Simple Harmonic Motion or SHM is defined as a motion in which the restoring force is directly proportional to the displacement of the body from its mean position. The direction of this restoring force is always towards the mean position. The acceleration of a particle executing simple harmonic motion is given by a(t) = -ω2 x(t). Here, ω is the angular velocity of the particle. Simple harmonic motion can be described as an oscillatory motion in which the acceleration of the particle at any position 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 not all oscillatory motions are SHM. Oscillatory motion is also called the harmonic motion of all the oscillatory motions, wherein the most important one is simple harmonic motion (SHM). In this type of oscillatory motion, displacement, velocity and acceleration, and force vary (w.r.t time) in a way that can be described by either sine (or) the cosine functions collectively called sinusoids. Also Read: The study of Simple Harmonic Motion is very useful and forms an important tool in understanding the characteristics of sound waves, light waves and alternating currents. Any oscillatory motion which is not simple Harmonic can be expressed as a superposition of several harmonic motions of different frequencies. Conditions for SHM: \(\begin{array}{l}\begin{matrix} \overrightarrow{F}\propto -\overrightarrow{x} \\ \overrightarrow{a}\,\,\propto -\overrightarrow{x} \\ \end{matrix}\end{array} \) SHM or Simple Harmonic Motion can be classified into two types: When a particle moves to and fro about a fixed point (called equilibrium position) along with a straight line then its motion is called linear Simple Harmonic Motion. For Example spring-mass system The restoring force or acceleration acting on the particle should always be proportional to the displacement of the particle and directed towards the equilibrium position. \(\begin{array}{l}\begin{matrix} \overrightarrow{F}\propto -\overrightarrow{x} \\ \overrightarrow{a}\,\,\propto -\overrightarrow{x} \\ \end{matrix}\end{array} \) \(\begin{array}{l}\overrightarrow{x}- \text{displacement of particle from equilibrium position}\end{array} \) \(\begin{array}{l}\overrightarrow{F}- \text{Restoring force}\end{array} \) \(\begin{array}{l}\overrightarrow{a}- \text{acceleration}\end{array} \) When a system oscillates angular long with respect to a fixed axis, then its motion is called angular simple harmonic motion. The restoring torque (or) Angular acceleration acting on the particle should always be proportional to the angular displacement of the particle and directed towards the equilibrium position. Τ ∝ θ or α ∝ θ Where, The point at which net force acting on the particle is zero. From the mean position, the force acting on the particle is, \(\begin{array}{l}\overrightarrow{F}\propto -\overrightarrow{x}\end{array} \) \(\begin{array}{l}\overrightarrow{a}\,\propto -\overrightarrow{x}\end{array} \) \(\begin{array}{l}\overrightarrow{{{F}_{net}}}=0\\\overrightarrow{a}=0\end{array} \) The force acting on the particle is negative of the displacement. So, this point of equilibrium will be a stable equilibrium. It is the maximum displacement of the particle from the mean position. Time Period and Frequency of SHMThe minimum time after which the particle keeps on repeating its motion is known as the time period (or) the shortest time taken to complete one oscillation is also defined as the time period. T = 2π/ω Frequency: The number of