Simple Harmonic Motion (SHM)

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JEE Main 2021

In the given figure, a body of mass $M$ is held between two massless springs, on a smooth inclined plane. The free ends of the springs are attached to firm supports. If each spring has spring constant $k$ , then the frequency of oscillation of given body is

\frac{1}{2 π} \frac{k}{M}

(B)

\frac{1}{2 π} \frac{2 k}{M g s i n α}

(C)

\frac{1}{2 π} \frac{2 k}{M}

(D)

\frac{1}{2 π} \frac{k}{M g s i n α}

JEE Main 2021

LEVELJEE Main

The motion of a mass on a spring, with spring constant $k$ is as shown in figure. The equation of motion is given by $x (t) = A sin ω t + B cos ω t$ with $ω = \frac{k}{m}$ . Suppose that at time $t = 0$ , the position of mass is $x (0)$ and velocity $v (0)$ , then its displacement can also be represented as $x (t) = C cos (ω t - ϕ)$ , where $C$ and $ϕ$ are

(A)

C = \frac{2 v ( 0 ) ^{2}}{ω ^{2}} + x (0)^{2}, ϕ = tan^{- 1} (\frac{v ( 0 )}{x ( 0 ) ω})

(B)

C = \frac{2 v ( 0 ) ^{2}}{ω ^{2}} + x (0)^{2}, ϕ = tan^{- 1} (\frac{x ( 0 ) ω}{2 v ( 0 )})

(C)

C = \frac{v ( 0 ) ^{2}}{ω ^{2}} + x (0)^{2}, ϕ = tan^{- 1} (\frac{x ( 0 ) ω}{v ( 0 )})

(D)

C = \frac{v ( 0 ) ^{2}}{ω ^{2}} + x (0)^{2}, ϕ = tan^{- 1} (\frac{v ( 0 )}{x ( 0 ) ω})

LEVELJEE Main

The maximum velocity of a particle, executing simple harmonic motion with an amplitude $7 mm$ , is $4.4 ms^{- 1}$ . The period of oscillation is

(A)

0.01 s

(B)

10 s

(C)

0.1 s

(D)

100 s

JEE Main 2016

LEVELJEE Main

A particle performs simple harmonic motion with amplitude $A$ . Its speed is trebled at the instant that it is at a distance $\frac{2}{3} A$ from equilibrium position. The new amplitude of the motion is

(A)

\frac{A}{3} 41

(B)

3 A

(C)

A 3

(D)

\frac{7}{3} A

LEVELJEE Main

A wooden cube (density of wood $d$ ) of side $l$ floats in a liquid of density $ρ$ with its upper and lower surfaces horizontal. If the cube is pushed slightly down and released, it performs simple harmonic motion of period, $T$ . Then, $T$ is equal to

(A)

2 π \frac{l ρ}{( ρ - d ) g}

(B)

2 π \frac{l d}{ρ g}

(C)

2 π \frac{l ρ}{d g}

(D)

2 π \frac{l d}{( ρ - d ) g}

LEVELJEE Main

The function $sin^{2} (ω t)$ represents

(A)

a periodic but not simple harmonic motion with a period

2 π / ω

(B)

a periodic but not simple harmonic motion with a period

π / ω

(C)

a simple harmonic motion with a period

2 π / ω

(D)

a simple harmonic motion with a period

π / ω

JEE Main 2021

LEVELJEE Main

A particle performs simple harmonic motion with a period of 2 s. The time taken by the particle to cover a displacement equal to half of its amplitude from the mean position is $1/ a$ s. The value of $a$ to the nearest integer is ......... .

LEVELJEE Main

The length of a simple pendulum executing simple harmonic motion is increased by 21%. The percentage increase in the time period of the pendulum of increased length is

Two particles A and B of equal masses are suspended from two massless springs of spring constants $k_{1}$ and $k_{2}$ , respectively. If the maximum velocities, during oscillations are equal, the ratio of amplitudes of A and B is

(A)

\frac{k _{1}}{k _{2}}

(B)

\frac{k _{1}}{k _{2}}

(C)

\frac{k _{2}}{k _{1}}

(D)

\frac{k _{2}}{k _{1}}

JEE Main 2021

LEVELJEE Advanced

In the reported figure, two bodies A and B of masses $200 g$ and $800 g$ are attached with the system of springs. Springs are kept in a stretched position with some extension when the system is released. The horizontal surface is assumed to be frictionless. The angular frequency will be ......... $rad/s$ when $k = 20 N/m$ .

