Understanding the Concepts
* Simple Harmonic Motion: A type of periodic motion where the restoring force is proportional to the displacement from equilibrium. Examples include a mass on a spring or a pendulum swinging with a small angle.
* Amplitude (A): The maximum displacement of the object from its equilibrium position.
* Velocity (v): The rate of change of displacement.
* Energy Conservation: In SHM, the total mechanical energy (potential + kinetic) remains constant.
Derivation
1. Energy Conservation: At any point in SHM, the total energy (E) is the sum of potential energy (PE) and kinetic energy (KE):
E = PE + KE
2. Potential Energy: At the maximum displacement (amplitude, A), the velocity is zero, and all the energy is potential:
PE(max) = 1/2 * k * A^2 (where k is the spring constant)
3. Kinetic Energy: At half amplitude (A/2), the potential energy is:
PE(A/2) = 1/2 * k * (A/2)^2 = 1/8 * k * A^2
4. Using Energy Conservation: Since total energy is constant:
E = PE(max) = PE(A/2) + KE(A/2)
1/2 * k * A^2 = 1/8 * k * A^2 + 1/2 * m * v^2 (where m is the mass)
5. Solving for Velocity: Simplify the equation and solve for v:
* 3/8 * k * A^2 = 1/2 * m * v^2
* v^2 = (3/4) * (k/m) * A^2
* v = √[(3/4) * (k/m) * A^2]
Important Notes:
* Angular Frequency (ω): You can express the velocity in terms of angular frequency (ω = √(k/m)):
* v = √[(3/4) * ω^2 * A^2] = (√3/2) * ω * A
* Phase: The above formula assumes the mass is at its maximum displacement when time t = 0. If the mass is at a different phase, you'll need to consider the sinusoidal nature of the motion.
Example
Let's say a mass of 0.5 kg is attached to a spring with a spring constant of 20 N/m. The amplitude of oscillation is 0.1 m. To find the velocity at half amplitude:
1. Calculate angular frequency: ω = √(k/m) = √(20 N/m / 0.5 kg) ≈ 6.32 rad/s
2. Calculate velocity: v = (√3/2) * ω * A = (√3/2) * 6.32 rad/s * 0.1 m ≈ 0.55 m/s
Therefore, the velocity of the mass at half amplitude is approximately 0.55 m/s.
