Simple Harmonic Motion Example Problems with Solutions PDF

Simple Harmonic Motion Example Problems with Solutions PDF – Physics

Periodic Motion

Any motion which repeats itself after regular interval of time is called periodic or harmonic motion. Motion of hands of a clock, motion of earth around the sun, motion of the needle of a sewing machine are the examples of periodic motion.

Oscillatory Motion

If a particles repeats its motion after a regular interval about a fixed point, motion is said to be oscillatory or vibratory. Oscillatory motion is a constrained periodic motion between precisely fixed limits. Motion of piston in an automobile engine, motion of balance wheel of a watch are the examples of oscillatory motion.

Time Period

Time taken in one complete oscillation is called time period. Or, Time after which motion is repeated is called time period.

Frequency = Frequency is the no. of oscillations completed by oscillating body in unit time interval. Its SI unit is Hertz.

If n = frequency, T = time period, the nT = 1.

Simple Harmonic Motion

If a particle repeats its motion about a fixed point after a regular time interval in such a way that at any moment the acceleration of the particle is directly proportional to its displacement from the fixed point at that moment and is always directed towards the fixed point, then the motion of the particle is called simple harmonic motion.

The fixed point is called mean point or equilibrium point.

Characteristics of SHM

When a particle executing SHM passes through the mean position :

  1. No force acts on the particle.
  2. Acceleration of the particle is zero.
  3. Velocity is maximum.
  4. Kinetic energy is maximum.
  5. Potential energy is zero.

When a particle executing SHM is at the extreme end, then :

  1. Acceleration of the particle is maximum.
  2. Restoring force acting on particle is maximum.
  3. Velocity of particle is zero.
  4. Kinetic energy of particle is zero.
  5. Potential energy is maximum.

Simple Pendulum

If a point mass is suspended from a fixed support with help of a massless and inextensible string, the arrangement is called simple pendulum. The above is an ideal definition. Practically a simple pendulum is made by suspending a small ball (called bob) from a fixed support with the help of a light string.

If the bob of a simple pendulum is slightly displaced from its mean position and then released, it start oscillating in simple harmonic motion. Time period of oscillation of a simple pendulum is given as

T = 2π√(l/g), where l is the effective length of the pendulum and g is the acceleration due to gravity.

Simple Pendulum Example

Let the bob of the pendulum be displaced through a linear displacement ‘x’ and angular displacement θ from its equilibrium position O to A.

When displacement x is taken:

Simple Pendulum Example

  • At position A, the forces acting on the bob after resolving mg, are:
  • (i) mg cosθ, the horizontal component
  • (ii) mg sinθ, the vertical component
  • Since mg cosθ equal and opposite to the tension T, hence they cancel each other.
  • The net force acting on the bob towards its equilibrium = -mg cosθ = F.
  • This force is called Restoring force F = -mg sinθ
  • If θ is very small, sinθ = θ

Simple Pendulum & Compound Pendulum Differences

In case of Simple Pendulum, the distance between the centre of gravity of the suspended body and the axis of suspension is large compared to the dimensions of the suspended body.

In case compound Pendulum, the dimensions of the suspended body are comparable to the distance between the body’s centre of gravity and the axis of suspension.

Simple Harmonic Motion (SHM) Questions and Answer

Question 1 – The velocity of a particle moving with simple harmonic motion is . . . . at the mean position.

(a) zero

(b) minimum

(c) maximum

(d) none

Ans – (c) At mean the value of x = 0. Therefore, it is maximum at mean position. Vmax = ω.r.

Question 2 – The periodic time (tp) is given by

(a) ω / 2 π
(b) 2 π / ω
(c) 2 π × ω
(d) π/ω

Ans – (b) Periodic time is the time taken for one complete revolution of the particle. ∴ Periodic time, tp = 2 π/ω seconds.

Question 3 – The velocity of a particle (v) moving with simple harmonic motion, at any instant is given by

(a) ω √r2 − x2
(b) ω √x2 − r2
(c) ω2 √r2 − x2
(d) ω2√x2 − r2

Ans – (a) Velocity of any particle vN = vsinθ = ω.rsinθ = ω √r2 − x2.

Questions 4 – The maximum acceleration of a particle moving with simple harmonic motion is

a) ω
b) ω.r
c) ω2.r
d) ω2/r

Ans – (c) Acceleration, aN = ω2.rcosθ = ω2.r.

Simple Harmonic Motion PDF 

Candidates can download the Simple Harmonic Motion (SHM) PDF by clicking on below link.

SHM PDF Link


Practice make a man perfect. All the best for your upcoming exam!

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