Oscillations Test

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Oscillations Test - 31

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Weekly Quiz Competition

Question 1
1 / -0

Which one of the following will take place when a watch based on oscillating spring is taken to a deep mine?

A
It will indicate the same time on earth

B
It will become fast

C
It will become slow

D
It will stop working

Solution
Question 2
1 / -0

A particle executing simple harmonic motion with time period T. The time period with which its kinetic energy oscillates is

A
T

B
$$2$$T

C
$$4$$T

D
$$\frac{T}{2}$$

Solution

Kinetic energy of the particle executing simple harmonic motion is periodic with period $$\frac{T}{2}$$.
Question 3
1 / -0

A block of mass $$m$$ is hanging vertically by spring of spring constant $$k$$. If the mass is made to oscillate vertically, its total energy is:

A
Maximum at the extreme position

B
Maximum at the mean position

C
Minimum at the mean position

D
Same at all positions

Solution

The block executes SHM. In SHM, the total energy remains constant at all positions.
Question 4
1 / -0

Frequency of variation of kinetic energy of a simple harmonic motion of frequency n is

A
$$2$$n

B
$$n$$

C
$$\frac{n}{2}$$

D
$$3$$n

Solution

The velocity of a simple harmonic oscillator varies as $$v_0sin(2\pi nt+\phi)$$

Kinetic energy is,

$$K=\dfrac 12mv^2=\dfrac 12 mv_0^2 sin^2(2\pi nt+\phi)$$

Now,

$$sin^2 \theta=\dfrac 12 (1-cos(2\theta))$$

So,

$$K=\dfrac 12 mv_0^2(\dfrac 12-\dfrac 12 cos(4\pi nt+2\phi))$$

$$K=\dfrac 14 mv_0^2-\dfrac 14 mv_0^2 cos(2\pi(2n)t+2\phi)$$

Here, the time dependence comes from the last term and its frequency is $$2n$$.
Question 5
1 / -0

A body of mass $$20$$g connected to a spring of spring constant k, executes simple harmonic motion with a frequency of $$(\dfrac{5 }{\pi})$$Hz. The value of spring constant is

A
$$4 Nm^{-1}$$

B
$$3 Nm^{-1}$$

C
$$2 Nm^{-1}$$

D
$$5 Nm^{-1}$$

Solution

Here, $$m = 20g = 20 \times 10^{-3} kg, v = \dfrac{5}{\pi} Hz$$
$$ v = \dfrac{1}{2 \pi} \sqrt{\dfrac{k}{m}}$$
$$k = 4 \pi^2 v^2 m = 4 \pi^2 [\frac{5}{\pi}]^2 \times 20 \times 10^{-3} = 2 N m^{-1}$$
Question 6
1 / -0

A particle executing simple harmonic motion with an amplitude $$5$$ cm and a time period $$0.2$$s. The velocity and acceleration of the particle when the displacement is $$5$$ cm is

A
$$0.5 \pi m s^{-1}, 0 m s^{-2}$$

B
$$0.5 m s^{-1}, -5 \pi^2 m s^{-2}$$

C
$$0 m s^{-1}, -5 \pi^2 m s^{-2}$$

D
$$0.5\pi m s^{-1}, -0.5 \pi^2 m s^{-2}$$

Solution

A particle executing simple harmonic motion with an amplitude$$=5cm=\cfrac{5}{100}=0.05m$$
Time period$$=0.2s$$
The velocity and acceleration of the particle when the displacement$$=5cm$$
$$\omega=\cfrac{2\pi}{T}\\ \quad=\cfrac{2\pi}{0.2}=\cfrac{2\pi}{2}\times10=10\pi rad/s$$
And the displacement in $$y$$, then acceleration, $$A=\omega^2y$$
And velocity$$v=\omega\sqrt{r^2-y^2}$$
$$v=10\pi\sqrt{(0.05)^2-(0.05)^2}=0$$
When $$y=0, A=(10\pi)^2\times0=0$$
$$v=10\pi\times\sqrt{(0.05)^2-0^2}=10\pi\times0.05\\ \quad=0.5\pi m/s$$
Question 7
1 / -0

When the displacement of a particle executing SHM is one - fourth of its amplitude, what fraction of the total energy is the kinetic energy?

