Oscillations Test

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Oscillations Test - 46

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Weekly Quiz Competition

Question 1
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A simple harmonic wave in a medium is given by,(y is in centimeters) $$ y=\frac { 10 }{ \pi } sin (2000{ \pi }t- \frac { \pi x }{ 17 } )$$
The maximum velcocity of a particle executing tha wave is

A
$$330ms^-1$$

B
$$20ms^-1$$

C
$$200ms^-1$$

D
$$2000ms^-1$$

Solution
Question 2
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A particle starts simple harmonic motion from the mean position. Its amplitude is a and total energy E. At one instant its kinetic energy is 3E/4. Its displacement at the instat is-

A
$$a/\sqrt { 2 } $$

B
$$a/2$$

C
$$\dfrac { a }{ \sqrt { 3/2 } } $$

D
$$a/\sqrt { 3 } $$

Solution

The equation of the simple harmonic motion is $$y=A sin \omega t$$ with usual notations.
The total energy is $$\left( {\frac{1}{2}} \right)m{\omega ^2}{a^2}$$ and the kinetic energy is $$\left( { \frac { 1 }{ 2 } } \right) m{ \omega ^{ 2 } }\left( { { a^{ 2 } }-{ y^{ 2 } } } \right) .$$
Therefore we have $$\left( {\frac{1}{2}} \right)m{\omega ^2}{a^2} = E$$ and
$$\left( {\frac{1}{2}} \right)m{\omega ^2}\left( {{a^2} - {y^2}} \right) = \dfrac{{3E}}{4}.$$
Dividing the second equation by the first , $$1 - \dfrac{{{y^2}}}{{{a^2}}} = \dfrac{3}{4}$$ from which $$y=\dfrac{a}{2}$$.
Question 3
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A mass is suspended from a wire and is pulled along the length of wire, resulting in oscillations of time period $$T_{1}$$. The same mass is next attached to wire of the same material and length but double the cross-sectional area. The time period this time $$T_{2}$$. Then $$T_{1}/T_{2}$$ is equal to

A
$$1 : 2$$

B
$$2 : 1$$

C
$$1 : \sqrt{2}$$

D
$$\sqrt {2} : 1$$
Question 4
1 / -0

Oscillatory or vibrating motion means to and fro motion

A
Which does not repeat

B
which repeats after unequal interval of time

C
which repeats after equal interval of time

D
which remains unchanged

Solution
Question 5
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Two particles $$P$$ and $$Q$$ start from origin and execute simple harmonic motion along $$X-$$ axis with same amplitude but with periods $$3\ s$$ and $$6\ s$$ respectively. The ratio of the velocities of $$P$$ and $$Q$$ when they meet is

A
$$1 : 2$$

B
$$2 : 1$$

C
$$2 : 3$$

D
$$3 : 2$$

Solution

The particles will meet at the mean position when $$P$$ completes one oscillation and $$Q$$ completes half an oscillation

The angular velocity for first particle is $$\omega_1=\dfrac{2\pi}{3}$$.

The angular velocity for second particle is $$\omega_2=\dfrac{2\pi}{6}=\dfrac{\pi}{3}$$

So, the velocities at different points will be:
$$v_1=\omega_1A$$ and $$v_2=\omega_2A$$

So, $$\dfrac{v_1}{v_2}=\dfrac{\omega_1}{\omega_2}$$

$$\dfrac{v_1}{v_2}=\dfrac{2\pi/3}{\pi/3}$$

$$\dfrac{v_1}{v_2}=2:1$$
Question 6
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The time period of a particle executing $$SHM$$ is $$8\ s$$. At $$t = 0$$ it is at the mean position. The ratio of distance covered by the particle in $${ 1 }^{ st }$$ second to the $${ 2 }^{ nd }$$ second is

