Content Curator
Simple Harmonic Motion(SHM) is the simplest form of oscillatory motion. An oscillatory motion, also called vibratory motion is a special type of periodic motion in which the motion of the particle is to and fro about a mean position in a regular interval of time.
- In SHM, the restoring force on the moving object is considered to be directly proportional to the magnitude of the object’s displacement and constantly acts toward the object’s equilibrium position.
- This results in oscillation, if not stopped by friction, that continues indefinitely.
- The force responsible for motion is directly proportional to distance. Thus, it is
F = −kx
where k is a constant known as restoring force constant. This is commonly known as Hooke’s Law.
The negative sign shows the direction of restoring force is opposite to that of the displacement.
The acceleration of a particle executing Simple Harmonic Motion is given by,
a(t) = -ω2x(t)
Here, ω is the angular velocity of the particle.
Key Takeaways: Simple Harmonic Motion, Periodic Motion, Oscillatory Motion, Linear, Angular, Amplitude, Period, Frequency, Phase, Kinetic Energy, Potential Energy
Simple Harmonic Motion Definition
[Click Here for Sample Questions]
Simple harmonic motion (SHM) is an oscillatory motion. In SHM, the acceleration of a particle at any position is found to be directly proportional to its displacement from the original position.
- SHM can be defined as a special case of oscillatory motion.
- Hence, all the simple harmonic motions can be oscillatory and periodic in nature.
- However, the reverse is not true.
- All oscillatory motions are not SHM.
In SHM, parameters like displacement, velocity, acceleration, and force vary with respect to time. This is represented by sinusoids, commonly known as sine, and the cosine functions. Simple Harmonic Motion elucidates the special characteristics of sound and light waves as well as of alternating currents.
Simple Harmonic Motion
The video below explains this:
Simple Harmonic Motion Detailed Video Explanation:
Difference Between Periodic, Oscillation, and Simple Harmonic Motion
[Click Here for Sample Questions]
Simple Harmonic Motion is a special case of oscillatory motion. Periodic motion, on the other hand, repeats itself regularly at equal time intervals. These types of motion differ with respect to their equilibrium position of the object and restoring force acting on the object. Tabulated below is the difference between periodic, oscillation, and simple harmonic motion.
| Periodic Motion | Oscillatory Motion | Simple Harmonic Motion |
|---|---|---|
| The motion repeats itself at regular time intervals. | Particle motion is to and fro about a mean position. The particle moves on either side of the mean position. | This is a type of oscillatory motion wherein the particles move along a straight line between the two extreme points. The path of SHM thus remains a constant. |
| There is no equilibrium position of the particle. | The particle motion is between two extreme points. | The path of the particle is a straight line. |
| There is no restoring face acting on the particle. | The restoring force is directed toward the equilibrium position. | The restoring force is directed toward the equilibrium position. |
Also Read:
Types of Simple Harmonic Motion
[Click Here for Sample Questions]
There are two types of Simple Harmonic Motion:
- Linear Simple Harmonic Motion
- Angular Simple Harmonic Motion
Linear Simple Harmonic Motion
Linear Simple Harmonic Motion can be defined as, when a particle moves from to and fro from an equilibrium position along with its motion is known as Linear SHM.
- One of the major examples of Linear Simple Harmonic Motion is the spring-mass system.
- The acceleration on a particle must be directly proportional to the displacement of the particle and should be directed to the equilibrium position.
\(\color{red}{\begin{aligned} \vec{F} &\propto-\vec{x} \\ \vec{a}\ &\propto-\vec{x} \end{aligned}}\)
Where
- \(\vec{x}\) is the displacement of the particle from the equilibrium
- \(\vec{F}\) is the Restoring force
- \(\vec{a}\) is the acceleration
Linear Harmonic Oscillator
Angular Simple Harmonic Motion
When a particle is a fixed axis and when a system oscillates angular long along with the fixed axis, then the motion is known as Angular Harmonic Motion.
Hence, Τ ∝ θ or α ∝ θ
Where,
- Τ - Torque
- α - angular acceleration
- θ - angular displacement
Also Read:
Concepts of Simple Harmonic Motion
[Click Here for Sample Questions]
Simple harmonic motion is governed by the following important characteristic concepts.
Amplitude
Amplitude is defined as the maximum displacement of the particle from its mean position. The direction of motion of the particle is away from the mean position. The SI unit of amplitude is meter, m. The dimensions are given by L1M0T0.
Period
Period is the time the particle takes to complete one oscillation. For a simple harmonic motion, the period can be defined as the minimum time taken for the motion to repeat itself. This can be represented as nT where n is an integer and T is the time. The SI unit is seconds, s. The dimensions are given by L0M0T1.
Frequency
Frequency is defined as the total number of oscillations performed by the particle per second. The SI unit is Hertz, Hz or rotations per second, rps. The dimensions are given by L0M0T-1.
Phase
Phase is defined as the state of oscillation for a simple harmonic motion. The magnitude and direction of displacement of particles are hence given by the phase. The phase at the beginning of the motion is termed Epoch(α).
