Oscillations Test

Result

Oscillations Test - 70

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Weekly Quiz Competition

Question 1
1 / -0

Maximum tension to the minimum tension in the string during oscillations is?(Consider amplitude of oscillation =$$ \theta $$)

A
$$(1+\theta^2):(1-\theta^2)$$

B
$$(1+\theta^2):(1-\dfrac{\theta^2}{2})$$

C
$$(1+\theta^2):1$$

D
$$(1+\dfrac{\theta^2}{2}):(1-\theta^2)$$

Solution

At extrem position velocity of ball will be zero.
$$T_{min}=Mg\cos\theta$$

For velocity at mean position
Form conservation of energy:
Gain in K.E = Loss in PE
$$\dfrac{1}{2}mv^2=mg(l-l\cos\theta)$$

$$\dfrac{mv^2}l=2mg(1-\cos\theta)$$

$$T=mg+\dfrac{mv^2}{l}=mg+2gm(1-\cos\theta)$$

$$T_{max}=mg[3-2\cos\theta]$$

$$\dfrac{T_{max}}{T_{min}}=\dfrac{3-2\cos\theta}{\cos\theta}=\dfrac{3-2\left[1-2\sin^2\dfrac{\theta}{2}\right]}{1-2\sin^2\dfrac{\theta}{2}}$$

For small angular displacement
$$\dfrac{T_{max}}{T_{min}}=\dfrac{3-2\left[1-\dfrac{\theta^2}{2}\right]}{1-\dfrac{2\theta^2}{4}}=\dfrac{1+\theta^2}{1-\dfrac{\theta^2}{2}}$$
Question 2
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A particle rests in equilibrium under two forces of repulsion whose centres are at a distance of $$a$$ and $$b$$ from the particle. The forces vary as the cube of the distance. The forces per unit mass are $$k$$ and $$k'$$ respectively. If the particle is slightly displaced towards one of them then, the motion is simple harmonic with the time period equal to:

A
$$\cfrac { 2\pi }{ \sqrt { 3\left( \cfrac { k }{ { a }^{ 3 } } +\cfrac { { k }^{ ' } }{ { b }^{ 3 } } \right) } } $$

B
$$\cfrac { 2\pi }{ \sqrt { \left( \cfrac { k }{ { a }^{ 3 } } +\cfrac { { k }^{ ' } }{ { b }^{ 3 } } \right) } } $$

C
$$\cfrac { 2\pi }{ \sqrt { \left( \cfrac { k }{ { a }^{ 4 } } +\cfrac { { k }^{ ' } }{ { b }^{ 4 } } \right) } }$$

D
$$\cfrac { 2\pi }{ \sqrt { 3\left( \cfrac { k }{ { a }^{ 4 } } +\cfrac { { k }^{ ' } }{ { b }^{ 4 } } \right) } } $$

Solution
Question 3
1 / -0

The period of a conical pendulum of string length $$\sqrt 2$$ is $$2 \ s$$. The radius of the bob's orbit is about horizontal is $$[g=9.8 \ m/s^2]$$

A
$$\cfrac 1{\sqrt 2} \ m$$

B
$$1 \ m$$

C
$$\sqrt 2 \ m$$

D
$$\sqrt 3 \ m$$

Solution
Question 4
1 / -0

A particle executing SHM while moving from one extreme is found at distance $$x_1,\, x_2$$ and $$x_3$$ from the centre at the end of three successive seconds. The time period of oscillation is: Here $$\theta = {\cos ^{ - 1}}\left( {\frac{{{x_1} + {x_3}}}{{2{x_2}}}} \right)$$

A
$$\dfrac {2 \pi}{\theta}$$

B
$$\dfrac { \pi}{\theta}$$

C
$$\theta$$

D
$$\dfrac { \pi}{2\theta}$$

Solution
Question 5
1 / -0

Three measurements of the time for $$20$$ oscillations of a pendulum give $$t_1 = 39.6_s , t_2 = 39.9_s and t_3 = 39.5_s$$. The precision in the measurement, is

A
$$0.005 sec$$

B
$$0.01 sec$$

C
$$1 sec$$

D
$$2 sec$$

Solution
Question 6
1 / -0

A particle of mass 0.1 kg executes SHM under a force F= (-10x)N. Speed of particle at mean position is 6 m/s. Then amplitude of oscillations is

A
0.6 m

B
0.2 m

C
0.4 m

D
0.1 m

Solution
Question 7
1 / -0

If two wires of same length $$\ell $$ and area of cross section A with young modulus y and 2y connect in series and one end is fixed on roof and other end with mass m. Make simple harmonic motion, then the time period is-

A
$$2\pi \sqrt { \dfrac { m\ell }{ YA } } $$

B
$$2\pi \sqrt { \dfrac { m\ell }{ 3YA } } $$

C
$$2\pi \sqrt { \dfrac { 3m\ell }{ 2YA } } $$

D
$$2\pi \sqrt { \dfrac { m\ell }{ 2YA } } $$

Solution
Question 8
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A linear harmonic oscillator of force constant $$6 \times 10^5 N/m$$ and amplitude $$4 $$ cm, has total energy $$600 J$$. Select the correct statement.

A
Maximum potential energy is $$600$$ J

B
Maximum kinetic energy is $$480$$ J

C
Minimum potential energy is $$120$$ J

D
None

Solution
Question 9
1 / -0

Equation of SHM is x = 10 sin 10 $$\pi t$$. Find the distance between the two points where speed is $$50\pi$$ m/s. x is in cm and t is in seconds.

A
10 cm

B
14 cm

C
17.32 cm

D
8.66 cm

Solution
Question 10
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A particle executes SHM along the line AB . IFC divided AB in the ratio 3 : 1, the ration of the time taken to travel AC and CB is :

A
3:1

B
1:1

C
2:1

D
4:1