Problem 18 - Physics

Question

A person bounces up and down on a trampoline, while always staying in contact with it. The motion is simple harmonic motion, and it takes 1.90 s to complete one cycle. The height of each bounce above the equilibrium position is \(45.0 \mathrm{~cm} .\) Determine (a) the amplitude and (b) the angular frequency of the motion. (c) What is the maximum speed attained by the person?

Step-by-Step Solution

Answer

(a) Amplitude = 0.45 m; (b) Angular Frequency = 3.31 rad/s; (c) Maximum Speed = 1.49 m/s.

Step 1: Identify the Known Values

We know the period of the motion, \(T = 1.90 \, \text{s}\), and the maximum displacement from the equilibrium position (i.e., amplitude), \(A = 45.0 \, \text{cm} = 0.45 \, \text{m}\).

Step 2: Calculate the Amplitude

The amplitude of the motion is simply the maximum displacement from the equilibrium position. Given that the height of each bounce above the equilibrium position is \(45.0\, \text{cm}\), the amplitude \(A\) is \(0.45\, \text{m}\).

Step 3: Calculate the Angular Frequency

Angular frequency \(\omega\) for simple harmonic motion is given by the formula \(\omega = \frac{2\pi}{T}\). Substitute the known value of the period:\[\omega = \frac{2\pi}{1.90\, \text{s}} = \frac{2\pi}{1.90}\, \text{s}^{-1}\]. Evaluating this gives:\[\omega \approx 3.31 \, \text{rad/s}\].

Step 4: Determine the Maximum Speed

The maximum speed \(v_{\text{max}}\) in simple harmonic motion is given by the formula \(v_{\text{max}} = \omega A\). Using the previously calculated values:\[v_{\text{max}} = 3.31\, \text{rad/s} \times 0.45\, \text{m} \approx 1.49\, \text{m/s}\].

Key Concepts

Amplitude

In simple harmonic motion, the amplitude is the maximum distance the object travels from its equilibrium position. This is a crucial aspect because it helps us understand the extent or range of the motion. The equilibrium position can be thought of as the center point from which the object moves back and forth.

When a person is bouncing on a trampoline, the equilibrium position would be the height at which they are not moving up or down. The amplitude in this context is noted as the height of the bounce above this resting point. Here, we are given that the amplitude is 45 cm, meaning the person moves 0.45 meters above their equilibrium position at the height of each bounce.

This concept is important not only in understanding how high someone can bounce but also in knowing how much energy is involved in each bounce. The greater the amplitude, the higher the bounce and the more energetic the motion.

Angular Frequency

Angular frequency is a measure of how quickly an object goes through its cycles in simple harmonic motion. It's the rate of rotation or oscillation, explaining how often an object completes a cycle in a given time period. This is different from linear frequency, which just counts the number of cycles per second and is measured in hertz.

To calculate angular frequency, use the formula:

• \( \omega = \frac{2\pi}{T} \)

where \( \omega \) is the angular frequency and \( T \) is the period of one complete cycle.

In our case, we have a period \( T = 1.90 \, \text{s} \), meaning it takes 1.9 seconds for the person to complete one bounce cycle. Using the formula, the angular frequency works out to approximately 3.31 \( \text{rad/s} \), which means the person experiences over three radians of motion per second. Understanding \( \omega \) helps us predict motion patterns and ensure the safety and stability of the structure involved in the harmonic motion.

Maximum Speed

In simple harmonic motion, the concept of maximum speed is associated with the velocity an object reaches as it moves through its equilibrium position. This is when the object is changing direction and has accumulated the most kinetic energy.

The maximum speed can be determined using the relationship between angular frequency and amplitude:

• \( v_{\text{max}} = \omega A \)

where \( \omega \) is the angular frequency and \( A \) is the amplitude.

Given \( \omega = 3.31 \, \text{rad/s} \) and \( A = 0.45 \, \text{m} \), we find the maximum speed to be approximately 1.49 \( \text{m/s} \). This means that at the equilibrium crossing, where the person is neither moving up nor down, the speed peaked due to the dynamic nature of the motion. Maximum speed is an important factor in evaluating the performance and energy efficiency in systems undergoing simple harmonic motion.