Representation of Amplitude, Period, and Frequency
Also Read:
Expression for Simple Harmonic Motion
[Click Here for Sample Questions]
Mathematically, a simple harmonic motion can be expressed as
x(t) = A (sinωt + Φo) or x(t) = A (cosωt + Φo)
Where
- x = displacement of the body from the mean position at any instant t
- A = maximum displacement or amplitude of displacement of the body
- ω = angular frequency
- Φo = initial phase angle
Velocity of the Particle in SHM
[Click Here for Sample Questions]
The velocity of a particle executing simple harmonic motion is given by
\(v=\omega \sqrt{A^2-x^2}\)
Cases:
- When x = 0, the displacement of the particle is zero or the body is at the mean position. In this case, velocity is maximum, also known as velocity amplitude.
vmax = ωA
- When x = A, the displacement of the particle is maximum or the body is at extreme position, then velocity is minimum.
vmin = 0
Acceleration of the Particle in SHM
[Click Here for Sample Questions]
The acceleration of a particle executing simple harmonic motion is given by
a = – ω2x
Cases:
- When x = 0, the displacement of the particle is zero or the body is at the mean position. In this case, acceleration is minimum.
amin = 0
- When x = A, the displacement of the particle is maximum or the body is at the extreme position, then acceleration is maximum.
amax = – ω2A
Kinetic Energy of a Particle Performing SHM
[Click Here for Sample Questions]
Kinetic Energy can be described as the energy posted by any object when it's in motion. To better understand how an object possesses kinetic energy, consider a particle that possesses it.
Simple Harmonic Motion (SHM) with mass m performing along the path of AD. Let F be the mean position. Therefore,
FA = FB = a.
The instantaneous velocity of the particle performing SHM (Simple Harmonic Motion) at a distance x from the mean position taken as F is given by
v = ±ω √a2–x2
∴ v2 = ω2(a2–x2)
∴ Kinetic energy= ½ mv2 = ½ mω2(a2 –x2)
As,
k/m = ω2
∴ k = mω2
Potential Energy of a Particle Performing SHM
[Click Here for Sample Questions]
Potential Energy can be described as the energy possessed by an object when it is at rest.
This is how to calculate the Potential Energy of a particle performing SHM.
Consider a particle of any mass M performing SHM at a distance of x from its mean position, and consider it as y.
∫dw = ∫kxdx = k ∫x dx
Total work done by a particle = 1/2 kx2 = 1/2 m ω2x2
Here, the potential energy is stored in the form of the total work done.
Therefore Potential Energy = 1/2 kx2 = 1/2 m ω2x2
Characteristics of Simple Harmonic Motion
[Click Here for Sample Questions]
The following are the characteristics of simple harmonic motion
- A restoring force should act on the body performing SHM.
- The direction of the acceleration must be in the opposite direction of the displacement and is directly proportional to it.
- The system must have inertia.
- It is a type of oscillatory motion and a particular case of periodic motion.
- The Energy of the system oscillates between kinetic energy and potential energy but the total energy remains constant, if there is no loss of energy.
Also Read:
Applications of Simple Harmonic Motion
[Click Here for Sample Questions]
Some of the applications of simple harmonic motion in our everyday life are as follows:
- Pendulum or vibrating quartz in clocks
- Wide range of musical instruments like violin and guitar
- Shock absorbers in vehicles
- Adventure activities like bungee jumping
- Diving boards
- Hearing aids
- Metronome
- Making earthquake-proof buildings and other construction sites
Also Read:
Things to Remember
- Simple harmonic motion is a special case of oscillation along a straight line between the two extreme points (the path of SHM is a constraint).
- According to Hooke’s Law, F = -kx.
- A periodic motion repeats itself at regular time intervals.
- In oscillatory motion, the particle motion is to and fro about a mean position. The Path of the object always needs to be a straight line.
- There will be a restoring force directed towards the equilibrium position (or) mean position.
- The mean position in Simple harmonic motion is a stable equilibrium.
- The kinetic energy of a particle performing SHM is given by, k = mω2
- The potential energy of a particle performing SHM is given by, k = 1/2 m ω2x2
Sample Questions
Ques. What is the Geometrical Interpretation of Simple Harmonic Motion? (3 marks)
Ans. If a particle is moving with uniform speed along the circumference of a circle then the straight-line motion of the foot of the perpendicular drawn from the particle on the diameter of the circle is called simple harmonic motion.
Representation of Simple Harmonic Motion
Ques. What is the Problem-Solving Strategy in Horizontal Phasor? (5 marks)
Ans. Following is the problem-solving strategy in a horizontal phasor:
- Let us assume a circle of radius equal to the amplitude of SHM.
- Assume a particle rotating in a circular path moving with constant same as that of simple harmonic motion in the clockwise direction.
- Angle made by the particle at t = 0 with the upper vertical axis equal to φ (phase constant).
- The horizontal component of the velocity of a particle gives you the velocity of a particle performing the simple harmonic motion.
- The component of the acceleration of a particle in the horizontal direction is equal to the acceleration of the particle performing SHM. [In uniform circular acceleration centripetal only ac = ω2A].
