SIMPLE HARMONIC MOTION (S.H.M) in One Shot || JEE Main & Advanced || PRAYAS 2025 || RAHUL YADAV

Summary

Rahul Yadav presents a broad JEE-focused revision of simple harmonic motion (SHM), covering its defining conditions, equations, graphs, energy, time-period calculations, spring systems, phasors, angular SHM, and superposition. He emphasizes identifying the mean position and measuring displacement from it, since many common mistakes arise from confusing displacement with the particle’s absolute coordinate.

1. What Qualifies as SHM?

• The motion must be periodic.
• A restoring force must act toward the mean position.

• The force magnitude must be proportional to displacement from the mean position: [ F=-ky. ] Here, (y) is displacement from equilibrium; the minus sign indicates that the force points back toward equilibrium.

• For a mass (m), this gives: [ \ddot y=-\omega^2y,\qquad \omega=\sqrt{\frac{k}{m}},\qquad T=\frac{2\pi}{\omega}. ]

• A standard form of the motion is: [ y=A\sin(\omega t+\phi), ] where (A) is the amplitude and (\phi) is the initial phase.

• If equilibrium is not at the origin, the particle’s position is the equilibrium coordinate plus its SHM displacement. The force expression and equation of motion must be written relative to the shifted mean position.

2. Position, Velocity, Acceleration, and Their Graphs

Let (y) denote displacement from the mean position.

• Velocity as a function of time: [ v=A\omega\cos(\omega t+\phi). ]

• Velocity as a function of displacement: [ v^2=\omega^2(A^2-y^2). ] This is a key problem-solving relation. Use displacement from equilibrium—not the particle’s absolute coordinate.

• Speed is greatest at the mean position: [ v_{\max}=A\omega. ] It is zero at either extreme.

• Acceleration: [ a=-\omega^2y. ] Thus, acceleration is zero at the mean position and has maximum magnitude (\omega^2A) at the extremes.

• Graph relationships:

  • (v) versus (y): ellipse.
  • (a) versus (v): ellipse.
  • (a) versus (y): straight line with slope (-\omega^2).
• If the mean position shifts, graphs expressed against absolute position shift as well.

3. Energy in SHM

The restoring force is conservative, so mechanical energy remains constant.

• Kinetic energy: [ K=\frac12mv^2=\frac12k(A^2-y^2). ]

• Potential energy, allowing a choice of reference level: [ U=\frac12ky^2+C, ] where (C) is the potential energy at the mean position.

• Total mechanical energy: [ E=K+U=\frac12kA^2+C. ]

• At the mean position, kinetic energy is maximum and the displacement-dependent part of potential energy is minimum.

• At an extreme position, kinetic energy is zero and potential energy is maximum.
• The kinetic-energy graph versus displacement is a downward-opening parabola; the potential-energy graph is an upward-opening parabola; total energy is constant.
• Potential energy need not be zero at the mean position. Average kinetic and potential energies are equal over a full cycle only with the usual zero-at-equilibrium potential-energy reference; they are not necessarily equal over an arbitrary time interval.

4. Finding the Time Period in Unfamiliar Systems

For systems that are not immediately recognizable as standard spring oscillators, use this procedure:

1. Find the equilibrium position by setting the net force to zero.
2. Displace the body slightly from equilibrium.
3. Find the net restoring force caused by that small displacement.
4. Apply Newton’s second law.

5. Put the acceleration into the form: [ a=-\omega^2y. ]

6. Read off (\omega), then use (T=2\pi/\omega).

A constant applied force can shift the equilibrium of a spring system without changing its time period, provided the spring’s effective stiffness and the oscillating mass remain unchanged. The instructor applies the same small-displacement reasoning to examples involving a floating block, a weighing scale in vertical SHM, and other force systems.

5. Spring Combinations

• Series: [ \frac1{k_{\rm eq}}=\frac1{k_1}+\frac1{k_2}+\cdots. ] In ideal springs in series, the force is the same in each spring, and their extensions add.

• Parallel: [ k_{\rm eq}=k_1+k_2+\cdots. ] The restoring forces add when the arrangement makes both springs respond to the displacement.

