Oscillations Test

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Oscillations Test - 58

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Weekly Quiz Competition

Question 1
1 / -0

A block is placed on a frictionless horizontal table. The mass of the block is $$m$$ and springs are attached on either side with force constants $$K_1$$ and $$K_2$$. If the block is displaced a little and left to oscillate, then the angular frequency of oscillation will be

A
$$\left( \dfrac{K_1+K_2}{m}\right)^{1/2}$$

B
$$\left[ \dfrac{K_1K_2}{m(K_1+K_2)}\right]^{1/2}$$

C
$$\left[ \dfrac{K_1K_2}{(K_1-K_2)m}\right]^{1/2}$$

D
$$\left[ \dfrac{K_1^2+K_2^2}{(K_1+K_2)m}\right]^{1/2}$$

Solution

In this case springs are parallel, so $$k_{eq}=k_1+k_2$$

and $$\omega =\sqrt{\dfrac{k_{eq}}{m}}=\sqrt{\dfrac{k_1+k_2}{m}}$$
Question 2
1 / -0

In a simple pendulum, the period of oscillation $$T$$ is released to length of the pendulum $$l$$ as

A
$$\dfrac {l}{T}=$$ constant

B
$$\dfrac {l^2}{T}=$$ constant

C
$$\dfrac {l}{T^2}=$$ constant

D
$$\dfrac {l^2}{T^2}=$$ constant

Solution
Question 3
1 / -0

Graph between velocity and displacement of a particle, executing S.H.M. is

A
A straight line

B
A parabola

C
A hyperbola

D
An ellipse

Solution

We know that the displacement function of a particle performing SHM is given by,
$$y=Asin(\omega t+\phi)$$
or, $$ sin(\omega t+\phi)=\frac{y}{A}$$ (1)
The velocity can be found by differentiating the displacement function with respect to time,
$$v=\frac{dy}{dt}=A\omega cos(\omega t + \phi)$$
or, $$cos(\omega t+\phi)= \frac{v}{A\omega}$$ (2)
Squaring and adding equations (1) and (2) we get,
$$\frac{y^2}{A^2}+\frac{v^2}{(A\omega)^2}=1$$ ($$\because sin^2\theta +cos^2\theta =1$$)
This equation resembles the equation of a ellipse.
(equation of ellipse is $$\frac{x^2}{a^2} + \frac{y^2}{b^2} =1$$)
Question 4
1 / -0

A simple pendulum is executing simple harmonic motion with a time period $$T$$. If the length of the pendulum is increased by $$21\%$$, the percentage increase in the time period of the pendulum of increased length is

A
$$10\%$$

B
$$21\%$$

C
$$30\%$$

D
$$50\%$$

Solution

If initial length $$l_1=100$$ then $$l_2=121$$
By using $$T=2\pi \sqrt {\dfrac {l}{g}}\Rightarrow \dfrac {T_1}{T_2}=\sqrt {\dfrac {l_1}{l_2}}$$

Hence, $$\dfrac {T_1}{T_2}=\sqrt {\dfrac {100}{121}}\Rightarrow T_2 =1.1 T_1$$

$$\%$$ increase $$=\dfrac {T_2 -T_1}{T_1}\times 100=10\%$$
Question 5
1 / -0

The force constants of two springs are $$K_1$$ and $$K_2$$. Both are stretched till their elastic energies are equal. If the stretching forces are $$F_1$$ and $$F_2$$, then $$F_1 : F_2$$ is

A
$$K_1: K_2$$

B
$$K_2: K_1$$

C
$$\sqrt{K_1}: \sqrt{K_2}$$

D
$$K_1^2: K_2^2$$

Solution

Given elastic energies are equal

i.e. $$\dfrac 12 k_1x_1^2=\dfrac 12 k_2x_2^2$$

$$\Rightarrow \dfrac{k_1}{k_2}=\left( \dfrac{x_2}{x_1}\right)^2$$ and using $$F=kx$$

$$\Rightarrow \dfrac{F_1}{F_2}=\dfrac{k_1x_1}{k_2x_2}=\dfrac{k_1}{k_2}\times \sqrt{\dfrac{k_2}{k_2}}=\sqrt{\dfrac{k_1}{k_2}}$$
Question 6
1 / -0

If a simple harmonic oscillator has got a displacement of $$0.02\ m $$ and acceleration equal to $$2.0ms^{-2}$$ at any time, the angular frequency of the oscillator is equal to

A
$$10\ rad\ s^{-1}$$

B
$$0.1\ rad\ s^{-1}$$

C
$$100\ rad\ s^{-1}$$

D
$$1\ rad\ s^{-1}$$

Solution

When a particle undergoes SHM, its acceleration is given by,
$$\alpha = \omega^2 x$$
Given, $$\alpha = 2, x=0.02$$. Using these values
$$\omega =\sqrt{\frac{\alpha}{x}}$$
or, $$\omega=\sqrt{\frac{2}{0.02}}$$
or, $$\omega=10\space rad/s$$ is our required answer.
Question 7
1 / -0

