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Simple Harmonic Motion chapter is the part of mechanics and also acts as a backbone for other chapters or topics like Wave Motion, Kinematics etc. These chapters are interrelated and if you skip one chapter or topic then you might face difficulty in solving the questions. Other than that you have to pay special attention in learning the equations and applications of Simple Harmonic Motion so that you can easily understand the diagrams or identify variables in the questions during the exam.

- The weightage of questions from SHM is not very high. Around one to two can be expected from this chapter.
- The major topics from this chapter are SHM equation, Oscillation Motion, Time Period of Simple Harmonic Motion, Energy in SHM, etc.
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The chapter involves a lot of formulas and the key to learn and understand all those formulas is that you have to take it as an important chapter and get completely immersed in it.

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Simple Harmonic Motion is a motion in which the restoring force is directly proportional to the displacement of the body from its mean position. The direction of this force is always towards the mean position.

Acceleration of a particle executing SHM is given by,

a(t) = -ω^{2} x(t)

ω is the angular velocity of the particle.

- SHM is an oscillatory motion where the acceleration of the particle at any point is directly proportional to the displacement from the mean position. It is a special case of oscillatory motion.
- All the Simple Harmonic Motions are oscillatory and also periodic but vice versa is not the same.
- Oscillatory motion is also known as the harmonic motion.

**Question: Let us consider that two particles execute Simple Harmonic Motion of the same amplitude and frequency along the same straight line. It has been seen that they pass one another when going in opposite directions, and it has been noticed that each time their displacement is half of their amplitude. The phase difference between them is **

- 30?
- 60?
- 90?
- 120?

**Solution: (4)**

Terms | Periodic Motion | Oscillation Motion | Simple Harmonic Motion |
---|---|---|---|

Definition | A motion repeats itself after an equal interval of time. | To and fro motion of a particle about its mean position is called an oscillatory motion in which a particle moves on either side mean position is an oscillatory motion. | It is a special case of oscillation along with straight line between the two extreme points (the path of SHM is a constraint) |

Example | Uniform circular motion. | Oscillation of Simple Pendulum, Spring-Mass System. | - |

Force | There is no restoring force. | There will be a restoring force directed towards equilibrium position (or) mean position. The net force on the particle is zero at the mean position of oscillatory motion. | There will be a restoring force directed towards equilibrium position (or) mean position |

Equilibrium Positions | There is no stable equilibrium position | The mean position is a stable equilibrium position. | Mean position in Simple harmonic motion is also called a stable equilibrium. |

Other Features | - | It is a kind of periodic motion which is bounded between two extreme points. The object that keeps on moving between two extreme points about a fixed point is called mean position. | The path of the object needs to be a straight line. |

**Question: The displacement of a particle varies with time as x=12sinωt−16sin**^{3}**ωt(in cm). If its motion is Simple Harmonic Motion then its maximum acceleration is**

- 12 ω
^{2} - 36 ω
^{2} - 144 ω
^{2} - 29 ω
^{2}

**Solution: (2)**

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SHM or Simple Harmonic Motion can be classified into two types,

- Linear SHM
- Angular SHM

**Linear SHM:** When a particle moves to and fro about a fixed point along with a straight line then its motion is called linear Simple Harmonic Motion.

**Example:** spring-mass system

**Angular SHM:** When a system oscillates angular long with respect to a fixed axis then its motion is called angular angular simple harmonic motion.

**1. Mean Position:** The point at which net force acting on the particle is zero.

From the mean position, the force acting on the particle is,

**F∝−x a∝−x**

The force acting on the particle is negative of the displacement. So this will be a stable equilibrium.

**2. Amplitude in SHM**

It is the maximum displacement of the particle from the mean position.

**3. Time Period and Frequency of SHM:** The minimum time after which the particle keeps on repeating its motion is known as the time period or the smallest time taken to complete one oscillation is also defined as the time period.

T = 2π\ω

**4. Frequency:** The number of oscillations per second is defined as the frequency.

Frequency = 1/T and, angular frequency ω = 2πf = 2π/T

**5: Phase in SHM: **The phase of a vibrating particle at any instant is the state of the oscillating particle regarding its displacement and direction of vibration at that particular instant.

