The 15 g head of a bobble-head doll oscillates in SHM at a frequency of 4.0 Hz. 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. Step 1: Recognize that the problem involves damped simple harmonic motion (SHM). The amplitude of oscillations decreases over time due to damping, and the damping constant can be determined using the exponential decay formula for amplitude: \( A(t) = A_0 e^{-bt/2m} \), where \( A_0 \) is the initial amplitude, \( A(t) \) is the amplitude at time \( t \), \( b \) is the damping constant, and \( m \) is the mass of the oscillating object.
2. Step 2: Identify the given values from the problem: \( A_0 = 0.5 \, \text{cm} \), \( A(t) = 0.5 \, \text{cm} \), \( t = 4.0 \, \text{s} \), and \( m = 15 \, \text{g} = 0.015 \, \text{kg} \). Note that \( A_0 \) and \( A(t) \) are the same in this case, which suggests the damping constant is being calculated for the given time interval.
3. Step 3: Rearrange the exponential decay formula to solve for the damping constant \( b \): \( b = \frac{-2m}{t} \ln\left(\frac{A(t)}{A_0}\right) \). Substitute the known values into this equation.
4. Step 4: Perform the logarithmic operation \( \ln\left(\frac{A(t)}{A_0}\right) \). Since \( A(t) = A_0 \), the ratio \( \frac{A(t)}{A_0} \) equals 1, and \( \ln(1) = 0 \). This simplifies the equation for \( b \).
5. Step 5: Conclude that the damping constant \( b \) is zero in this case, as there is no decrease in amplitude over the given time interval. This indicates that the system is not experiencing damping during the specified time period.

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 amplitude of the head's oscillations decreases, indicating that damping is occurring. The damping constant quantifies the rate at which the oscillations diminish.

Damping Constant

The damping constant is a parameter that describes how quickly the oscillations of a damped system decrease in amplitude. It is typically denoted by the symbol 'b' and is related to the time it takes for the amplitude to reduce to a certain fraction of its initial value. In this problem, calculating the damping constant involves analyzing the change in amplitude over a specified time period.