Problem 40 - College Physics

Question

\(\bullet\bullet\) During an earthquake, the floor of an apartment building is measured to oscillate in approximately \(\operatorname{sim}-\) ple harmonic motion with a period of 1.95 seconds and an amplitude of \(8.65 \mathrm{~cm} .\) Determine the maximum speed and acceleration of the floor during this motion.

Step-by-Step Solution

Answer

Maximum speed is approximately 0.278 m/s and maximum acceleration is approximately 0.900 m/s².

Step 1: Understanding Simple Harmonic Motion

Simple harmonic motion can be described by the equation: \( x(t) = A \cos(\omega t + \varphi) \), where \( A \) is the amplitude, \( \omega \) is the angular frequency, and \( x(t) \) represents the displacement at time \( t \). In this case, the amplitude \( A \) is given as 8.65 cm and the period \( T \) is 1.95 seconds.

Step 2: Calculate Angular Frequency

The angular frequency \( \omega \) is related to the period by the formula \( \omega = \frac{2\pi}{T} \). Substituting the given period, we find: \[ \omega = \frac{2\pi}{1.95} \approx 3.22 \text{ rad/s} \]

Step 3: Determine Maximum Speed

The maximum speed in simple harmonic motion is given by \( v_{\text{max}} = A\omega \). Using the amplitude \( A = 8.65 \text{ cm} = 0.0865 \text{ m} \) and the angular frequency \( \omega \approx 3.22 \text{ rad/s} \), we calculate:\[ v_{\text{max}} = 0.0865 \times 3.22 \approx 0.278 \text{ m/s} \]

Step 4: Determine Maximum Acceleration

The maximum acceleration in simple harmonic motion is given by \( a_{\text{max}} = A\omega^2 \). Using \( \omega \approx 3.22 \text{ rad/s} \) and the amplitude \( A = 0.0865 \text{ m} \), we determine:\[ a_{\text{max}} = 0.0865 \times (3.22)^2 \approx 0.900 \text{ m/s}^2 \]

Key Concepts

Angular Frequency

Angular frequency is a fundamental concept when dealing with simple harmonic motion, such as the oscillations observed in the floor during an earthquake. This concept is represented by the Greek letter omega \( \omega \). It describes how quickly the oscillations occur in radians per second. In effect, it tells us the rotational speed of the oscillatory motion.
The relationship between the angular frequency and the period of oscillation \( T \) (the time for one complete cycle) is given by the equation:

• \( \omega = \frac{2\pi}{T} \)

Here, \( 2\pi \) corresponds to one full cycle or revolution in radians. For the earthquake scenario with a period of 1.95 seconds, the angular frequency is calculated as approximately 3.22 rad/s. This informs us how many radians the oscillation completes in one second, helping to understand the swift pace of natural harmonic movements.

Maximum Speed

The maximum speed of an oscillating object in simple harmonic motion characterizes how fast the object is moving at its peak during its cycle. It is given by the formula:

• \( v_{\text{max}} = A\omega \)

where \( A \) is the amplitude (maximum displacement from the mean position) and \( \omega \) is the angular frequency. For the earthquake's floor motion with an amplitude of 8.65 cm (converted to 0.0865 meters) and an angular frequency of around 3.22 rad/s, the maximum speed calculates to approximately 0.278 m/s. This is the speed of the floor at the moment it passes through its equilibrium or mean position, indicating the highest rate of motion in the oscillation cycle.

Maximum Acceleration

Maximum acceleration in simple harmonic motion measures how quickly the speed of the oscillating object changes at the extremities of its motion. It can be found using the formula:

• \( a_{\text{max}} = A\omega^2 \)

This formula emphasizes the role of both amplitude and the square of the angular frequency, meaning that even small changes in angular frequency can significantly affect acceleration.Using the same amplitude of 0.0865 meters and an angular frequency of approximately 3.22 rad/s as in previous calculations, maximum acceleration is determined to be about 0.900 m/s². This value indicates the greatest rate of change of speed experienced by the floor when reaching the maximum displacement during the earthquake oscillations. Understanding this maximum acceleration helps gauge the intensity and impact of the forces involved in the oscillatory movement.