The expression, position of a particle as a function of time.

x = A sin (ωt + Φ)

Where (ωt + Φ) is the phase of the particle

The phase angle at time t = 0 is known as the initial phase.

**6: Phase Difference:** The difference of total phase angles of two particles executing simple harmonic motion with respect to the mean position is known as the phase difference.

**Question: Let us consider that a linear harmonic oscillator of force constant 2×106N/m times and amplitude 0.01 m has a total mechanical energy of 160 joules. Its **

- Maximum potential energy is 100 J
- Maximum K.E. is 100 J
- Maximum P.E. is 160 J
- Minimum P.E. is zero

**Solution: (2)**

Consider a particle of mass (m) executing Simple Harmonic Motion along a path and the mean position at O. Let the speed of the particle be v0 when it is at position p (at a distance no from O)

At t = 0 the particle at P (moving towards the right)

At t = t the particle is at Q (at a distance x from O)

With a velocity (v)the restoring force F at Q is given by

⇒ F =−Kx, where K – is positive constant

⇒ F = ma, where a - is the acceleration at Q

⇒ ma =−Kx⇒ a=−(mK)x

Put, mK = ω^{2}

⇒ ω = √mK

⇒ a = −(mK)/m =−ω^{2}x

Since, [a = d^{2}x/dt^{2}]

Therefore d^{2}x/dt^{2} = −ω^{2}xd^{2}x/dt^{2} + ω^{2}x = 0, which is the differential equation for linear simple harmonic motion.

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A body free to rotate along an axis can make angular oscillations. For example, a photo frame or a calendar suspended from a nail on the wall. When it will be slightly pushed from its mean position and then released, it makes angular oscillations.

**Example:**

- Simple pendulum
- Seconds pendulum
- The physical pendulum
- Torsional pendulum

Consider a particle executing SHM between A and A1 about passing through the mean position (O). Its analysis is as follows

Displacement | x = -A | x = 0 | x = +A |

Acceleration | |a| = Max | a = 0 | |a| = max |

Speed | |v| = 0 | |v| = Max | |v| = 0 |

Kinetic energy | KE = 0 | KE = Max | KE = 0 |

Potential energy | PE = Max | PE = Min | PE = Max |

**Question: A sphere has a radius r is kept on a concave mirror that has a radius of curvature R. Then this arrangement is kept on a horizontal table. Now ff the sphere is displaced from its equilibrium position and left, then it executes Simple Harmonic Motion. The period of oscillation will be**

**(4 options are given in the image below)**

**Solution: (2)**

The coefficient of t is ω. So the time period T = 2π/ω

ω =2π/T = 2πf

ωt = angular frequency of SHM.

From the expression of particle position as a function of time: We can find particles, displacement (x), velocity (v), and acceleration as follows.

Velocity in SHM is given by v = dx/dt,

x = A sin (ωt + Φ)

v = d/dt Asin(ωt+ϕ) = ωAcos(ωt+ϕ)

v = Aω√1−sinωt

Since, x = A sin ωt

x^{2}/A^{2} = sin2ωt

⇒ v = Aω√1−x^{2}/A^{2}

⇒ v = ωA^{2}−x^{2}

On squaring both sides

⇒v^{2} = ω^{2}(A^{2}−x^{2})

⇒v^{2}/ω^{2} = (A^{2}−x^{2})

⇒v^{2}/ω^{2}A^{2} = (1−x^{2}/A^{2})

⇒v^{2}/A^{2} + v^{2}/ω^{2}A^{2} = 1 this is an equation of Ellipse

**Eclipse is the curve between displacement and velocity of a particle executing the simple harmonic motion.**

When ω = 1 then, the curve between v and x will be circular.

a = dv/dt = d/dt(Aωcos/ωt + ϕ)

⇒ a = −ω^{2}Asin(ωt+ϕ)

⇒ ∣a∣ = −ω^{2}x

**Expression for displacement, velocity, and acceleration in linear simple harmonic motion is**

- x = A sin ( ωt + Φ)
- v = Aωcos(ωt+ϕ) = ω√A
^{2}−x^{2}and - a = −Aω
^{2}sin(ωt+ϕ) = −ω^{2}x

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The system that executes SHM is called the harmonic oscillator.

