Oscillation and Simple Harmonic Motion | Physics JEE previous year questions with Solutions

Q 1 :
A simple pendulum doing small oscillations at a place R height above earth surface has time period of $T_{1} = 4 s .$ $T_{2}$ would be it's time period if it is brought to a point which is at a height 2R from earth surface.

Choose the correct relation [R = radius of earth] [2024]

$2 T_{1} = 3 T_{2}$
$2 T_{1} = T_{2}$
$T_{1} = T_{2}$
$3 T_{1} = 2 T_{2}$

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(4)

$T = 2 π \sqrt{\frac{l}{g}} = 2 π \sqrt{\frac{l r^{2}}{GM}} = 2 π \sqrt{\frac{l}{GM}} r$

$\Rightarrow \frac{T_{2}}{T_{1}} = \frac{r_{2}}{r_{1}} = \frac{3 R}{2 R} \Rightarrow 2 T_{2} = 3 T_{1}$

Q 2 :
A mass m is suspended from a spring of negligible mass and the system oscillates with a frequency $f_{1}$ . The frequency of oscillations if a mass 9 m is suspended from the same spring is $f_{2} .$ The value of $\frac{f_{1}}{f_{2}}$ is _______ . [2024]

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(3)

Q 3 :
In simple harmonic motion, the total mechanical energy of a given system is E. If the mass of oscillating particle P is doubled, then the new energy of the system for same amplitude is [2024]

$E \sqrt{2}$
$E$
$2 E$
$E / \sqrt{2}$

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(2)

Total energy $E = K E + P E$

$E = \frac{1}{2} m ω^{2} (A^{2} - x^{2}) + \frac{1}{2} m ω^{2} x^{2}$

$E = \frac{1}{2} k A^{2}, since A is same, T . E . will be same$

Q 4 :
The time period of simple harmonic motion of mass M in the given figure is $π \sqrt{\frac{α M}{5 k}},$ where the value of $α$ is ________ . [2024]

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(12)

$k_{eq} = \frac{2 k \cdot k}{3 k} + k = \frac{5 k}{3}$

Angular frequency of oscillation $(ω) = \sqrt{\frac{k_{eq}}{m}}$

$(ω) = \sqrt{\frac{5 k}{3 m}}$

Period of oscillation $(τ) = \frac{2 π}{ω} = 2 π \sqrt{\frac{3 m}{5 k}}$

$= π \sqrt{\frac{12 m}{5 k}}$

Q 5 :
Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason (R).

Assertion (A) : A simple pendulum is taken to a planet of mass and radius, 4 times and 2 times, respectively, than the Earth. The time period of the pendulum remains same on earth and the planet.

Reasosn (R) : The mass of the pendulum remains unchanged at Earth and the other planet.

In the light of the above statements, choose the correct answer from the options given below: [2025]

Both (A) and (R) are true but (R) is NOT the correct explanation of (A).
(A) is true but (R) is false.
(A) is false but (R) is true.
Both (A) and (R) are true and (R) is the correct explanation of (A).

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(1)

$g = \frac{G M}{R^{2}}$

$g' = \frac{G (4 M)}{{(2 R)}^{2}} = g$

A is correct, R is correct; but since $T = 2 π \sqrt{\frac{l}{g}}$

Doesn't depend on mass; R doesn't explain A.

Q 6 :
Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R).

Assertion (A) : Time period of a simple pendulum is longer at the top of a mountain than that at the base of the mountain.

Reason (R) : Time period of a simple pendulum decreases with increasing value of acceleration due to gravity and vice-versa.

In the light of the above statements, choose the most appropriate answer from the options given below: [2025]

Both (A) and (R) are true but (R) is not the correct explanation of (A).
Both (A) and (R) are true and (R) is the correct explanation of (A).
(A) is true but (R) is false.
(A) is false but (R) is true.

