Springs in Parallel and Series

Q.

The displacement of an oscillating particle varies with time (in seconds) according to the equation y (cm) = sin $\frac{\pi }{2}(\frac{t}{2}+\frac{1}{3})$ . The maximum acceleration of the particle is approximately

1. 5.21 cm/s²

2. 3.62 cm/s²

3. 1.81 cm/s²

4. 0.62 cm/s²

Q. integration of e^-2t dt with limits infinity and 0 is ?

Q.

The Volume of air increases by $5$% in its Adiabatic Expansion. The percentage decrease in its pressure will be

1. $6\%$

2. $7\%$

3. $8\%$

4. $9\%$

Q. what is MKS and FPS system of units?

Q.

Two S.H.M's of unequal amplitude with same frequency (act) on a particle at right angle, with a phase difference of $\frac{\pi }{2}$. The resultant motion is:

1. A straight line at 45° to x- axis

2. An ellipse

3. A circle

4. Data is not sufficient

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. What is the difference between d²x/dt² and dx²/dt²?

Q.

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

1. 

2. 

3. 

4. 

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. Two identical springs of spring constant $k$ each are connecetd in series and parallel as shown in figure. A mass $M$ is suspended from them. The ratio of their frequencies of vertical oscillation will be

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

Q.

A smooth wedge of mass m and angle of inclination 60 degrees rests unattached between two springs of spring constant k and 4k, on a smooth horizontal plane, both springs in the un-extended position. The time period of small oscillations of the wedge (assuming that the springs are constrained to get compressed along their length) equals

1. $\pi (1+\frac{1}{2})\sqrt{\frac{m}{k}}$

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

3. $\pi (1+\frac{1}{\sqrt{3}})\sqrt{\frac{m}{k}}$

4. None of above

Q.

Initially the spring is un-deformed. Now the force 'F' is applied to 'B' as shown in the figure, When the displacement of 'B' with respect to 'A' is 'x' towards right in some time then the relative acceleration of 'B' with respect to 'A' at that moment is:

1. none of these

2. 

3. 

4. 

Q. An air chamber of volume V has a neck of cross – sectional area a into which a light ball of mass m can move without friction. The diameter of the ball is equal to that of the neck of the chamber. The ball is pressed down a little and released. If the bulk modulus of air in B, the time period of the resulting oscillation of the ball is given by

1. $T=2\pi \sqrt{\frac{B{a}^2}{mV}}$
2. $T=2\pi \sqrt{\frac{BV}{m{a}^2}}$
3. $T=2\pi \sqrt{\frac{mB}{V{a}^2}}$
4. $T=2\pi \sqrt{\frac{mV}{B{a}^2}}$

Q.

A particle is subjected to two simple harmonic motions given by

$x_1=2.0sin\left(100\pi t\right)$ and $x_2=2.0sin(120\pi t+\frac{\pi }{3})$,

where x is in centimeter and t in second. Find the displacement of the particle at

(a) t = 0.0125,

(b) t = 0.025

1. 241 cm, -0.27 cm
2. -241 cm, 0.27 cm
3. 1.61 cm, -0.71 cm
4. -1.61 cm, -0.71 cm

Q. Three simple harmonic motions of equal amplitudes A and equal time periods in the same direction combine. The phase of the second motion is ${60}^{\circ }$ ahead of the first and the phase of the third motion is ${60}^{\circ }$ ahead of the second. Find the amplitude of the resultant motion.

1. 2A

2. A

3. $\frac{A}{2}$

4. 3A

Q.

The springs shown in the figure are all unstretched in the beginning when a man starts pulling the block. The man exerts a constant force F on the block. Find the amplitude and the frequency of the motion of the block.

1. $\frac{F(k_2+k_3)}{{k_1}^2{k_2}^2+{k_3}^3},\frac{1}{2\pi }\sqrt{\frac{{k_1}^2{k_2}^2+{k_3}^3}{M(k_2+k_3)}}$

2. $\frac{F{k}_2}{k_1{k}_2+k_2{k}_3+k_1+k_3},\frac{1}{2\pi }\sqrt{\frac{k_1{k}_2+k_2{k}_3+k_1+k_3}{M(k_2+k_3)}}$

3. $\frac{F(k_2+k_3)}{k_1{k}_2+k_2{k}_3+k_1+k_3},\frac{1}{2\pi }\sqrt{\frac{k_1{k}_2+k_2{k}_3+k_1+k_3}{M(k_2+k_3)}}$

4. None of these

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.

The given setup is in equilibrium and springs were in natural length before force was applied. If I have to replace the above setup by 1 block and 1 spring then what will be the spring constant of that new spring, such that the block is displaced by the same amount previously by the applied force.

1. 

2. 

3. 

4. 

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. 0

3. 4a

4. 2a

Q. the component of rbar along x axis will have maximum value when the angle between r bar and i^[icap] will be ?

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

1. True
2. False

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

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