Ques. The total mechanical energy of a spring-mass system in simple harmonic motion is E = ½ m ⍵2A2. Let an oscillating particle be replaced by another particle having double the mass. The amplitude A remains the same. Calculate the new mechanical energy. (3 marks)
Ans. The energy of a spring-mass system is given by
E = ½ m ⍵2A2
The Time period of the oscillation is given by,
\(\mathrm{T} = 2\pi \sqrt{\frac{\mathrm{m}}{\mathrm{K}}}\)
Therefore,
\(\frac{\mathrm{T}}{2 \pi}=\sqrt{\frac{\mathrm{m}}{\mathrm{K}}} \Rightarrow \mathrm{w}=\sqrt{\frac{\mathrm{K}}{\mathrm{m}}}\)
Hence,
\(\begin{aligned} &\mathrm{E}=\frac{1}{2} \mathrm{~m} \times \frac{\mathrm{K}}{\mathrm{m}} \times \mathrm{A}^{2} \\ &\mathrm{E}=\frac{1}{2} \mathrm{KA}^{2} \end{aligned}\)
The new mechanical energy will hence remain the same as E itself.
Ques. List the characteristics of oscillatory motion. (5 marks)
Ans. The characteristics are as follows:
- The to and fro motion of a particle about a mean position is called an oscillatory motion in which a particle moves on either side of the equilibrium (or) mean position is an oscillatory motion.
- It is a kind of periodic motion bounded between two extreme points. For example, the Oscillation of a Simple Pendulum, Spring-Mass System.
There is no equilibrium position. - The object will keep on moving between two extreme points about a fixed point called mean position (or) equilibrium position along any path. (the path is not a constraint).
There is no restoring force. - There will be a restoring force directed towards the equilibrium position (or) mean position.
- In an oscillatory motion, the net force on the particle is zero at the mean position.
- The mean position is a stable equilibrium position.
Ques. Derive the equation for the frequency of small-angle oscillations. (5 marks)
Ans. Any motion below 15° is termed small-angle oscillation. This is with respect to the equilibrium point as the displacements are approximately linear.
For example, a grandfather clock is a small-angle oscillation The period is T of the pendulum, the lengths L and g the acceleration due to gravity are related as
\(\mathrm{T} = 2\pi \sqrt{\frac{\mathrm{L}}{\mathrm{g}}}\)
Frequency is the reciprocal of the time period.
Therefore,
\(f = \frac{1}{T}\) and \(T = \frac{1}{f}\)
Frequency can hence be given by the equation,
\(\begin{aligned} f &=\frac{1}{T} \\ &=\frac{1}{2 \pi \sqrt{\frac{L}{g}}} \\ &=\frac{\sqrt{g}}{2 \pi \sqrt{L}} \end{aligned}\)
Ques. If a pendulum clock is taken to the top of a mountain, does the clock lose or gain time under the assumption that the clock works perfectly at a lower height? (2 marks)
Ans. The time period of a simple pendulum, T is inversely proportional to the square root of the center of gravity, g.
\(T \propto \frac{1}{\sqrt{g}}\)
The value of g decreases as we move up and is lower at the mountain top. From the above relation, we can say that value of T increases. This causes the pendulum to take a longer time for a single oscillation making the clock slow. Thus the pendulum clock loses time on the mountaintop.
Ques. Find an expression for the time period and frequency of a liquid in a U-tube. (5 marks)
Ans. Consider liquid in a U-tube having a uniform cross-sectional area, A as follows:
The liquid density is ⍴, liquid in both limbs is the same and the total tube length is L.
Mass of liquid in tube = LA⍴
If the liquid level in the left limb is depressed from P to Q by a small distance y, the liquid rises from P1 to Q1 in the right limb by the same distance y.
The difference in liquid level of the two limbs is QQ1 = 2y
The restoring force is given by
\(F=-(A 2 y) \rho g=-(2 A \rho g) y=-K y\)
The periodic time is given by
\(T=2 \pi \sqrt{\frac{\text { Intertia factor }}{\text { Spring factor }}}=2 \pi \sqrt{\frac{L A \rho}{2 A \rho g}}\)
\(T=2 \pi \sqrt{\frac{L}{2 g}} \quad \text { and frequency, } v=\frac{1}{T}=\frac{1}{2 \pi} \sqrt{\frac{2 g}{L}}\)
When h is the height of the undisturbed liquid in each limb, L= 2h.
\(\therefore T=2 \pi \sqrt{\frac{2 h}{2 g}}=2 \pi \sqrt{\frac{h}{g}}\)
Ques. The bob of a vibrating pendulum is made of ice. How does the period of the swing change when the ice starts melting? (2 marks)
Ans. The period of swing of the simple pendulum stays the same until the location of the bob’s center of gravity post-melting of ice remains at a fixed distance from the point of suspension. So, after the melting of ice, if the center of gravity is raised upwards, the pendulum length decreases. The time period of the swing hence decreases. When the center of gravity of the ice shifts on the lower side, the swing’s time period increases.
Also Read:
Comments