JEE Main 2019

LEVELJEE Advanced

A rod of mass 'M' and length '2L' is suspended at its middle by a wire. It exhibits torsional oscillations. If two masses each of 'm' are attached at distance 'L/2' from its centre on both sides, it reduces the oscillation frequency by 20%. The value of ratio m/M is close to

Two particles are executing simple harmonic motion of the same amplitude $A$ and frequency $ω$ along the x-axis. Their mean position is separated by distance $X_{0}$ ( $X_{0} > A$ ). If the maximum separation between them is ( $X_{0} + A$ ), the phase difference between their motion is

(A)

\frac{π}{3}

(B)

\frac{π}{4}

(C)

\frac{π}{6}

(D)

\frac{π}{2}

JEE Main 2021

LEVELJEE Main

The variation of displacement with time of a particle executing free simple harmonic motion is shown in the figure. The potential energy $U (x)$ versus time ( $t$ ) plot of the particle is correctly shown in figure :

A simple pendulum oscillating in air has period $T$ . The bob of the pendulum is completely immersed in a non-viscous liquid. The density of the liquid is $\frac{1}{16}$ th of the material of the bob. If the bob is inside liquid all the time, its period of oscillation in this liquid is

(A)

2 T \frac{1}{10}

(B)

2 T \frac{1}{14}

(C)

4 T \frac{1}{14}

(D)

4 T \frac{1}{15}

JEE Main 2021

LEVELBoard

$T_{0}$ is the time period of a simple pendulum at a place. If the length of the pendulum is reduced to $\frac{1}{16}$ times of its initial value, then the modified time period is

(A)

T_{0}

(B)

8 π T_{0}

(C)

4 T_{0}

(D)

\frac{1}{4} T_{0}

JEE Main 2021

LEVELJEE Main

For a body executing SHM A. potential energy is always equal to its kinetic energy. B. average potential and kinetic energy over any given time interval are always equal. C. sum of the kinetic and potential energy at any point of time is constant. D. average kinetic energy in one time period is equal to average potential energy in one time period. Choose the most appropriate option from the options given below.

The displacement of a particle varies according to the relation $x = 4 (cos π t + sin π t)$ . The amplitude of the particle is

(A)

- 4

(B)

4

(C)

42

(D)

8

LEVELJEE Main

The displacement of an object attached to a spring and executing simple harmonic motion is given by $x = 2 \times 1 0^{- 2} cos π t$ metre. The time at which the maximum speed first occurs is

(A)

0.5 s

(B)

0.75 s

(C)

0.125 s

(D)

0.25 s

JEE Main 2005

LEVELJEE Main

If a simple harmonic motion is represented by $\frac{d ^{2} x}{d t ^{2}} + α x = 0$ , its time period is

(A)

\frac{2 π}{α}

(B)

\frac{2 π}{α}

(C)

2 π α

(D)

2 π α

LEVELJEE Main

A coin is placed on a horizontal platform which undergoes vertical simple harmonic motion of angular frequency $ω$ . The amplitude of oscillation is gradually increased. The coin will leave contact with the platform for the first time

(A)

at the mean position of the platform

(B)

for an amplitude of

\frac{g}{ω ^{2}}

(C)

for an amplitude of

\frac{g ^{2}}{ω ^{2}}

(D)

at the highest position of the platform

JEE Main 2007

LEVELJEE Main

A particle of mass $m$ executes simple harmonic motion with amplitude $a$ and frequency $ν$ . The average kinetic energy during its motion from the position of equilibrium to the end is

(A)

π^{2} m a^{2} ν^{2}

(B)

\frac{1}{4} m a^{2} ν^{2}

(C)

