A 350 g mass on a 45-cm-long string is released at an angle of 4.5° from vertical. It has a damping constant of 0.010 kg/s. After 25 s, (a) how many oscillations has it completed and (b) what fraction of the initial energy has been lost?

Verified step by step guidance

1. Step 1: Convert the given values into SI units. The mass is 350 g, which is equivalent to 0.35 kg. The string length is 45 cm, which is equivalent to 0.45 m. The damping constant is already in SI units (0.010 kg/s).
2. Step 2: Calculate the angular frequency of the undamped pendulum using the formula ω₀ = √(g / L), where g is the acceleration due to gravity (9.8 m/s²) and L is the length of the string. Substitute the values to find ω₀.
3. Step 3: Determine the damped angular frequency using the formula ω = √(ω₀² - (b / (2m))²), where b is the damping constant and m is the mass of the pendulum. Substitute the values to find ω.
4. Step 4: Calculate the number of oscillations completed in 25 seconds using the formula N = (ω / (2π)) × t, where t is the time in seconds. Substitute the values to find N.
5. Step 5: Determine the fraction of the initial energy lost using the formula E/E₀ = e^(-bt/m), where b is the damping constant, t is the time, and m is the mass. Substitute the values to find the fraction of energy remaining, and subtract this from 1 to find the fraction lost.

Key Concepts

Simple Harmonic Motion (SHM)

Simple Harmonic Motion refers to the oscillatory motion of an object where the restoring force is directly proportional to the displacement from its equilibrium position. In the case of a pendulum, this motion occurs when the angle of displacement is small, allowing the motion to be approximated as sinusoidal. The period of oscillation depends on the length of the string and the acceleration due to gravity.

Damping

Damping is the effect that reduces the amplitude of oscillations in a system over time, often due to friction or resistance. In this scenario, the damping constant quantifies how quickly the energy of the oscillating mass decreases. The presence of damping leads to a gradual loss of energy, affecting both the number of oscillations completed and the fraction of initial energy lost.

Energy Conservation in Oscillations

In oscillatory systems, energy is exchanged between kinetic and potential forms. Initially, the mass has maximum potential energy at its highest point, which converts to kinetic energy as it swings down. However, due to damping, some energy is lost as heat or sound, which can be calculated to determine how much of the initial energy remains after a certain number of oscillations.