What will be the force constant of the spring system shown in the figure

A
$$\dfrac{K_1}{2}+K_2$$

B
$$\left[ \dfrac{1}{2K_1}+\dfrac{1}{K_2}\right]^{-1}$$

C
$$\dfrac{1}{2K_1}+\dfrac{1}{K_2}$$

D
$$\left[ \dfrac{2}{K_1}+\dfrac{1}{K_1}\right]^{-1}$$

Solution

In series combination
$$\dfrac{1}{k_s}=\dfrac{1}{2k_1}+\dfrac{1}{k_2}$$
$$\Rightarrow $$
$$k_S=\left[ \dfrac {1}{2k_1}+\dfrac{1}{k_2}\right]^{-1}$$
Question 8
1 / -0

A simple pendulum oscillates in air with time period $$T$$ and amplitude $$A$$. As the time passes

A
$$T$$ and $$A$$ both decrease

B
$$T$$ increases and $$A$$ is constant

C
$$T$$ increases and $$A$$ decreases

D
$$T$$ decreases and $$A$$ is constant

Solution
Question 9
1 / -0

Two simple pendulum of length $$1.44\ m$$ and $$1\ m$$ start swinging together. After how many vibrates will they again start swinging together

A
$$5$$ oscillations of smaller pendulum

B
$$6$$ oscillations of smaller pendulum

C
$$4$$ oscillation of bigger pendulum

D
$$6$$ oscillation of bigger pendulum

Solution

$$n\propto \dfrac{1}{\sqrt{l}}$$
$$\Rightarrow \dfrac{n_{2}}{n_{1}}=\sqrt{\dfrac{1.44}{1}}=\dfrac{1.2}{1}$$
$$\Rightarrow n_{2}=1.2\ n_{1}$$

For $$n_2$$ be integer minimum value of $$n_1$$ should be $$5$$ and then $$n_2=6$$ i.e., after $$6$$ oscillations of smaller pendulum both will be in phase.
Question 10
1 / -0

A simple pendulum is set into vibrations. The bob of the pendulum comes to rest after some time due to

A
Air friction

B
Moment of inertia

C
Weight of the bob

D
Combination of all the above

Solution

Air friction causes the oscillating pendulum to damp and thus eventually it comes to rest after some time.

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Oscillations Test - 58

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Oscillations Test - 58

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Question 1

$$\left( \dfrac{K_1+K_2}{m}\right)^{1/2}$$

$$\left[ \dfrac{K_1K_2}{m(K_1+K_2)}\right]^{1/2}$$

$$\left[ \dfrac{K_1K_2}{(K_1-K_2)m}\right]^{1/2}$$

$$\left[ \dfrac{K_1^2+K_2^2}{(K_1+K_2)m}\right]^{1/2}$$

Solution

Question 2

$$\dfrac {l}{T}=$$ constant

$$\dfrac {l^2}{T}=$$ constant

$$\dfrac {l}{T^2}=$$ constant

$$\dfrac {l^2}{T^2}=$$ constant

Solution

Question 3

A straight line

A parabola

A hyperbola

An ellipse

Solution

Question 4

$$10\%$$

$$21\%$$

$$30\%$$

$$50\%$$

Solution

Question 5

$$K_1: K_2$$

$$K_2: K_1$$

$$\sqrt{K_1}: \sqrt{K_2}$$

$$K_1^2: K_2^2$$

Solution

Question 6

$$10\ rad\ s^{-1}$$

$$0.1\ rad\ s^{-1}$$

$$100\ rad\ s^{-1}$$

$$1\ rad\ s^{-1}$$

Solution

Question 7

$$\dfrac{K_1}{2}+K_2$$

$$\left[ \dfrac{1}{2K_1}+\dfrac{1}{K_2}\right]^{-1}$$

$$\dfrac{1}{2K_1}+\dfrac{1}{K_2}$$

$$\left[ \dfrac{2}{K_1}+\dfrac{1}{K_1}\right]^{-1}$$

Solution

Question 8

$$T$$ and $$A$$ both decrease

$$T$$ increases and $$A$$ is constant

$$T$$ increases and $$A$$ decreases

$$T$$ decreases and $$A$$ is constant

Solution

Question 9

$$5$$ oscillations of smaller pendulum

$$6$$ oscillations of smaller pendulum

$$4$$ oscillation of bigger pendulum

$$6$$ oscillation of bigger pendulum

Solution

Question 10

Air friction

Moment of inertia

Weight of the bob

Combination of all the above

Solution

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