• Do not treat springs as connected in parallel merely because they touch a block. If a spring is only touching rather than attached, it may act only in one direction. Analyze which spring is actually compressed or stretched during each part of the motion.

• Once (k_{\rm eq}) is known, use: [ T=2\pi\sqrt{\frac{m}{k_{\rm eq}}}. ]

6. Phasor Diagrams and Timing

• SHM can be represented as the projection of uniform circular motion: the circle’s radius is the amplitude, and the rotating radius vector’s angle represents phase.
• To find the time for a particle to move between two positions, identify the phase at each position and find the angle through which the phasor rotates.

• Convert the phase change to elapsed time: [ \Delta t=\frac{\Delta\phi}{2\pi}T, ] with (\Delta\phi) expressed in radians. Equivalently, a phase rotation of (\theta) degrees takes ((\theta/360^\circ)T).

• The quadrant matters when assigning phase. A particular displacement can occur while moving in either direction, so displacement alone may not fully specify the particle’s state.

• Phasors are used in the session to solve timing questions and combine SHMs.

7. Sudden Changes and Disturbed SHM

When an event changes the mass, forces, or equilibrium:

• Find the system’s state immediately before and after the event.
• Use momentum conservation only when the external force during the event is non-impulsive or negligible over the short event interval.
• Determine the new equilibrium position after the change.
• Find the particle’s displacement and velocity relative to that new equilibrium.

• Use the SHM relation [ v^2=\omega^2(A^2-y^2) ] to determine the new amplitude.

• Recalculate the time period using the post-event mass and effective stiffness.

• The mean position, amplitude, and period may change by different amounts; do not assume they all remain as before.

The instructor also considers cases where a block is added to a moving oscillator, a mass detaches, or a contact force becomes zero. These examples reinforce the need to distinguish the state before and after the event and to check the new equilibrium.

8. Angular SHM and Pendulums

Angular SHM follows the same structure as linear SHM, with torque replacing force and angular displacement replacing linear displacement.

• The restoring torque must satisfy: [ \tau=-C\theta, ] where (C) is a torsional constant.

• Since (\tau=I\alpha): [ \alpha=-\frac{C}{I}\theta,\qquad \omega=\sqrt{\frac{C}{I}},\qquad T=2\pi\sqrt{\frac{I}{C}}. ]

• The session reviews:

  • Simple pendulum: (T=2\pi\sqrt{L/g_{\rm eff}}).
  • Physical pendulum: (T=2\pi\sqrt{I/(mgd)}), where (d) is the distance from the pivot to the centre of mass.
  • Torsion pendulum: (T=2\pi\sqrt{I/C}).
• In an accelerating frame, use the appropriate effective gravitational acceleration when calculating a pendulum’s period.

9. Superposition of SHMs

• If two SHMs act along the same line with the same angular frequency, their displacements can be added to obtain another SHM.
• The instructor uses phasor/vector addition:
  1. Represent each oscillation by a vector whose length is its amplitude.
  2. Use the phase difference as the angle between the vectors.
  3. Add the vectors to obtain the resultant amplitude and phase.

• For amplitudes (A_1) and (A_2), with phase difference (\delta), the resultant amplitude is: [ A_R=\sqrt{A_1^2+A_2^2+2A_1A_2\cos\delta}. ]

• Before combining terms written as sine and cosine, rewrite them in a consistent trigonometric form and account for the phase shift.

10. Practice and Exam Approach

The lecture interleaves theory with JEE-style multiple-choice and previous-year questions. Its recurring problem-solving advice is to:

• Read the question’s final instruction and each option carefully.
• Identify the mean position, amplitude, and relevant displacement before substituting into formulas.
• Check whether an event shifts equilibrium or changes the effective mass or spring constant.
• For graph questions, use the known relationships among displacement, velocity, and acceleration.
• For timing questions, use a phasor diagram rather than guessing the fraction of a cycle.
• Verify the physical meaning of results, such as zero speed at an extreme or maximum speed at equilibrium.

Speakers and Sources Featured

• Rahul Yadav — the instructor and principal speaker.
• Students/viewers in the live chat — their names and brief comments are occasionally read or addressed; they are not presented as sustained speakers.
• JEE Main and JEE Advanced previous-year questions — used as problem sources during the teaching session.