The 15 g head of a bobble-head doll oscillates in SHM at a frequency of 4.0 Hz. b. The amplitude of the head's oscillations decreases to 0.5 cm in 4.0 s. What is the head's damping constant?

Verified step by step guidance

1. Identify the given parameters: mass of the head (m) = 15 g = 0.015 kg, frequency (f) = 4.0 Hz, initial amplitude (A_0) = 0.5 cm = 0.005 m, and time (t) = 4.0 s.
2. Understand that the amplitude of an oscillating system experiencing damping decreases exponentially over time, described by the equation A(t) = A_0 e^{-eta t}, where A(t) is the amplitude at time t, A_0 is the initial amplitude, and eta is the damping constant.
3. Rearrange the damping equation to solve for the damping constant eta. The equation becomes eta = -\frac{1}{t} \ln\left(\frac{A(t)}{A_0}\right).
4. Substitute the values of A(t), A_0, and t into the equation to find eta. Since the amplitude decreases to zero, assume A(t) approaches a very small value close to zero for practical calculation.
5. Calculate eta using the values substituted into the rearranged damping equation. This will give the damping constant in units of s^{-1}.

Key Concepts

Simple Harmonic Motion (SHM)

Simple Harmonic Motion is a type of periodic motion where an object oscillates around an equilibrium position. The motion is characterized by a restoring force proportional to the displacement from the equilibrium, leading to sinusoidal oscillations. In this context, the bobble-head doll's head moves back and forth in a regular pattern, defined by its frequency and amplitude.

Damping

Damping refers to the effect of reducing the amplitude of oscillations in a system over time, often due to energy loss from friction or air resistance. In the case of the bobble-head doll, the damping causes the head's oscillations to decrease in amplitude from its initial value. The damping constant quantifies the rate at which the oscillations diminish, providing insight into the system's energy dissipation.

Damping Constant

The damping constant is a parameter that describes how quickly the amplitude of oscillations decreases in a damped harmonic oscillator. It is typically denoted by the symbol 'b' and is related to the time it takes for the oscillations to decay. In this scenario, calculating the damping constant involves analyzing the amplitude reduction over a specified time, allowing us to understand the efficiency of energy loss in the system.