oscillations per second is defined as the frequency. Frequency = 1/T and, angular frequency ω = 2πf = 2π/T Phase in SHMThe phase of a vibrating particle at any instant is the state of the vibrating (or) oscillating particle regarding its displacement and direction of vibration at that particular instant. The expression, and 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. Phase DifferenceThe difference of total phase angles of two particles executing simple harmonic motion with respect to the mean position is known as the phase difference. Two vibrating particles are said to be in the same phase; the phase difference between them is an even multiple of π. ΔΦ = nπ where n = 0, 1, 2, 3, . . . . . Two vibrating particles are said to be in the opposite phases if the phase difference between them is an odd multiple of π. ΔΦ = (2n + 1) π where n = 0, 1, 2, 3, . . . . . Simple Harmonic Motion Equation and its SolutionConsider a particle of mass (m) executing Simple Harmonic Motion along a path x o x; 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)  \(\begin{array}{l}\text{The restoring force}\ \overrightarrow{F}\ \text{at Q is given by}\end{array} \) \(\begin{array}{l}\overrightarrow{F}=-K\overrightarrow{x}\end{array} \) K – is a positive constant \(\begin{array}{l}\Rightarrow \overrightarrow{F}=m\overrightarrow{a}\end{array} \) Here, \(\begin{array}{l}\overrightarrow{a}- \text{acceleration at Q}\end{array} \) \(\begin{array}{l}\Rightarrow m\overrightarrow{a}=-K\overrightarrow{x}\end{array} \) \(\begin{array}{l}\Rightarrow \overrightarrow{a}=-\left( \frac{K}{m} \right)\overrightarrow{x}\end{array} \) \(\begin{array}{l}Put\ \frac{K}{m}={{\omega }^{2}}\end{array} \) \(\begin{array}{l}\Rightarrow \omega =\sqrt{\frac{K}{m}}\end{array} \) \(\begin{array}{l}\Rightarrow \overrightarrow{a}=-\left( \frac{K}{m} \right)\overrightarrow{m}=-{{\omega }^{2}}\overrightarrow{x}\end{array} \) Since,\(\begin{array}{l}\left[ \overrightarrow{a}=\frac{{{d}^{2}}x}{d{{t}^{2}}} \right]\end{array} \) \(\begin{array}{l}\frac{{{d}^{2}}\overrightarrow{x}}{d{{t}^{2}}}=-{{\omega }^{2}}\overrightarrow{x}\end{array} \) d2x/dt2 + ω2x = 0, which is the differential equation for linear simple harmonic motion. Solutions of Differential Equations of SHMThe differential equation for the Simple harmonic motion has the following solutions:
These solutions can be verified by substituting these x values in the above differential equation for the linear simple harmonic motion. Angular Simple Harmonic MotionA body free to rotate about an axis can make angular oscillations. For example, a photo frame or a calendar suspended from a nail on the wall. If it is slightly pushed from its mean position and released, it makes angular oscillations. Conditions for an Angular Oscillation to be Angular SHMThe body must experience a net Torque that is restored in nature. If the angle of oscillation is small, this restoring torque will be directly proportional to the angular displacement. Τ ∝ – θ Τ = – kθ Τ = Iα α = – kθ \(\begin{array}{l}I\frac{{{d}^{2}}\theta }{d{{t}^{2}}}=-K\theta\end{array} \) \(\begin{array}{l}\frac{{{d}^{2}}\theta }{d{{t}^{2}}}=-\left( \frac{K}{I} \right)\theta =-\omega _{0}^{2}\theta\end{array} \) \(\begin{array}{l}\frac{{{d}^{2}}\theta }{d{{t}^{2}}}=-\omega _{0}^{2}\theta =0\end{array} \) This is the differential equation of an angular Simple Harmonic Motion. Solution of this equation is the angular position of the particle with respect to time. \(\begin{array}{l}\theta ={{\theta }_{0}}\sin \left( {{\omega }_{0}}t+\phi \right)\end{array} \) Then angular velocity, \(\begin{array}{l}\omega ={{\theta }_{0}}.\,{{\omega }_{0}}\cos \left( {{\omega }_{0}}t+\phi \right)\end{array} \) θ0 – amplitude of the angular SHM Example:
