_{0}sin\omega t$ and $s=s_{0}sin\omega t$

Find the amplitude of the resultant motion.

1. $\sqrt{x_0^2+s_0^2+\sqrt{2}{x}_0{s}_0}$

2. $x_0+s_0$

3. $\frac{x_0+s_0}{2}$

4. None of these