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

Q 11 :

A particle moves according to the law $x = a \cos \frac{π t}{2}$ . The distance covered by it in the time interval between $t = 0$ to $t = 3$ sec is

2a
3a
4a
a

1

2

3

4

(2)

T = 4 sec, Body starts at extreme position and ends at mean position as shown.

Q 12 :

Time period of a particle executing SHM is 16 s. At time $t = 2 s$ , it crosses the mean position its amplitude of motion is $\frac{32 \sqrt{2}}{π}$ m. Its velocity at $t = 4 s$ is

$1 m s^{- 1}$
$2 m s^{- 1}$
$4 m s^{- 1}$
$8 m s^{- 1}$

1

2

3

4

(3)

$For SHM, x = A \sin ω t = A \sin (\frac{2 π t}{T}), v = ω \sqrt{A^{2} - x^{2}}$

$When t = 4 s, time taken by the particle to travel from the mean position to the given position = 4 - 2 = 2 s$

$x = A \sin (\frac{2 π t}{T}) = A \sin (\frac{2 π \times 2}{16}) = \frac{A}{\sqrt{2}}$

$v = ω \sqrt{A^{2} - x^{2}}$

$v = ω \sqrt{A^{2} - \frac{A^{2}}{2}} = \frac{ω A}{\sqrt{2}}$

$ω = \frac{2 π}{T} = \frac{2 π}{16} = \frac{π}{8}$

$∴$ $v = \frac{π A}{8 \sqrt{2}}$

Q 13 :

The displacement of simple harmonic oscillator after 3 seconds starting from its mean position is equal to half of its amplitude. The time period of harmonic motion is:

6s
8s
12s
36s

1

2

3

4

(4)

$Time taken by the harmonic oscillator to move from mean position to half of the amplitude is \frac{T}{12}$

$so, \frac{T}{12} = 3$

$T = 36 sec$

Q 14 :

The equation of motion of a particle is given by $x = a \sin (50 t + \frac{π}{3}) cm .$ The particle will come to rest at time $t_{1}$ and it will have zero acceleration at time $t_{2}$ . The $t_{1}$ and $t_{2}$ respectively are ______. [2026]

$\frac{π}{300} s, \frac{π}{75} s$
$\frac{π}{75} s, \frac{π}{300} s$
$\frac{π}{300} s, \frac{π}{25} s$
$\frac{π}{50} s, \frac{π}{100} s$

1

2

3

4

(1)

$Phase of the motion of particle is as follows:$

[IMAGE 40]

$v = 0, when particle cross A .$

$∴$ $t_{1} = \frac{π}{6 ω} = \frac{π}{300}$

$a = 0, when particle cross B .$

$∴$ $t_{2} = \frac{2 π}{3 ω} = \frac{π}{75}$

Q 15 :

Match List-I with List-II. [2026]

List-I List-II

A. $\sin^{2} ω t$ I. Periodic with time period $T = \frac{π}{ω}$ but not simple harmonic motion (SHM)

B. $\sin^{3} (2 ω t)$ II. Periodic with time period $T = \frac{2 π}{ω}$ but not SHM

C. $\sin (ω t) + \cos (π ω t)$ III. Periodic with time period $T = \frac{π}{ω}$ and SHM

D. $\cos ω t + \cos 2 ω t$ IV. Non-periodic

Choose the correct answer from the options given below.

A-III, B-I, C-IV, D-II
A-II, B-I, C-III, D-IV
A-III, B-II, C-IV, D-I
A-II, B-I, C-IV, D-III

1

2

3

4

(1)

$(A)$ $\sin^{2} ω t = \frac{1}{2} - \frac{1}{2} \cos 2 ω t, it represents SHM motion having$ $period T = \frac{π}{ω}$

$(B)$ $\sin^{3} 2 ω t = \frac{3}{4} \sin 2 ω t - \frac{1}{4} \sin 6 ω t, the common period \frac{π}{ω} but not SHM.$

$(C)$ $\sin ω t + \cos π ω t \to no common period hence it is a nonperiodic function$

$(D)$ $\cos ω t + \cos 2 ω t common period \frac{2 π}{ω} but it do not represent SHM.$

Q 16 :

A particle is executing simple harmonic motion. Its amplitude is A and time period is 5 sec. The time required by it to move from $x = A$ to $x = \frac{A}{\sqrt{2}}$ is _________ sec. [2026]

$\frac{1}{4}$
$\frac{5}{4}$
$\frac{5}{8}$
$\frac{3}{8}$

1

2

3

4

(3)

[IMAGE 41]

$x = 0 \Rightarrow Phase ϕ = 0 °$

$x = \frac{A}{\sqrt{2}} \Rightarrow Phase ϕ = \frac{π}{4}$

$∴$ $phase covered Δ ϕ = \frac{π}{4}$

$time t = \frac{Phase}{ω} = \frac{π}{4 ω} = \frac{π}{4} \times \frac{T}{2 π} = \frac{T}{8}$

$The time required by it to move from x = A to x = \frac{A}{\sqrt{2}},$

$t' = \frac{T}{4} - \frac{T}{8} = \frac{T}{8} = \frac{5}{8} s$

Q 17 :