A
$$\dfrac{16}{15}$$

B
$$\dfrac{15}{16}$$

C
$$\dfrac{3}{4}$$

D
$$\dfrac{4}{3}$$

Solution

$$x=\cfrac{A}{4}$$
$$v=\omega \sqrt{A^2-x^2}$$
$$v=\omega \sqrt{A^2-\cfrac{A^2}{16}}$$
$$v=\omega \sqrt{\cfrac{15A^2}{16}}$$
$$KE=\cfrac{1}{2}mv^2=\cfrac{1}{2}m\omega ^2(\cfrac{15A^2}{16})=\cfrac{15}{16}(TE)$$ $$[TE=\cfrac{1}{2}m\omega ^2A^2]$$
Question 8
1 / -0

Time period of oscillation of a spring is $$12$$s on earth. What shall be the time period if it is taken to moon?

A
$$6$$s

B
$$12$$s

C
$$36$$s

D
$$72$$s

Solution

Period of oscillation of the spring does not depend on g.
$$T=2 \pi \sqrt{\dfrac{m}{K}}$$
Question 9
1 / -0

Two blocks each of mass m is connected to the spring of spring constant k as shown in the figure.
If the blocks are displaced slightly in opposite directions and released, they will execute simple harmonic motion. The time period of oscillation is

A
$$2 \pi \sqrt{\frac{m}{k}}$$

B
$$2 \pi \sqrt{\frac{m}{2k}}$$

C
$$2 \pi \sqrt{\frac{m}{4k}}$$

D
$$2 \pi \sqrt{\frac{2m}{k}}$$

Solution

Reduced mass of the system
$$ \mu = \dfrac{(m)(m)}{m + m} = \dfrac{m}{2}$$
$$\therefore Time period, T = 2 \pi \sqrt{\dfrac{\mu}{k}} = 2 \pi \sqrt{\dfrac{m}{2k}}$$
Question 10
1 / -0

Match the Column I with Column II

A
A - p, B - q, C - s. D - r

B
A - s, B - r, C - p, D - q

C
A - r, B - p, C - s. D - q

D
A - p, B - r, C - q, D - s

Solution

Time period of spring block system is given by
$$T=2\pi \sqrt { \cfrac { m }{ K{ e }_{ ff } } } $$ .....(1)
$$K{ e }_{ ff }=$$ effective
In series $$K{ e }_{ ff }=\cfrac { 1 }{ { K }_{ 1 } } +\cfrac { 1 }{ { K }_{ 2 } } =\cfrac { { K }_{ 1 }+{ K }_{ 2 } }{ { K }_{ 1 }{ K }_{ 2 } } $$ ....(2)
In parallel $$K{ e }_{ ff }={ K }_{ 1 }+{ K }_{ 2 }$$ .......(3)

In Diagram (A) and (B) spring are connected in parallel, whereas in (C) and (D) Spring is connected in series.

using equation 1,2,3
option $$B$$ will be correct.

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Oscillations Test - 31

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Oscillations Test - 31

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SHARING IS CARING

Question 1

It will indicate the same time on earth

It will become fast

It will become slow

It will stop working

Solution

Question 2

T

$$2$$T

$$4$$T

$$\frac{T}{2}$$

Solution

Question 3

Maximum at the extreme position

Maximum at the mean position

Minimum at the mean position

Same at all positions

Solution

Question 4

$$2$$n

$$n$$

$$\frac{n}{2}$$

$$3$$n

Solution

Question 5

$$4 Nm^{-1}$$

$$3 Nm^{-1}$$

$$2 Nm^{-1}$$

$$5 Nm^{-1}$$

Solution

Question 6

$$0.5 \pi m s^{-1}, 0 m s^{-2}$$

$$0.5 m s^{-1}, -5 \pi^2 m s^{-2}$$

$$0 m s^{-1}, -5 \pi^2 m s^{-2}$$

$$0.5\pi m s^{-1}, -0.5 \pi^2 m s^{-2}$$

Solution

Question 7

$$\dfrac{16}{15}$$

$$\dfrac{15}{16}$$

$$\dfrac{3}{4}$$

$$\dfrac{4}{3}$$

Solution

Question 8

$$6$$s

$$12$$s

$$36$$s

$$72$$s

Solution

Question 9

$$2 \pi \sqrt{\frac{m}{k}}$$

$$2 \pi \sqrt{\frac{m}{2k}}$$

$$2 \pi \sqrt{\frac{m}{4k}}$$

$$2 \pi \sqrt{\frac{2m}{k}}$$

Solution

Question 10

A - p, B - q, C - s. D - r

A - s, B - r, C - p, D - q

A - r, B - p, C - s. D - q

A - p, B - r, C - q, D - s

Solution

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