4 π^{2} m a^{2} ν^{2}

(D)

2 π^{2} m a^{2} ν^{2}

JEE Main 2015

LEVELJEE Advanced

A pendulum made of a uniform wire of cross-sectional area $A$ has time period $T$ . When an additional mass $M$ is added to its bob, the time period changes $T_{M}$ . If the Young's modulus of the material of the wire is $Y$ , then $\frac{1}{Y}$ is equal to ( $g =$ gravitational acceleration)

(A)

[(\frac{T _{M}}{T})^{2} - 1] \frac{A}{M g}

(B)

[(\frac{T _{M}}{T})^{2} - 1] \frac{M g}{A}

(C)

[1 - (\frac{T _{M}}{T})^{2}] \frac{A}{M g}

(D)

[1 - (\frac{T}{T _{M}})^{2}] \frac{A}{M g}

LEVELJEE Main

If $x$ , $v$ and $a$ denote the displacement, the velocity and the acceleration of a particle executing simple harmonic motion of time period $T$ , then which of the following does not change with time?

(A)

a^{2} T^{2} + 4 π^{2} v^{2}

(B)

\frac{a T}{x}

(C)

a T + 2 π v

(D)

\frac{a T}{v}

JEE Main 2021

LEVELJEE Main

Time period of a simple pendulum is $T$ . The time taken to complete $\frac{5}{8}$ oscillations starting from mean position is $\frac{α}{β} T$ . The value of $α$ is ......... .

LEVELJEE Main

A point mass oscillates along the x-axis according to the law $x = x_{0} cos (ω t - π /4)$ . If the acceleration of the particle is written as $a = A cos (ω t + δ)$ , then

(A)

A = x_{0}, δ = - \frac{π}{4}

(B)

A = x_{0} ω^{2}, δ = \frac{π}{4}

(C)

A = x_{0} ω^{2}, δ = - \frac{π}{4}

(D)

A = x_{0} ω^{2}, δ = \frac{3 π}{4}

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The maximum velocity of a particle, executing simple harmonic motion with an amplitude 7 mm, is 4.4 ms−1. The period of oscillation is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The function sin2(ωt) represents

The length of a simple pendulum executing simple harmonic motion is increased by 21%. The percentage increase in the time period of the pendulum of increased length is

A rod of mass 'M' and length '2L' is suspended at its middle by a wire. It exhibits torsional oscillations. If two masses each of 'm' are attached at distance 'L/2' from its centre on both sides, it reduces the oscillation frequency by 20%. The value of ratio m/M is close to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } T0​ is the time period of a simple pendulum at a place. If the length of the pendulum is reduced to 161​ times of its initial value, then the modified time period is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The displacement of a particle varies according to the relation x=4(cosπt+sinπt). The amplitude of the particle is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If a simple harmonic motion is represented by dt2d2x​+αx=0, its time period is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Time period of a simple pendulum is T. The time taken to complete 85​ oscillations starting from mean position is βα​T. The value of α is ......... .

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A point mass oscillates along the x-axis according to the law x=x0​cos(ωt−π/4). If the acceleration of the particle is written as a=Acos(ωt+δ), then

The maximum velocity of a particle, executing simple harmonic motion with an amplitude $7 mm$ , is $4.4 ms^{- 1}$ . The period of oscillation is

The function $sin^{2} (ω t)$ represents

$T_{0}$ is the time period of a simple pendulum at a place. If the length of the pendulum is reduced to $\frac{1}{16}$ times of its initial value, then the modified time period is

The displacement of a particle varies according to the relation $x = 4 (cos π t + sin π t)$ . The amplitude of the particle is

If a simple harmonic motion is represented by $\frac{d ^{2} x}{d t ^{2}} + α x = 0$ , its time period is

Time period of a simple pendulum is $T$ . The time taken to complete $\frac{5}{8}$ oscillations starting from mean position is $\frac{α}{β} T$ . The value of $α$ is ......... .

A point mass oscillates along the x-axis according to the law $x = x_{0} cos (ω t - π /4)$ . If the acceleration of the particle is written as $a = A cos (ω t + δ)$ , then