Question

Which one of the following equations of motion represents simple harmonic motion ?
(where, $$k,{k_0},{k_1}$$  and $$a$$ are all positive.)

A. Acceleration $$ = - {k_0}x + {k_1}{x^2}$$
B. Acceleration $$ = - k\left( {x + a} \right)$$  
C. Acceleration $$ = k\left( {x + a} \right)$$
D. Acceleration $$ = kx$$
Answer :   Acceleration $$ = - k\left( {x + a} \right)$$
Solution :
As we know that, the condition for a body executing $$SHM$$  is $$F = - kx$$
$$\eqalign{ & {\text{So,}}\,a = \frac{F}{m} = - \frac{k}{m}x \cr & {\text{or}}\,a = - {\omega ^2}x \cr & {\text{Acceleration}} \propto - \left( {{\text{displacement}}} \right) \cr & A \propto - y \cr & A = - {\omega ^2}y \cr & A = - \frac{k}{m}y \cr & A = - ky \cr & {\text{Here,}}\,y = x + a \cr & \therefore {\text{Acceleration}} = - k\left( {x + a} \right) \cr} $$

Releted MCQ Question on
Oscillation and Mechanical Waves >> Simple Harmonic Motion (SHM)

Releted Question 1

Two bodies $$M$$ and $$N$$ of equal masses are suspended from two separate massless springs of spring constants $${k_1}$$ and $${k_2}$$ respectively. If the two bodies oscillate vertically such that their maximum velocities are equal, the ratio of the amplitude of vibration of $$M$$ to that of $$N$$ is

A. $$\frac{{{k_1}}}{{{k_2}}}$$
B. $$\sqrt {\frac{{{k_1}}}{{{k_2}}}} $$
C. $$\frac{{{k_2}}}{{{k_1}}}$$
D. $$\sqrt {\frac{{{k_2}}}{{{k_1}}}} $$
View Answer
Releted Question 2

A particle free to move along the $$x$$-axis has potential energy given by $$U\left( x \right) = k\left[ {1 - \exp \left( { - {x^2}} \right)} \right]$$      for $$ - \infty \leqslant x \leqslant + \infty ,$$    where $$k$$ is a positive constant of appropriate dimensions. Then

A. at points away from the origin, the particle is in unstable equilibrium
B. for any finite nonzero value of $$x,$$ there is a force directed away from the origin
C. if its total mechanical energy is $$\frac{k}{2},$$  it has its minimum kinetic energy at the origin.
D. for small displacements from $$x = 0,$$  the motion is simple harmonic
View Answer
Releted Question 3

The period of oscillation of a simple pendulum of length $$L$$ suspended from the roof of a vehicle which moves without friction down an inclined plane of inclination $$\alpha ,$$ is given by

A. $$2\pi \sqrt {\frac{L}{{g\cos \alpha }}} $$
B. $$2\pi \sqrt {\frac{L}{{g\sin \alpha }}} $$
C. $$2\pi \sqrt {\frac{L}{g}} $$
D. $$2\pi \sqrt {\frac{L}{{g\tan \alpha }}} $$
View Answer
Releted Question 4

A particle executes simple harmonic motion between $$x = - A$$  and $$x = + A.$$  The time taken for it to go from 0 to $$\frac{A}{2}$$ is $${T_1}$$ and to go from $$\frac{A}{2}$$ to $$A$$ is $${T_2.}$$ Then

A. $${T_1} < {T_2}$$
B. $${T_1} > {T_2}$$
C. $${T_1} = {T_2}$$
D. $${T_1} = 2{T_2}$$
View Answer

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Simple Harmonic Motion (SHM)

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Basic PhysicsUnit and MeasurementKinematicsLaws of MotionFrictionUniform Circular MotionImpulseMomentumWork Energy and PowerRotational MotionGravitationMechanical Properties of Solids and Fluids
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Modern PhysicsDual Nature of Matter and RadiationAtoms And NucleiAtoms or Nuclear Fission and FusionRadioactivityModern Physics MiscellaneousSemiconductors and Electronic Devices
Oscillation and Mechanical WavesSimple Harmonic Motion (SHM)Waves