A
$$(\dfrac{1}{\sqrt { 2 } -1})$$

B
$$\sqrt { 2 } $$

C
$$(\sqrt { 2 } +1)$$

D
$$\dfrac { 1 }{ \sqrt { 2 } } $$

Solution

$$\begin{array}{l} Y'=A\sin w=A\sin \dfrac { { 2\pi } }{ 8 } =\dfrac { A }{ { \sqrt { 2 } } } \\ Dis\tan ce\, \, when\, \, t=2\, \, \sec \, \, is \\ { Y^{ 2 } }=A\sin w=A\sin \frac { \pi }{ 2 } =A \\ Dis\tan ce\, \, { { cov } }ered\, \, in\, \, 2\sec \, \, is\, \, \dfrac { { A-A } }{ { \sqrt { 2 } } } \\ \dfrac { { { Y^{ 1 } } } }{ { { Y^{ 2 } } } } =\dfrac { { \left( { \dfrac { A }{ { \sqrt { 2 } } } } \right) } }{ { \left( { A-\dfrac { A }{ { \sqrt { 2 } } } } \right) } } \\ =\dfrac { 1 }{ { \sqrt { 2 } -1 } } \end{array}$$
Question 7
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Initially the spring is undeformed. Now the force $$F$$ is applied to $$B$$ as shown. When the displacement of $$B$$ w.r.t $$A$$ is $$x$$ towards right in some time then the relative acceleration of $$B$$ w.r.t $$A$$ at the moment is:

A
$$\cfrac{F}{2m}$$

B
$$\cfrac{F-kx}{m}$$

C
$$\cfrac{F-2kx}{m}$$

D
$$none\ of\ these$$

Solution

A moment is $$\frac{{F - 2kx}}{m}$$.
Option $$C$$ is correct answer.
Question 8
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Two charges each $$+5\mu C$$ are at $$(0, \pm 3)$$. A third charge of $$1\mu C$$ and of mass $$2g$$ is at $$(4, 0)$$. What is the minimum velocity to be given to the $$1\mu C$$ charge such that it just reaches the origin (in $$ms^{-1}$$)?

A
$$3$$

B
$$12$$

C
$$\sqrt12$$

D
$$4$$

Solution
Question 9
1 / -0

A particle is executing SHM about $$\gamma =0$$ along y- axis.Its position at an instant is given by $$\gamma =(7m)$$ sin(πt) its average velcocity for a time interval 0 to 0.5 s is

A
$$14m/s$$

B
$$7m/s$$

C
$$\dfrac { 1}{ 7 } m/s$$

D
$$28m/s$$

Solution

at t=0, position of particle , r=0
at t=0.5, position of particle, r=7m
Total displacement = 7m
Total time = 0.55
$$Average\,velocity =\frac{Total\, displacement}{Total\, time}$$
$$ = \frac{7}{0.5}= 14\, m/s$$
Question 10
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Two springs of force constants $${k}_{1}$$ and $${k}_{2}$$ are connected to a mass $$m$$ as shown. The frequency of oscillation of the mass is $$f$$. If both $${k}_{1}$$ and $${k}_{2}$$ are made four times their original values, the frequency of oscillation becomes:

A
$$f/2$$

B
$$f/4$$

C
$$4f$$

D
$$2f$$

Solution

$$\textbf{Hint:}$$ In parallel combination of spring, spring constant added.
$$\textbf{Step 1:}$$ The equivalent spring constant can be find out as,
$$K_{eq}=k_1+k_2$$
And, the frequency,
$$f=\dfrac{1}{2\pi}\sqrt{\dfrac{k_{eq}}{m}}$$
$$\textbf{Step 2:}$$ The value of $$k_1$$ and $$k_2$$ is four times their original value, So,
$$f'=\dfrac{1}{2\pi}\sqrt{\dfrac{k'_{eq}}{m}}$$
$$k'_{eq}=4k_1+4k_2$$
$$f'=\dfrac{1}{2\pi}\sqrt{\dfrac{4(k_1+k_2)}{m}}$$
So, from the previous frequency,
$$f'=2\times \dfrac{1}{2\pi}\sqrt{\dfrac{k_1+k_2}{m}}$$
$$f'=2\times f$$