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(2)

As h increases, g decteases, T increases

$T = 2 π \sqrt{\frac{l}{g}}$

$g = \frac{g_{0} R^{2}}{{(R + h)}^{2}}$

Q 7 :
Two bodies A and B of equal mass are suspended from two massless springs of spring constant $k_{1}$ and $k_{2}$ , respectively. If the bodies oscillate vertically such that their amplitudes are equal, the ratio of the maximum velocity of A to the maximum velocity of B is [2025]

$\sqrt{\frac{k_{1}}{k_{2}}}$
$\frac{k_{1}}{k_{2}}$
$\frac{k_{2}}{k_{1}}$
$\sqrt{\frac{k_{2}}{k_{1}}}$

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(1)

Here $ω = \sqrt{\frac{k}{m}}$ and maximum velocity $V = A ω = A \sqrt{\frac{k}{m}}$

So, $\frac{V_{A}}{V_{B}} = \sqrt{\frac{k_{1}}{k_{2}}}$

Q 8 :
Two blocks of masses m and M, (M > m), are placed on a frictionless table as shown in figure. A massless spring with spring constant k is attached with the lower block. If the system is slightly displaced and released then ( $μ$ = coefficient of friction between the two blocks)

(A) The time period of small oscilation of the two blocks is $T = 2 π \sqrt{\frac{(m + M)}{k}}$

(B) The acceleration of the blocks is $a = \frac{k x}{M + m}$

(x = displacement of the blocks from the mean position)

(C) The magnitude of the frictional force on the upper block is $\frac{m μ | x |}{M + m}$

(D) The maximum amplitude of the upper block, if it does not slip, is $\frac{μ (M + m) g}{k}$

(E) Maximum frictional force can be $μ (M + m) g$

Choose the correct answer from the options given below: [2025]

A, B, D Only
B, C, D Only
C, D, E Only
A, B, C Only

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(1)

A. Assuming no slipping, $T = 2 π \sqrt{\frac{m_{total}}{k}} = 2 π \sqrt{\frac{m + M}{k}} A$ is correct.

B. Let block is displaced by x in (+ve) direction so force on block will be in (–ve) direction

$F = - K x \Rightarrow (M + m) a = - K x$

$\Rightarrow a = - \frac{K x}{(M + m)}$ , B is correct.

C. As upper block is moving due to friction thus

$f = m a = \frac{m K x}{(M + m)}$ , C is not correct.

D. If both the blocks moves together at maximum amplitude, the friction force on the block of mass m is also mazimum,

For no slipping $\frac{m K a}{m + M} \leq μ m g$

$\Rightarrow A = \frac{μ (M + m g}{K}$ , D is correct.

E. Maximum friction, $f_{\max} = μ m g$ , E is incorrect.

Q 9 :
Two simple pendulums having lengths $l_{1}$ and $l_{2}$ with negligible string mass undergo angular displacements $θ_{1}$ and $θ_{2}$ , from their mean positions, respectively. If the angular accelerations of both pendulums are same, then which expression is correct? [2025]

$θ_{1} l_{2}^{2} = θ_{2} l_{1}^{2}$
$θ_{1} l_{1} = θ_{2} l_{2}$
$θ_{1} l_{1}^{2} = θ_{2} l_{2}^{2}$
$θ_{1} l_{2} = θ_{2} l_{1}$

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(4)

Angular frequency of simple pendulum, $ω = \sqrt{\frac{g}{l}}$

Angular acceleration, $α = - ω^{2} θ$

$∴ \frac{g}{l_{1}} θ_{1} = \frac{g}{l_{2}} θ_{2}$

$\Rightarrow θ_{1} l_{2} = θ_{2} l_{1}$

Topic Question Set

Q 1 :
A simple pendulum doing small oscillations at a place R height above earth surface has time period of $T_{1} = 4 s .$ $T_{2}$ would be it's time period if it is brought to a point which is at a height 2R from earth surface.

