Oscillations Test

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Oscillations Test - 55

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Weekly Quiz Competition

Question 1
1 / -0

An elastic ball of density $$d$$ is released and it falls through a height $$h$$ before striking the surface of liquid of density $$\rho(d < \rho)$$. The motion of ball is:

A
Periodic

B
S.H.M.

C
Circular

D
Parabolic

Solution

The density of the ball is less than the density of the liquid. So it will float in the liquid and a buoyant force will act on the body which will decelerate the body and after going some deep inside the liquid the buoyant force will be greater than the gravitational force and the body will start to go up. In this way it will execute periodic motion.
Question 2
1 / -0

The motion of a particle is given by
$$y=4\sin\omega t+8\sin \left(\omega t+\dfrac{\pi}{3}\right)$$
The motion of particle is:

A
S.H.M.

B
Not S.H.M.

C
Periodic but not S.H.M.

D
None of the above

Solution
Question 3
1 / -0

The displacement of a particle in simple harmonic motion in one time period is

A
$$A$$

B
$$2A$$

C
$$4A$$

D
zero

Solution

During a Simple harmonic motion, in one oscillation the final and initial point of particle is at same position . Thus there is no displacement of particle .
Option D is correct.
Question 4
1 / -0

There is a clock which gives correct time at $$20^o$$C is subjected to $$40^o$$C. The coefficient of linear expansion of the pendulum is $$12\times 10^{-6}$$ per $$^oC$$, how much is gain or loss in time?

A
$$10.3$$ sec/day

B
$$19$$ sec/day

C
$$5.5$$ sec/day

D
$$6.8$$ sec/day

Solution
Question 5
1 / -0

A spring-mass system oscillation in a car. if the car accelerates on a horizontal road, the frequency of oscillation will

A
increase

B
decrease

C
remain same

D
become zero

Solution

The frequency of a spring-mass system depends on the mass attached to the spring and the spring constant $$K$$ by the equation:
$$\omega = \sqrt{\dfrac Km}$$
Angular frequency is independent of acceleration of system ,Hence remains same.
Option C is correct.
Question 6
1 / -0

$$m_1$$ and $$m_2$$ are connected with a light inextensible string with $$m_1$$ lying in smooth table and $$m_2$$ hanging as shown in figure. $$m_1$$ is also connected to a light spring which is initially unstretched and the system is released form rest:

A
system performs S.H.M with angular frequency given by $$\sqrt{\dfrac{k(m_1+m_2)}{m_1m_2}}$$

B
maximum displacement of $$m_1$$ will be $$\dfrac{m_2g}{k}$$

C
tension in string will be $$0$$ when the system is released

D
system performs S.H.M with angular frequency given by $$\sqrt{\dfrac{k}{m_1+m_2}}$$

Solution
Question 7
1 / -0

A cubical block of side 'a' is floating in a fixed and closed cylindrical container of radius $$2a$$ kept on the ground. Density of the block is $$\rho$$, whereas the density of liquid is $$2\rho$$. Container is made up of conducting wall so that the temperature remains constant. A piston is mounted in the cylinder which can move inside the cylinder without friction. If piston oscillates with large amplitude A.

A
The cube will remain stationary

B
The cube will oscillate with very small amplitude in same phase with piston

C
The cube will oscillate with very small amplitude in opposite phase with piston

D
The cube will oscillate with amplitude A

Solution

We know cube has all sides and faces are same.
So the pressure it is given will be distributed from all its surfaces equally at all direction.
Temperature remains constant so the variation in temperature does not affect the cube inside the cylinder.

It is floating on a constant density so when the piston oscillates but cube due to its structure it will remain stationary.

Option A is correct.
Question 8
1 / -0

A hollow sphere is filled with water. It is hung by a long thread to make it a simple pendulum. As the water flows out of a hole at the bottom of the sphere, the frequency of oscillation will

A
go on increasing

B
go on decreasing

C
first increases and then decreases

D
first decreases and then increases

Solution

Frequency of a simple pendulum
$$f=\dfrac {1}{2\pi}\sqrt {\dfrac gl}$$
Evidently, $$f\infty \dfrac {1}{\sqrt l}$$ at any place.
The effective length $$l$$ is the distance of the point of suspension $$O$$' from the cenre of gravity $$(CG)$$ of the bob. As, water flows out of the hole at the bottom, the $$CG$$ descends from centre towards the bottom, increasing the effective length and consequently $$f$$ decreases.]
However, when all the water has flows out the $$CG$$ of a hollow sphere, is once again at its centre and hence the effective.
Question 9
1 / -0

A particle of mass $$m$$ moving along the $$x-$$ axis as a potential energy $$U(x)=a+bx^2$$ where $$a$$ and $$b$$ are positive constants. It will execute simple harmonic motion with a frequency determined by the value of :

A
$$b$$ alone

B
$$b$$ and $$a$$ alone

C
$$b$$ and $$m$$ alone

D
$$b,a $$ and $$m$$ alone

Solution

$$U=a+bx^2$$
$$F=-\dfrac {dU}{dx}=-[0+2bx]$$
$$F=ma=-2bx \ \Rightarrow \ a=-\left(\dfrac {2b}{m}\right)x$$
$$\omega =2\pi f =\left(\dfrac {2b}{m}\right)^{1/2}$$
Hence frequency depends upon $$b$$ and $$m$$
Question 10
1 / -0

A particle executes SHM starting from its mean position at $$t=0$$. If its velocity is $$\sqrt 3 b\omega$$, when it is at a distance $$b$$ from the mean position, when $$\omega==2\pi /T$$, the time taken by the particle to move from $$b$$ to the extreme position on the same side is

A
$$\dfrac{5\pi}{6\omega}$$

B
$$\dfrac{\pi}{3\omega}$$

C
$$\dfrac{\pi}{2\omega}$$

D
$$\dfrac{\pi}{4\omega}$$

Solution

$$v^2=\omega^2 (A^2-b^2)=3\omega^2 b^2$$

$$ \Rightarrow A^2 =4b^2$$

$$b=\dfrac {A}{2} =A\sin \omega t \ \Rightarrow t=\dfrac {\pi}{6\omega}$$

Required time, $$t_1 =\dfrac {T}{4}-t=\dfrac {2\pi}{4\omega}-\dfrac {\pi}{6\omega}=\dfrac {\pi}{3\omega}$$