Lesson1: Simple Harmonic Motion: Why is x(t) written as A cos(ωt + φ) in SHM?

1. What is the difference between simple periodic motion and Simple Harmonic Motion (SHM)?

Periodic motion is simply any motion that repeats at regular intervals. For it to be Simple Harmonic Motion, the restoring force must be proportional to displacement, and the motion must follow a sinusoidal function (sine or cosine) of time.

2. How are Frequency and Time Period related?

They are reciprocals of each other. Frequency (f) is the number of cycles completed in one second, while the time period (T) is the time taken for one single oscillation. Mathematically, T = 1/f.

3. Does the "Phase Angle" change the shape of the wave?

No. The sole purpose of the phase angle (φ) is to describe the motion of the particle accurately with reference to where it starts when we set time t = 0. For example, changing the sign of φ sets the initial direction of motion (left or right) but does not change the wave's shape.

4. How do I decide whether to use Sine or Cosine for the equation?

It depends on the starting position. If the mass starts at the extreme right (max displacement), it is standard to use Cosine because at t = 0, displacement is maximum. If the mass starts at x = 0, you would modify the equation with a phase shift (like adding π/2) or use a function that starts at zero.

5. How do I calculate the position of the mass at a random time, like t = 22 seconds?

You use the displacement equation x = A cos(2πt/T). You simply substitute the values for Amplitude (A), Time Period (T), and the specific time t into the equation to get the corresponding position.

Lesson 2: How to read SHM displacement and velocity graphs?

6. If the force is zero at the equilibrium position, why doesn't the object stop there?

This is due to Newton's First Law. At that instant, there is no net force acting on the mass, so nothing stops it; it simply continues in its state of motion carrying the kinetic energy it gained.

7. How do we get the velocity equation if we only have the position equation?

Velocity is the first derivative of the position. If the position is x = A cos(ωt + φ), differentiating it with respect to time gives v(t) = -ωA sin(ωt + φ).

8. Why is the acceleration maximum at the ends where the object is stopped?

At the extremes, the spring is fully stretched or compressed, meaning it offers maximum force. Since F = ma, maximum force implies maximum acceleration, even though the velocity is momentarily zero.

9. Is there a quick test to identify SHM just by looking at an acceleration equation?

Yes. Whenever you find that acceleration is proportional to displacement but opposite in direction (a = -k · x), you can confidently say the motion is Simple Harmonic Motion.

Lesson 3: Energy in Simple Harmonic Motion: Kinetic & Potential Energy vs. Time Graphs

10. What does the negative sign in F = -kx actually mean?

The minus sign tells us that the force always acts in a direction opposite to the displacement. If the displacement is positive (right), the force pulls left, trying to restore the particle to the center.

11. How do I calculate the spring constant (k) if I only know mass and frequency?

You can link them using the angular frequency formula. We know that k = mω². Since ω is determined by the mass and spring stiffness, knowing the mass and the frequency (which gives you ω) allows you to solve for k.

12. Where is Kinetic Energy maximum vs. Potential Energy?

They trade off perfectly. Potential energy (U) is maximum at the extremes where the spring is most stretched. Kinetic energy (K) is maximum at the equilibrium position where the spring is relaxed and speed is highest.

13. Why is the Total Mechanical Energy independent of time?

While potential and kinetic energy both change with time, their sum remains constant. The formula for total energy depends only on the spring constant and Amplitude (E = ½kA²), so time (t) does not appear in the final expression.

Lesson 4: Simple Harmonic Motion of a Simple Pendulum

14. Does the mass of the bob affect the period of a pendulum?

No. Gravity pulls harder on more massive objects (mg), but more massive objects are harder to accelerate (ma). These two effects cancel out perfectly, so mass does not appear in the period equation.

15. What is the "Small Angle Approximation" and why do we use it?

We assume the pendulum swings less than 10°. This allows us to assume sin(θ) ≈ θ (in radians) and cos(θ) ≈ 1. This approximation makes the math fit the Simple Harmonic Motion structure.

16. Why do we ignore vertical acceleration for a pendulum?

For small swings, the mass hardly moves up and down; the vertical displacement is much smaller than the horizontal displacement. The vertical acceleration is very small (about 3% of g), so we treat it as negligible (ay ≈ 0).

17. If I take a pendulum to the Moon, will it swing faster or slower?

It will swing in slow motion. The period depends on gravity (g) in the denominator (T = 2π√(L/g)). Since gravity on the Moon is weaker, the period gets bigger (longer).

18. Does the period stay the same if I swing the pendulum with a huge amplitude?

No. The formula is independent of amplitude only for small swings. For larger angles like 40° or 60°, the period gets a bit longer than the value predicted by the standard formula.

Lesson 5: Simple Harmonic Motion : The Reference Circle Method & Why We Use Omega

19. Why do we use "Angular Frequency" (ω) when nothing is actually rotating?

We use circular motion as a reference model. One full cycle of SHM corresponds to 2π radians on a reference circle. Omega (ω) measures how many radians of this reference circle are covered in a second, acting as a "hidden code" to unlock the math of SHM.

20. What is the relationship between the mass moving in a line and the dot on the reference circle?

The moving mass is like a projection or "shadow" of the dot moving on the circle. When the mass completes one oscillation, the dot completes one full circle (2π radians).

Lesson 6: Damped Harmonic Motion : Amplitude Decay & The -bv Drag Force

21. Does damping just reduce the height of the swing, or does it slow it down too?

It does both. The drag force steals energy (reducing height) but also lengthens the time period. A damped oscillator swings slightly slower than an undamped one because the frequency is reduced.

22. What does the scary equation for Damped Oscillation actually mean?

The equation x(t) = A e^(-bt/2m) cos(ω't + φ) has two main parts. The cosine part represents the oscillation back and forth. The exponential part (e raised to a negative power) acts as a shrinking "envelope" that squashes the amplitude over time.

23. Where does the energy go in a damped system?

The drag force does negative work on the system. It steals mechanical energy and turns it into heat.

24. What happens if the friction is extremely high (Critical Damping)?

If the friction is high enough that the damping term equals the natural frequency term (where b²/4m² = k/m), the system is Critically Damped. The system returns to equilibrium as fast as possible without oscillating at all.

25. What is "Overdamping"?

This occurs when the damping is even higher than critical damping (like moving in thick honey). The math gives a negative number inside the square root, meaning there is no oscillation at all—the system just slowly returns to equilibrium.