Choose the correct relation [R = radius of earth] [2024]

Q 2 :
A mass m is suspended from a spring of negligible mass and the system oscillates with a frequency $f_{1}$ . The frequency of oscillations if a mass 9 m is suspended from the same spring is $f_{2} .$ The value of $\frac{f_{1}}{f_{2}}$ is _______ . [2024]

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Q 3 :
In simple harmonic motion, the total mechanical energy of a given system is E. If the mass of oscillating particle P is doubled, then the new energy of the system for same amplitude is [2024]

Q 4 :
The time period of simple harmonic motion of mass M in the given figure is $π \sqrt{\frac{α M}{5 k}},$ where the value of $α$ is ________ . [2024]

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Q 7 :
Two bodies A and B of equal mass are suspended from two massless springs of spring constant $k_{1}$ and $k_{2}$ , respectively. If the bodies oscillate vertically such that their amplitudes are equal, the ratio of the maximum velocity of A to the maximum velocity of B is [2025]

Q 9 :
Two simple pendulums having lengths $l_{1}$ and $l_{2}$ with negligible string mass undergo angular displacements $θ_{1}$ and $θ_{2}$ , from their mean positions, respectively. If the angular accelerations of both pendulums are same, then which expression is correct? [2025]

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Topic Question Set

Q 1 : A simple pendulum doing small oscillations at a place R height above earth surface has time period of T1=4s. T2 would be it's time period if it is brought to a point which is at a height 2R from earth surface. Choose the correct relation [R = radius of earth] [2024]

Q 2 : A mass m is suspended from a spring of negligible mass and the system oscillates with a frequency f1. The frequency of oscillations if a mass 9 m is suspended from the same spring is f2. The value of f1f2 is _______ . [2024]

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Q 3 : In simple harmonic motion, the total mechanical energy of a given system is E. If the mass of oscillating particle P is doubled, then the new energy of the system for same amplitude is [2024]

Q 4 : The time period of simple harmonic motion of mass M in the given figure is παM5k, where the value of α is ________ . [2024]

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Q 7 : Two bodies A and B of equal mass are suspended from two massless springs of spring constant k1 and k2, respectively. If the bodies oscillate vertically such that their amplitudes are equal, the ratio of the maximum velocity of A to the maximum velocity of B is [2025]

Q 9 : Two simple pendulums having lengths l1 and l2 with negligible string mass undergo angular displacements θ1 and θ2, from their mean positions, respectively. If the angular accelerations of both pendulums are same, then which expression is correct? [2025]

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Q 1 :
A simple pendulum doing small oscillations at a place R height above earth surface has time period of $T_{1} = 4 s .$ $T_{2}$ would be it's time period if it is brought to a point which is at a height 2R from earth surface.

Choose the correct relation [R = radius of earth] [2024]

Q 2 :
A mass m is suspended from a spring of negligible mass and the system oscillates with a frequency $f_{1}$ . The frequency of oscillations if a mass 9 m is suspended from the same spring is $f_{2} .$ The value of $\frac{f_{1}}{f_{2}}$ is _______ . [2024]

Q 3 :
In simple harmonic motion, the total mechanical energy of a given system is E. If the mass of oscillating particle P is doubled, then the new energy of the system for same amplitude is [2024]

Q 4 :
The time period of simple harmonic motion of mass M in the given figure is $π \sqrt{\frac{α M}{5 k}},$ where the value of $α$ is ________ . [2024]

Q 7 :
Two bodies A and B of equal mass are suspended from two massless springs of spring constant $k_{1}$ and $k_{2}$ , respectively. If the bodies oscillate vertically such that their amplitudes are equal, the ratio of the maximum velocity of A to the maximum velocity of B is [2025]

Q 9 :
Two simple pendulums having lengths $l_{1}$ and $l_{2}$ with negligible string mass undergo angular displacements $θ_{1}$ and $θ_{2}$ , from their mean positions, respectively. If the angular accelerations of both pendulums are same, then which expression is correct? [2025]