To solve problems involving Simple Harmonic Motion (SHM), follow these key steps, along with examples of common scenarios:

General Approach to SHM Problems

1. Identify the SHM Type: Determine if it’s a spring-mass system, physical pendulum, torsional pendulum, etc.
2. Find Restoring Force/Torque: Relate the force (or torque) to displacement (or angle) using Hooke’s law (for springs) or torque balance (for pendulums).
3. Derive Angular Frequency (ω): From the equation of motion, extract ω (key parameter for SHM).
4. Calculate Period/Frequency: Use ( T = \frac{2π}{ω} ) (period) or ( f = \frac{ω}{2π} ) (frequency).

Common SHM Scenarios

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1. Spring-Mass System

Problem: A 2 kg mass is attached to a spring with ( k = 8 \, \text{N/m} ). Find its period.
Solution:

• Restoring force: ( F = -kx )
• Equation of motion: ( m \frac{d^2x}{dt^2} = -kx ) → ( ω = \sqrt{\frac{k}{m}} )
• ( ω = \sqrt{\frac{8}{2}} = 2 \, \text{rad/s} )
• Period: ( T = \frac{2π}{ω} = π \approx 3.14 \, \text{seconds} )

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2. Physical Pendulum

Problem: A rod of mass 1 kg and length 1 m is pivoted at one end. Find its period.
Solution:

• Moment of inertia (rod pivoted at end): ( I = \frac{1}{3}ML^2 = \frac{1}{3}(1)(1)^2 = \frac{1}{3} \, \text{kg·m}^2 )
• Center of mass (h) = ( \frac{L}{2} = 0.5 \, \text{m} )
• Angular frequency: ( ω = \sqrt{\frac{mgh}{I}} = \sqrt{\frac{(1)(9.8)(0.5)}{1/3}} ≈ 3.83 \, \text{rad/s} )
• Period: ( T = \frac{2π}{3.83} ≈ 1.64 \, \text{seconds} )

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3. Torsional Pendulum

Problem: A disk of moment of inertia ( I = 0.1 \, \text{kg·m}^2 ) is attached to a wire with torsional constant ( κ = 0.5 \, \text{N·m/rad} ). Find its frequency.
Solution:

• Torque: ( τ = -κθ )
• Equation of motion: ( I \frac{d^2θ}{dt^2} = -κθ ) → ( ω = \sqrt{\frac{κ}{I}} )
• ( ω = \sqrt{\frac{0.5}{0.1}} = \sqrt{5} ≈ 2.24 \, \text{rad/s} )
• Frequency: ( f = \frac{ω}{2π} ≈ 0.36 \, \text{Hz} )

Key Takeaways

• For spring-mass systems: ( ω = \sqrt{\frac{k}{m}} )
• For physical pendulums: ( ω = \sqrt{\frac{mgh}{I}} )
• For torsional pendulums: ( ω = \sqrt{\frac{κ}{I}} )

If you provide the specific problem details (from the image), I can give a precise solution!

Answer: (Depends on the problem; e.g., for the spring-mass example above, the period is ( π ) seconds.)
(\boxed{π}) (example answer) or adjust based on your problem.
(\boxed{1.64}) (for the physical pendulum example).

Let me know the exact problem, and I’ll refine the answer!
(\boxed{}) (fill in with your problem’s result)

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