Quantitative Analysis of SHMLet us consider a particle executing Simple Harmonic Motion between A and A1 about passing through the mean position (or) equilibrium position (O). Its analysis is as follows SHM about Position O
Equation of Position of a Particle as a Function of TimeLet us consider a particle, which is executing SHM at time t = 0, and the particle is at a distance from the equilibrium position. Necessary conditions for Simple Harmonic Motion
Some conditions we know: At point A v = 0 [x = A] the equation (1) becomes \(\begin{array}{l}\frac{{{v}^{2}}}{2}=\frac{-{{\omega }^{2}}{{A}^{2}}}{2}+c\end{array} \) Using, v = 0\(\begin{array}{l}0 =\frac{-{{\omega }^{2}}{{A}^{2}}}{2}+c\end{array} \) \(\begin{array}{l}c =\frac{{{\omega }^{2}}{{A}^{2}}}{2}\end{array} \) Sub the value of C in equation (1) \(\begin{array}{l}\frac{{{v}^{2}}}{2}=\frac{-{{\omega }^{2}}{{x}^{2}}}{2}+\frac{{{\omega }^{2}}{{A}^{2}}}{2}\end{array} \) \(\begin{array}{l}\Rightarrow {{v}^{2}}=-{{\omega }^{2}}{{x}^{2}}+{{\omega }^{2}}{{A}^{2}}\end{array} \) \(\begin{array}{l}\Rightarrow {{v}^{2}}={{\omega }^{2}}\left( {{A}^{2}}-{{x}^{2}} \right)\end{array} \) \(\begin{array}{l}v = \sqrt{{{\omega }^{2}}\left( {{A}^{2}}-{{x}^{2}} \right)}\end{array} \) \(\begin{array}{l}v = \omega \sqrt{{{A}^{2}}-{{x}^{2}}}….(2)\end{array} \) where v is the velocity of the particle executing simple harmonic motion from definition instantaneous velocity \(\begin{array}{l}v =\frac{dx}{dt}=\omega \sqrt{{{A}^{2}}-{{x}^{2}}}\end{array} \) \(\begin{array}{l}\Rightarrow \int{\frac{dx}{\sqrt{{{A}^{2}}-{{x}^{2}}}}}=\int\limits_{0}^{t}{\omega dt}\end{array} \) \(\begin{array}{l}\Rightarrow {{\sin }^{-1}}\left( \frac{x}{A} \right)=\omega t+\phi\end{array} \) x = Asin ( ωt + Φ) . . . . . (3) The equation (3) – equation of position of a particle as a function of time. Case 1: If at t = 0 The particle at x = x0 \(\begin{array}{l}\Rightarrow {{\sin }^{-1}}\left( \frac{x}{A} \right)=\omega t+\phi\end{array} \) \(\begin{array}{l}\Rightarrow {{\sin }^{-1}}\left( \frac{{{x}_{0}}}{A} \right)=\phi\end{array} \) Φ is the initial phase of the particle. Case 2: If at t = 0 The particle at x = 0 \(\begin{array}{l}{{\sin }^{-1}}\left( \frac{O}{A} \right)=\phi\end{array} \) i.e. Φ = 0 Case 3: If the particle is at one of its extreme positions x = A at t = 0 \(\begin{array}{l}\Rightarrow {{\sin }^{-1}}\left( \frac{A}{A} \right)=\phi\end{array} \) \(\begin{array}{l}\Rightarrow {{\sin }^{-1}}\left( 1 \right)=\phi\end{array} \) ⇒ π/2 = Φ So, the value can be anything depending upon the position of the particle at t = 0. That is why it is called the initial phase of the particle. Now, if we see the equation of position of the particle with respect to time π/2 = x = A sin ( ωt + Φ) sin (ωt + Φ) – is the periodic function, whose period is T = 2π/ω Which can be anything sine function or cosine function Time Period of SHMThe 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: \(\begin{array}{l}\text{We can find particles, displacement}\ \left( \overrightarrow{x} \right), \text{velocity}\ \left( \overrightarrow{v} \right) \text{and acceleration as follows}.