- Asked 2/16
- 2073 Chaitra · 5 marks
- 2072 Kartik · 5 marks
Torsional pendulum, modulus of rigidity
Short answer
Bar pendulum swings about horizontal axis under gravity; torsional pendulum twists about vertical wire with elastic torque. Motion is angular SHM: d²θ/dt² + (C/I)θ = 0. With C = πηr⁴/(2l) and T = 2π√(I/C), modulus of rigidity η = 8πl I/(r⁴ T²).
Differentiate between bar (compound) pendulum and torsional pendulum. Using a torsional pendulum, derive a relation for the modulus of rigidity of the metallic wire. (Show that the motion of a torsional pendulum is angular SHM.)
Difference between bar pendulum and torsional pendulum
| Bar (compound) pendulum | Torsional pendulum |
|---|---|
| Rigid body swings about a horizontal axis | Body twists about a vertical wire |
| Restoring torque is due to gravity: −mgl sinθ | Restoring torque is due to elasticity of wire: −Cθ |
| T = 2π√((k² + l²)/gl) | T = 2π√(I/C) |
| Depends on g | Independent of g |
| Small angles needed (sinθ ≈ θ) | Twist must stay within elastic limit |
| Used to find g | Used to find η and moment of inertia I |
Angular SHM of a torsional pendulum
A disc of moment of inertia I is hung from a wire of length l and radius r. Twist the disc by angle θ.
- Wire exerts restoring torque proportional to the twist: τ = −Cθ, where C = torsional constant (torque per unit twist).
- Newton's law for rotation: I d²θ/dt² = −Cθ.
- d²θ/dt² + (C/I) θ = 0, which is of the form of SHM, d²θ/dt² + ω²θ = 0, with ω = √(C/I).
- So the motion is angular SHM with period T = 2π√(I/C).
Modulus of rigidity
- Consider a thin cylindrical shell of radius x, thickness dx, length l. Twisting by θ shears its surface through angle φ = xθ/l.
- Shear stress = ηφ, force on the shell's cross-section = ηφ · 2πx dx, torque = force × x:
dτ = η (xθ/l)(2πx dx) x = (2πηθ/l) x³ dx
τ = (2πηθ/l) ∫₀ʳ x³ dx = πηr⁴ θ / 2l
- Compare with τ = Cθ: C = πηr⁴/2l.
- Substitute in T = 2π√(I/C): T² = 4π²I/C = 8π l I /(η r⁴).
η = 8π l I / (r⁴ T²)
Measure l, r, T and know I (e.g. I = ½MR² for a disc) to get η of the wire.
✓ Answer checked by Claude Opus 5.5 (high effort)
Related questions
- Bar vs torsional pendulum; minimum period · Engineering Physics, chapter 1
- Torsional pendulum time period · Engineering Physics, chapter 1 · asked 2×
- Four collinear points of compound pendulum · Engineering Physics, chapter 1 · asked 2×
- Moment of inertia by torsional pendulum · Engineering Physics, chapter 1
- Compound pendulum centres and minimum period · Engineering Physics, chapter 1
- Compound pendulum with added mass · Engineering Physics, chapter 1
- Torsional rigidity; hollow shaft vs solid shaft strength · Strength of Materials, chapter 6
- OFC pros and cons over metallic wire · Propagation and Antenna, chapter 6
- General plane motion; angular acceleration of arm DE · Applied Mechanics (Dynamics), chapter 4
- Double pendulum angular momentum, m1 = m2 = 1 kg · Applied Mechanics (Dynamics), chapter 3 · asked 3×