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

Q 1 :
The displacement of a particle executing SHM is given by $x = 10$ $\sin (ω t + \frac{π}{3}) m$ . The time period of motion is 3.14s. The velocity of the particle at $t = 0$ is _______ m/s. [2024]

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(10) $x = 10 \sin (ω t + \frac{π}{3})$

$ω = \frac{2 π}{T} = \frac{2 π}{3.14} = 2 rad / s$

$at t = 0, x = 10 s i n \frac{π}{3} = 5 \sqrt{3} m$

$n o w v = ω \sqrt{(A^{2} - x^{2})} = 2 \sqrt{10^{2} - {(5 \sqrt{3})}^{2}}$

$v = 2 \sqrt{100 - 75} = 10 m / s$

Q 2 :
A particle is doing simple harmonic motion of amplitude 0.06 m and time period 3.14 s. The maximum velocity of the particle is ________ cm/s. [2024]

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(12) $Given : A = 0.06 cm$

$T = \frac{2 π}{ω} = 3.14 \Rightarrow ω = 2 rad / s$

$V_{\max} = ω A = 2 \times \frac{6}{100} = 12 cm / s$

Q 3 :
An object of mass 0.2 kg executes simple harmonic motion along x axis with frequency of $(\frac{25}{π}) H z$ . At the position $x = 0.04 m$ the object has kinetic energy 0.5 J and potential energy 0.4 J. The amplitude of oscillation is ________ cm. [2024]

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(6) $Total energy = K . E . + P . E .$

$x = 0.04 m,$ $T . E . = 0.5 + 0.4 = 0.9 J$

$\Rightarrow \frac{1}{2} \times 0.2 {(2 π \times \frac{25}{π})}^{2} \times A^{2} = 0.9$

$\Rightarrow A = 0.06 m = 6 c m$

Q 4 :
The position, velocity and acceleration of a particle executing simple harmonic motion are found to have magnitudes of $4 m,$ $2$ $m s^{- 1}$ and $16$ $m s^{- 2}$ at a certain instant. The amplitude of the motion is $\sqrt{x} m$ where $x$ is ________. [2024]

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(17) $x = 4 m, v = 2 m / s, a = 16 {m / s}^{2}$

$| a | = ω^{2} x$

$\Rightarrow 16 = ω^{2} (4)$

$ω = 2 rad / s$

$v = ω \sqrt{A^{2} - x^{2}}$

$A = \sqrt{\frac{v^{2}}{ω^{2}} + x^{2}} \Rightarrow A = \sqrt{\frac{4}{4} + 16} = \sqrt{17} m$

Q 5 :
A particle of mass 0.50 kg executes simple harmonic motion under force $F = - 50 (N m^{- 1}) x$ . The time period of oscillation is $\frac{x}{35} s$ . The value of $x$ is _______ $(G i v e n π = \frac{22}{7})$ [2024]

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(22) Time period $T = 2 π \sqrt{\frac{m}{k}} = 2 \times \frac{22}{7} \times \sqrt{\frac{0.50}{50}} = \frac{22}{35} s$

On comparing $\frac{x}{35} = \frac{22}{35} \Rightarrow x = 22$

Q 6 :
A particle executes simple harmonic motion with an amplitude of 4 cm. At the mean position, velocity of the particle is 10 cm/s. The distance of the particle from the mean position when its speed becomes 5 cm/s is $\sqrt{α}$ cm, where $α =$ ______ . [2024]

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(12) $v_{m e a n} = A ω \Rightarrow 10 = 4 ω$

which gives, $ω = \frac{5}{2} r a d / s$

$v = ω \sqrt{A^{2} - x^{2}}$

$5 = \frac{5}{2} \sqrt{4^{2} - x^{2}} \Rightarrow x^{2} = 16 - 4 = 12$

$\Rightarrow x = \sqrt{12} c m$

Q 7 :
When the displacement of a simple harmonic oscillator is one third of its amplitude, the ratio of total energy to the kinetic energy is x/8, where x = __________. [2024]

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(9) $E = \frac{1}{2} K A^{2}$

$U = \frac{1}{2} K {(\frac{A}{3})}^{2} = \frac{K A^{2}}{2 \times 9} = \frac{E}{9}$

$K E = E - \frac{E}{9} = \frac{8 E}{9}$

Ratio = $\frac{E}{K E} = \frac{E}{\frac{8 E}{9}} = \frac{9}{8} \Rightarrow x = 9$

Q 8 :
A particle performs simple harmonic motion with amplitude A. Its speed is increased to three times at an instant when its displacement is 2A/3. The new amplitude of motion is $n A / 3$ . The value of $n$ is ______ . [2024]

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(7) $v = ω \sqrt{A^{2} - x^{2}}$

at $x = \frac{2 A}{3}$

$v = ω \sqrt{A^{2} - {(\frac{2 A}{3})}^{2}} = \frac{\sqrt{5} A ω}{3}$

New amplitude = $A^{'}$

$V^{'} = 3 v = \sqrt{5} A ω = ω \sqrt{{(A^{'})}^{2} - {(\frac{2 A}{3})}^{2}}$

$A^{'} = \frac{7 A}{3}$

Topic Question Set

Q 1 :
The displacement of a particle executing SHM is given by $x = 10$ $\sin (ω t + \frac{π}{3}) m$ . The time period of motion is 3.14s. The velocity of the particle at $t = 0$ is _______ m/s. [2024]

