Oscillations Test

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Oscillations Test - 66

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Weekly Quiz Competition

Question 1
1 / -0

The position vector of a particle that is moving in space is given by $$\vec {r} =(1 +2\cos 2\omega t)\hat {i} + (3\sin^{2} \omega t)\hat {j} + (3)\hat {k}$$ in the ground frame. All units are in $$SI$$. Choose the correct statement(s).

A
The particle executes SHM in the ground frame about the mean position $$\left (1, \dfrac {3}{2}, 3\right )$$

B
The particle executes SHM in a frame moving along the z-axis with a velocity of $$3\ m/s$$

C
The amplitude of the SHM of the particle is $$\dfrac {5}{2} m$$

D
The direction of the SHM of the particle is given by the vector $$\left (\dfrac {4}{5} \hat {i} - \dfrac {3}{5} \hat {j}\right )$$
Question 2
1 / -0

The string of a pendulum is horizontal initially. The mass of the bob attached is m. Now the string is released. 'R' is the length of the pendulum.
(i) The tension in the string at an angle $$30 ^\circ$$ with the vertical is ______

A
$$2 \sqrt{5} mg$$

B
$$2 \sqrt{3} mg$$

C
$$\dfrac{3 \sqrt{3}}{2} mg$$

D
$$4 \sqrt 3 mg$$
Question 3
1 / -0

A body is projected with an initial velocity $$20 \ m/s$$ at $$60^\circ$$ to the horizontal. The displacement after $$2 s$$ is:

A
$$20 [i + (\sqrt{3} - 1) j]$$

B
$$20 [i - (\sqrt{3} - 1) j]$$

C
$$10 [i + (\sqrt{3} + 1) j]$$

D
$$10 [i + (\sqrt{3} - 1) j]$$

Solution
Question 4
1 / -0

A simple pendulum with a brass bob has a time period T. The bob is now immersed in a non-viscous liquid and made to oscillate. The density of the liquid is (1/8)th that of the brass. The time period of the pendulum will be

A
$$\sqrt{\dfrac{8}{7}}$$ g

B
$$\dfrac{7}{8}$$ g

C
$$\dfrac{8^2}{7^2}$$ g

D
g

Solution
Question 5
1 / -0

If a body of mass $$36gm$$ moves with S.H.M of amplitude $$A=13cm$$ and $$T= 12\sec$$. At time $$t=0$$ the displacement is $$x=+12cm$$. The shortest time of passage from $$x=+6.5cm$$ to $$x=-6.5$$ is

A
$$4 \sec$$

B
$$2 \sec$$

C
$$6\sec$$

D
$$3 \sec$$

Solution
Question 6
1 / -0

A book is resting on shelf that is undergoing vertical simple harmonic oscillations with an amplitude of $$2.5 cm $$ . What is the minimum frequency of oscillations of the shell for which the book will the lose contact with the shelf?
(Assume that $$ g = m/s^2 $$ )

A
$$20 Hz $$

B
$$ 3.18 Hz $$

C
$$125.6 Hz $$

D
$$10 Hz $$

Solution

Book will lose contact with the shelf when acceleration of the shelf becomes equal to or greater than acceleration due to gravity.

We have acceleration of the shelf on which book is kept.

$$a = A{\omega ^2}$$

$$9.8 = 2.5 \times {10^{ - 2}} \times 4{\pi ^2}{f^2}$$

$${f^2} = \dfrac{{9.8 \times {{10}^2}}}{{10{\pi ^2}}}$$

$${f^2} = \dfrac{{9.8 \times {{10}^2}}}{{10 \times 9.869}}$$

$${f^2} = 10$$

$$f = 3.18\;{\rm{Hz}}$$

Thus frequency of oscillations should be equal to or greater than$$3.18\;{\rm{Hz}}$$.
Question 7
1 / -0

Two pendulums of lengths 121 cm and 100 cm start vibrating at the same instant. They are in the mean position and in the same phase. After how many vibrations of the shorter pendulum, the two will be in the same phase in the mean position?

A
10 vibrations

B
11 vibrations

C
21 vibrations

D
20 vibrations

Solution

Two pendulums of length $$121cm$$ and $$100cm$$.

Let,
$$L_1=121cm=\dfrac{121}{100}=1.21m$$

$$L_2=100cm=\dfrac{100}{100}=1m$$

We have to find the vibrations made by the shorter pendulum, such that both will be in same phase from the reaction,

$$T_1=longer\,pendulum$$

$$T_2=shorter\,pendulum$$

$$T=2\pi\sqrt{\dfrac{L}{g}}$$

$$T\propto \sqrt{L}$$

$$\dfrac{T_1}{T_2} \propto \sqrt{{L_1}/{L_2}}$$

$$\dfrac{T_1}{T_2}\propto \sqrt{\dfrac{1.21}{1}}$$

$$\dfrac{T_1}{T_2}=\dfrac{1.1}{1}$$

$$10T_1=11T_2$$

$$10$$ vibrations of longer pendulum= $$11$$ vibrations of shorter pendulum
Question 8
1 / -0

A particle performs harmonic oscillations along a straight line with a period of 6 s and amplitude 4 cm. The mean velocity of the particle averaged over the time interval during which it travel distance of 2 cm starting from the extreme position in cm /s is

A
1

B
2

C
4

D
8

Solution
Question 9
1 / -0

The motion of a particle is described by $$9\dfrac{d^{2}x}{dt^{2}}+25\ x=80$$ where $$x$$ is displacement and $$t$$ is time. Angular frequency of small oscillations is

A
$$5/3$$

B
$$4/3$$

C
$$3/4$$

D
$$5/4$$

Solution
Question 10
1 / -0

A particle executes SHM with amplitude $$0.5\ cm$$ and frequency $${100s}^{-1}$$. The maximum speed of the particle is (in m/s)

A
$$\pi $$

B
$$0.5$$

C
$$5\pi \times{10}^{-5}$$

D
$$100\pi$$

Solution