Consider a particle of mass m, executing linear simple harmonic motion of angular frequency (ω) and amplitude (A) the displacement (x), velocity (v) and acceleration (a) at any time t are given by

x = A sin (ωt + Φ)

v = Aωcos(ωt+ϕ) = ω√A^{2}−x^{2}

a = −ω^{2}Asin(ωt+ϕ) = −ω^{2}x

**The restoring force (F) acting on the particle is given by**

F = -kx where k = mω^{2}.

Kinetic Energy of SHM

Kinetic Energy = 1/2mv^{2}

[Since,v^{2} = A^{2}ω^{2}cos^{2}(ωt+ϕ)]

= 1/2mω^{2}A^{2}cos^{2}(ωt+ϕ)

= 1/2mω^{2}(A^{2}−x^{2})

Kinetic Energy = 1/2mω^{2}A^{2}cos^{2}(ωt+ϕ)=1/2mω^{2}(A^{2}−x^{2})

The total work done by the restoring force in displacing the particle from (x = 0) (mean position) to x = x:

When the particle has been displaced from x to x + dx the work done by restoring force is

dw = F dx = -kx dx

**Total Mechanical Energy of the Particle Executing SHM**

E = KE + PE

E = ½ mω^{2}(A^{2} - x^{2}) + ½ mω^{2}x^{2}

E = ½ mω^{2}A^{2}

Hence the total energy of the particle in SHM is constant and it is independent of the instantaneous displacement.

⇒ Relationship between Kinetic Energy, Potential Energy and time in Simple Harmonic Motion at t = 0, when x = ±A.

⇒ Variation of Kinetic Energy and Potential Energy in Simple Harmonic Motion with displacement:

**1. Question. A particle that has a mass of 0.1 kg executes SHM under a force F = (-10x) N. The speed of the particle at mean position is 6 m/s. Then the amplitude of oscillation is**

**Solution:** From the conservation of mechanical energy,

½ kA^{2} = ½ mv^{2}

Or, A = v√(m/k) = 6√(0.1/10) = 6/10 = 0.6 m

**2. Question: A particle of mass m is executing oscillations about the origin on the x-axis. The potential energy is U(x)=k[x]3 where k is the positive constant. Let us consider the amplitude of oscillation is a, then calculate its time period T?**

**Solution:** U=k|x|3⇒F==−dUdx=−3k|x|2...(i)

Also, for SHM

x=asinωt

and

d^{2}xdt^{2}+ω^{2}x=0

⇒acceleration =d^{2}xdt^{2}=−ω^{2}x⇒F=ma =md^{2}xdt^{2}=−mω^{2}x ...(ii)

From equation (i) & (ii) we get

ω=3k/xm

⇒T=2πω=2π√m3kx−−−−=2π√m/3k(asinωt)

⇒T∝1/√a

**3. Question: The displacement of a particle varies with time as x=12sinωt−16sin3ωt (in cm). If its motion is S.H.M, then calculate its maximum acceleration?**

**Solution:** x=12sinωt−16sin3ωt=4[3sinωt−4sin3ωt]

=4[sin3ωt]

(By using sin3θ=3sinθ−4sin3θ)

∴maximum acceleration

Amax=(3ω)2×4=36ω2

**4. Question: The total mechanical energy of a linear harmonic oscillator of force constant 2×106N/m times and amplitude 0.01 m is 160 joules. Its maximum potential energy or Kinetic energy is**

**Solution:** Harmonic oscillator has some initial elastic potential energy and amplitude of harmonic variation of energy is 12Ka2=12×2×106(0.01)2=100J

Kmax=100J

- Simple harmonic motion is a type of Vibratory motion
- The energy of SHM is maximum at the mean position, extreme position and in between mean and extreme
- A particle oscillates with undamped simple harmonic motion. It is least when the speed is greatest.
- The graph of acceleration against displacement is a straight line.
- The damping force is always in the opposite direction to the acceleration.

- Oscillating system that always experiences a force acting against the displacement of the system (restoring force).
- Restoring force always acts against the displacement of the system.
- SHM experiences a restoring force proportional to the displacement of the system.
- SHM is typified by the motion of a mass on a spring when it is subject to the linear elastic restoring force given by Hooke’s Law.

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