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Q 2 :
A particle is doing simple harmonic motion of amplitude 0.06 m and time period 3.14 s. The maximum velocity of the particle is ________ cm/s. [2024]

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Q 3 :
An object of mass 0.2 kg executes simple harmonic motion along x axis with frequency of $(\frac{25}{π}) H z$ . At the position $x = 0.04 m$ the object has kinetic energy 0.5 J and potential energy 0.4 J. The amplitude of oscillation is ________ cm. [2024]

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Q 4 :
The position, velocity and acceleration of a particle executing simple harmonic motion are found to have magnitudes of $4 m,$ $2$ $m s^{- 1}$ and $16$ $m s^{- 2}$ at a certain instant. The amplitude of the motion is $\sqrt{x} m$ where $x$ is ________. [2024]

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Q 5 :
A particle of mass 0.50 kg executes simple harmonic motion under force $F = - 50 (N m^{- 1}) x$ . The time period of oscillation is $\frac{x}{35} s$ . The value of $x$ is _______ $(G i v e n π = \frac{22}{7})$ [2024]

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Q 6 :
A particle executes simple harmonic motion with an amplitude of 4 cm. At the mean position, velocity of the particle is 10 cm/s. The distance of the particle from the mean position when its speed becomes 5 cm/s is $\sqrt{α}$ cm, where $α =$ ______ . [2024]

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Q 7 :
When the displacement of a simple harmonic oscillator is one third of its amplitude, the ratio of total energy to the kinetic energy is x/8, where x = __________. [2024]

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Q 8 :
A particle performs simple harmonic motion with amplitude A. Its speed is increased to three times at an instant when its displacement is 2A/3. The new amplitude of motion is $n A / 3$ . The value of $n$ is ______ . [2024]

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Topic Question Set

Q 1 : The displacement of a particle executing SHM is given by x=10 sin(ωt+π3)m. The time period of motion is 3.14s. The velocity of the particle at t=0 is _______ m/s. [2024]

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Q 2 : A particle is doing simple harmonic motion of amplitude 0.06 m and time period 3.14 s. The maximum velocity of the particle is ________ cm/s. [2024]

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Q 3 : An object of mass 0.2 kg executes simple harmonic motion along x axis with frequency of (25π)Hz. At the position x=0.04m the object has kinetic energy 0.5 J and potential energy 0.4 J. The amplitude of oscillation is ________ cm. [2024]

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Q 4 : The position, velocity and acceleration of a particle executing simple harmonic motion are found to have magnitudes of 4m, 2 ms-1 and 16 ms-2 at a certain instant. The amplitude of the motion is xm where x is ________. [2024]

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Q 5 : A particle of mass 0.50 kg executes simple harmonic motion under force F=-50(Nm-1)x. The time period of oscillation is x35s. The value of x is _______ (Given π=227) [2024]

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Q 6 : A particle executes simple harmonic motion with an amplitude of 4 cm. At the mean position, velocity of the particle is 10 cm/s. The distance of the particle from the mean position when its speed becomes 5 cm/s is α cm, where α= ______ . [2024]

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Q 7 : When the displacement of a simple harmonic oscillator is one third of its amplitude, the ratio of total energy to the kinetic energy is x/8, where x = __________. [2024]

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Q 8 : A particle performs simple harmonic motion with amplitude A. Its speed is increased to three times at an instant when its displacement is 2A/3. The new amplitude of motion is nA/3. The value of n is ______ . [2024]

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Q 1 :
The displacement of a particle executing SHM is given by $x = 10$ $\sin (ω t + \frac{π}{3}) m$ . The time period of motion is 3.14s. The velocity of the particle at $t = 0$ is _______ m/s. [2024]

Q 2 :
A particle is doing simple harmonic motion of amplitude 0.06 m and time period 3.14 s. The maximum velocity of the particle is ________ cm/s. [2024]

Q 3 :
An object of mass 0.2 kg executes simple harmonic motion along x axis with frequency of $(\frac{25}{π}) H z$ . At the position $x = 0.04 m$ the object has kinetic energy 0.5 J and potential energy 0.4 J. The amplitude of oscillation is ________ cm. [2024]

Q 4 :
The position, velocity and acceleration of a particle executing simple harmonic motion are found to have magnitudes of $4 m,$ $2$ $m s^{- 1}$ and $16$ $m s^{- 2}$ at a certain instant. The amplitude of the motion is $\sqrt{x} m$ where $x$ is ________. [2024]

Q 5 :
A particle of mass 0.50 kg executes simple harmonic motion under force $F = - 50 (N m^{- 1}) x$ . The time period of oscillation is $\frac{x}{35} s$ . The value of $x$ is _______ $(G i v e n π = \frac{22}{7})$ [2024]

Q 6 :
A particle executes simple harmonic motion with an amplitude of 4 cm. At the mean position, velocity of the particle is 10 cm/s. The distance of the particle from the mean position when its speed becomes 5 cm/s is $\sqrt{α}$ cm, where $α =$ ______ . [2024]

Q 7 :
When the displacement of a simple harmonic oscillator is one third of its amplitude, the ratio of total energy to the kinetic energy is x/8, where x = __________. [2024]

Q 8 :
A particle performs simple harmonic motion with amplitude A. Its speed is increased to three times at an instant when its displacement is 2A/3. The new amplitude of motion is $n A / 3$ . The value of $n$ is ______ . [2024]