Key concept — equilibrium position:
At the equilibrium (mean) position of a pendulum, the bob is at its lowest point. Potential energy (measured from this point) = 0.
Therefore: All energy = Kinetic Energy
Apply energy conservation:
Total energy = KE + PE = 0.02 J (given)
At equilibrium: PE = 0, so KE = 0.02 J
$$\frac{1}{2}mv^2 = 0.02 \text{ J}$$
Solve for v:
Mass m = 20 g = 0.02 kg
$$v^2 = \frac{2 \times 0.02}{0.02} = \frac{0.04}{0.02} = 2$$
$$v = \sqrt{2} \approx \mathbf{1.41 \text{ m/s}}$$ ✓
Simple Harmonic Motion (SHM) is a special type of periodic motion where the restoring force on the body is directly proportional to its displacement from the equilibrium position and is always directed towards the equilibrium. Mathematically: F = −kx, where k is the force constant and x is displacement. The negative sign indicates the force is always opposite to displacement.
Displacement in SHM: x = A sin(ωt + φ), where A is amplitude, ω is angular frequency, and φ is initial phase. Velocity: v = Aω cos(ωt + φ) = ω√(A² − x²). Acceleration: a = −ω²x (always towards equilibrium).
The total mechanical energy in SHM is constant and depends only on amplitude:
\(KE = \dfrac{1}{2}m\omega^2(A^2 - x^2)\)
\(PE = \dfrac{1}{2}m\omega^2 x^2\)
\(E_{total} = KE + PE = \dfrac{1}{2}m\omega^2 A^2\)
📌 At mean position (x = 0): KE = max = E_total, PE = 0
📌 At extreme (x = ±A): KE = 0, PE = max = E_total
📌 At x = A/√2: KE = PE = E_total/2
📌 Total energy ∝ A² (proportional to square of amplitude)
A simple pendulum consists of a point mass (bob) suspended by a massless, inextensible string of length L from a fixed point. For small angular displacements (θ < 10°), it undergoes SHM with:
\(T = 2\pi\sqrt{\dfrac{L}{g}}\)
\(\omega = \sqrt{\dfrac{g}{L}}\)
The time period is independent of: (1) mass of the bob, (2) amplitude (for small angles), (3) material of bob. It depends only on: length L and local gravity g. This isochronism (equal time periods) was first observed by Galileo.
For a pendulum bob at angular displacement θ, the height above the lowest point is h = L(1 − cosθ). PE = mgh = mgL(1 − cosθ). For small angles, PE ≈ ½mgLθ² ≈ ½mω²x², which is the SHM form. At equilibrium (θ = 0): PE = 0, KE = maximum = Total E. At extreme (θ = θ_max): KE = 0, PE = Total E = mgL(1 − cosθ_max).
\(v_{max} = A\omega = A\sqrt{\dfrac{g}{L}}\) (for pendulum)
The maximum velocity always occurs at the equilibrium position. In this problem, v_max = √2 ≈ 1.41 m/s. We can find the amplitude if we know ω (i.e., if L is given): A = v_max/ω.
📌 Increase in length L → T increases (T ∝ √L)
📌 Increase in g → T decreases (T ∝ 1/√g)
📌 On Moon (g = g/6): T_moon = √6 × T_earth
📌 In a falling lift (effective g = 0): T → ∞ (pendulum stops)
📌 In an accelerating lift (upward): effective g increases → T decreases