The velocity of a particle executing simple harmonic motion along $x$ -axis is described as $v^{2} = 50 - x^{2},$ where $x$ represents displacement. If the time period of motion is $\frac{x}{7} s,$ the value of $x$ is ______. [2026]

(44)

$Given velocity of a particle, v^{2} = 50 - x^{2}$

$By comparing this to the standard equation$

$v^{2} = ω^{2} A^{2} - ω^{2} x^{2}$

$∴$ $ω^{2} = 1 \Rightarrow ω = 1$

$Time period T = \frac{2 π}{ω} = \frac{2 π}{1} = 2 π sec$

$Now compare it with \frac{x}{7} = 2 π$

$\Rightarrow x = 2 \times \frac{22}{7} \times 7 = 44$

Topic Question Set

Q 11 :

A particle moves according to the law $x = a \cos \frac{π t}{2}$ . The distance covered by it in the time interval between $t = 0$ to $t = 3$ sec is

Q 12 :

Time period of a particle executing SHM is 16 s. At time $t = 2 s$ , it crosses the mean position its amplitude of motion is $\frac{32 \sqrt{2}}{π}$ m. Its velocity at $t = 4 s$ is

Q 13 :

The displacement of simple harmonic oscillator after 3 seconds starting from its mean position is equal to half of its amplitude. The time period of harmonic motion is:

Q 14 :

The equation of motion of a particle is given by $x = a \sin (50 t + \frac{π}{3}) cm .$ The particle will come to rest at time $t_{1}$ and it will have zero acceleration at time $t_{2}$ . The $t_{1}$ and $t_{2}$ respectively are ______. [2026]

Q 16 :

A particle is executing simple harmonic motion. Its amplitude is A and time period is 5 sec. The time required by it to move from $x = A$ to $x = \frac{A}{\sqrt{2}}$ is _________ sec. [2026]

Q 17 :

The velocity of a particle executing simple harmonic motion along $x$ -axis is described as $v^{2} = 50 - x^{2},$ where $x$ represents displacement. If the time period of motion is $\frac{x}{7} s,$ the value of $x$ is ______. [2026]

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	List-I		List-II
A.	$\sin^{2} ω t$	I.	Periodic with time period $T = \frac{π}{ω}$ but not simple harmonic motion (SHM)
B.	$\sin^{3} (2 ω t)$	II.	Periodic with time period $T = \frac{2 π}{ω}$ but not SHM
C.	$\sin (ω t) + \cos (π ω t)$	III.	Periodic with time period $T = \frac{π}{ω}$ and SHM
D.	$\cos ω t + \cos 2 ω t$	IV.	Non-periodic

Topic Question Set

Q 11 : A particle moves according to the law x=acosπt2. The distance covered by it in the time interval between t=0 to t=3 sec is

Q 12 : Time period of a particle executing SHM is 16 s. At time t=2s, it crosses the mean position its amplitude of motion is 322πm. Its velocity at t=4s is

Q 13 : The displacement of simple harmonic oscillator after 3 seconds starting from its mean position is equal to half of its amplitude. The time period of harmonic motion is:

Q 14 : The equation of motion of a particle is given by x=asin(50t+π3) cm. The particle will come to rest at time t1 and it will have zero acceleration at time t2. The t1 and t2 respectively are ______. [2026]

Q 16 : A particle is executing simple harmonic motion. Its amplitude is A and time period is 5 sec. The time required by it to move from x=A to x=A2 is _________ sec. [2026]

Q 17 : The velocity of a particle executing simple harmonic motion along x-axis is described as v2=50-x2, where x represents displacement. If the time period of motion is x7s,the value of x is ______. [2026]

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Q 11 :

A particle moves according to the law $x = a \cos \frac{π t}{2}$ . The distance covered by it in the time interval between $t = 0$ to $t = 3$ sec is

Q 12 :

Time period of a particle executing SHM is 16 s. At time $t = 2 s$ , it crosses the mean position its amplitude of motion is $\frac{32 \sqrt{2}}{π}$ m. Its velocity at $t = 4 s$ is

Q 13 :

The displacement of simple harmonic oscillator after 3 seconds starting from its mean position is equal to half of its amplitude. The time period of harmonic motion is:

Q 14 :

The equation of motion of a particle is given by $x = a \sin (50 t + \frac{π}{3}) cm .$ The particle will come to rest at time $t_{1}$ and it will have zero acceleration at time $t_{2}$ . The $t_{1}$ and $t_{2}$ respectively are ______. [2026]

Q 16 :

A particle is executing simple harmonic motion. Its amplitude is A and time period is 5 sec. The time required by it to move from $x = A$ to $x = \frac{A}{\sqrt{2}}$ is _________ sec. [2026]

Q 17 :

The velocity of a particle executing simple harmonic motion along $x$ -axis is described as $v^{2} = 50 - x^{2},$ where $x$ represents displacement. If the time period of motion is $\frac{x}{7} s,$ the value of $x$ is ______. [2026]