\end{array} \) Velocity in SHM is given by v = dx/dt, x = A sin (ωt + Φ) \(\begin{array}{l}v = \frac{d}{dt}A\sin \left( \omega t+\phi \right)=\omega A\cos \left( \omega t+\phi \right)\end{array} \) \(\begin{array}{l}v = A\omega \sqrt{1-{{\sin }^{2}}\omega t}\end{array} \) Since, x = A sin ωt \(\begin{array}{l}\frac{{{x}^{2}}}{{{A}^{2}}}={{\sin }^{2}}\omega \,t\end{array} \) \(\begin{array}{l}\Rightarrow v = A\omega \sqrt{1-\frac{{{x}^{2}}}{{{A}^{2}}}}\end{array} \) \(\begin{array}{l}\Rightarrow v = \omega \sqrt{{{A}^{2}}-{{x}^{2}}}\end{array} \) On squaring both sides \(\begin{array}{l}\Rightarrow {{v}^{2}}={{\omega }^{2}}\left( {{A}^{2}}-{{x}^{2}} \right)\end{array} \) \(\begin{array}{l}\Rightarrow \frac{{{v}^{2}}}{{{\omega }^{2}}}=\left( {{A}^{2}}-{{x}^{2}} \right)\end{array} \) \(\begin{array}{l}\Rightarrow \frac{{{v}^{2}}}{{{\omega }^{2}}{{A}^{2}}}=\left( 1-\frac{{{x}^{2}}}{{{A}^{2}}} \right)\end{array} \) \(\begin{array}{l}\Rightarrow \frac{{{v}^{2}}}{{{A}^{2}}}+\frac{{{v}^{2}}}{{{A}^{2}}{{\omega }^{2}}}=1\end{array} \) this is an equation of an ellipse.The curve between displacement and velocity of a particle executing the simple harmonic motion is an ellipse. When ω = 1, then the curve between v and x will be circular. Acceleration in SHM\(\begin{array}{l}\overrightarrow{a}=\frac{dv}{dt}=\frac{d}{dt}\left( A\omega \cos \omega t+\phi \right)\end{array} \) \(\begin{array}{l}\Rightarrow \overrightarrow{a}=-{{\omega }^{2}}A\sin \left( \omega t+\phi \right)\end{array} \) \(\begin{array}{l}\Rightarrow \left| a \right|=-{{\omega }^{2}}x\end{array} \) Hence the expression for displacement, velocity and acceleration in linear simple harmonic motion are
Energy in Simple Harmonic Motion (SHM)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), \(\begin{array}{l}\text{the displacement}\ \left( \overrightarrow{x} \right),\ \text{velocity}\ \left( \overrightarrow{v} \right)\ \text{and acceleration}\ \left( \overrightarrow{a} \right)\ \text{at any time t are given by}\end{array} \) x = A sin (ωt + Φ) \(\begin{array}{l}v = A\omega \cos \left( \omega t+\phi \right)=\omega \sqrt{{{A}^{2}}-{{x}^{2}}}\end{array} \) \(\begin{array}{l}a = -{{\omega }^{2}}A\sin \left( \omega t+\phi \right)=-{{\omega }^{2}}x\end{array} \) \(\begin{array}{l}\text{The restoring force}\ \left( \overrightarrow{F} \right)\ \text{acting on the particle is given by}\end{array} \) F = -kx, where k = mω2. Kinetic Energy of a Particle in SHMKinetic Energy \(\begin{array}{l}=\frac{1}{2}m{{v}^{2}}\ \left[ Since, \;{{v}^{2}}={{A}^{2}}{{\omega }^{2}}{{\cos }^{2}}\left( \omega t+\phi \right) \right]\end{array} \) \(\begin{array}{l}=\frac{1}{2}m{{\omega }^{2}}{{A}^{2}}{{\cos }^{2}}\left( \omega t+\phi \right)\end{array} \) \(\begin{array}{l}=\frac{1}{2}m{{\omega }^{2}}\left( {{A}^{2}}-{{x}^{2}} \right)\end{array} \) Therefore, the Kinetic Energy \(\begin{array}{l}=\frac{1}{2}m{{\omega }^{2}}{{A}^{2}}{{\cos }^{2}}\left( \omega t+\phi \right)=\frac{1}{2}m{{\omega }^{2}}\left( {{A}^{2}}-{{x}^{2}} \right)\end{array} \) Potential Energy of SHMThe 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 \(\begin{array}{l}w = \int{dw}=\int\limits_{0}^{x}{-kxdx=\frac{-k{{x}^{2}}}{2}}\end{array} \) \(\begin{array}{l}= -\frac{m{{\omega }^{2}}{{x}^{2}}}{2}\end{array} \) \(\begin{array}{l}\left[ \,k=m{{\omega }^{2}} \right]\end{array} \) \(\begin{array}{l}= -\frac{m{{\omega }^{2}}}{2}{{A}^{2}}{{\sin }^{2}}\left( \omega t+\phi \right)\end{array} \) Potential Energy = -(work done by restoring force) \(\begin{array}{l}=\frac{m{{\omega }^{2}}{{x}^{2}}}{2}=\frac{m{{\omega }^{2}}{{A}^{2}}}{2}{{\sin }^{2}}\left( \omega t+\phi \right)\end{array} \) E = KE + PE \(\begin{array}{l}E = \frac{1}{2}m{{\omega }^{2}}\left( {{A}^{2}}-{{x}^{2}} \right)+\frac{1}{2}m{{\omega }^{2}}{{x}^{2}}\end{array} \) \(\begin{array}{l}E = \frac{1}{2}m{{\omega }^{2}}{{A}^{2}}\end{array} \) Hence, the particle’s total energy in SHM is constant, 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: Geometrical Interpretation of Simple Harmonic MotionIf 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. SHM as a Projection of Circular MotionThe particle is at position P at t = 0 and revolves along a circle with a constant angular velocity (ω). The projection of P on the diameter along the x-axis (M). At the later time (t), the particle is at Q. Now its projection on the diameter along the x-axis is N. As the particle P revolves around in a circle anti-clockwise, its projection M follows it up, moving back and forth along the diameter such that the displacement of the point of projection at any time (t) is the x-component of the radius vector (A). x = A cos ( ωt + Φ) . . . . . . . (1) y = A sin ( ωt + Φ) . . . . . (2) Thus, we see that the uniform circular motion is the combination of two mutually perpendicular linear harmonic oscillation. It implies that P is under uniform circular motion, (M and N) and (K and L) are performing simple harmonic motion about O with the same angular speed ω as that of P. P is under uniform circular motion, which will have centripetal acceleration along A (radius vector) \(\begin{array}{l}\overrightarrow{{{a}_{c}}}=A{{\omega }^{2}}\end{array} \) (towards the centre)It can be resolved into two components:
aN and aL acceleration corresponding to the points N and L, respectively. In the above discussion, the foot of the projection on the x-axis is called a horizontal phasor. Similarly, the foot of the perpendicular on the y-axis is called the vertical phasor. Already we know the vertical and horizontal phasor will execute the simple harmonic motion of amplitude A and angular frequency ω. The phases of the two SHM differ by π/2. Video Lessons
Simple Harmonic Motion – Top 12 Most Important and Expected JEE QuestionsProblem-Solving Strategy in Horizontal Phasor
Simple harmonic motion is a periodic motion in which the particle acceleration is directly proportional to its displacement and is directed towards the mean position.
The restoring force is given by the formula F = -kx The negative sign shows that the force is in the opposite direction. k is the force constant. x is the displacement of the string from the equilibrium position.
The harmonic motion is when the restoring force is proportional to the displacement but in opposite directions. Simple harmonic motion is a harmonic motion with a constant amplitude and the same frequency.
The periodic motion of gradually decreasing amplitude is called the damped simple harmonic motion.
Distance travelled by the particle from its mean position at any instant is called the displacement of the particle executing simple harmonic motion.
At the mean position, the energy is entirely kinetic energy. At the extreme position, the energy is entirely potential energy.
The phase difference between displacement and acceleration of the particle executing simple harmonic motion is π radian.
The maximum displacement of the particle is called the amplitude of motion.
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