Problem 25 - College Physics

Question

\(\bullet\) If an object on a horizontal frictionless surface is attached to a spring, displaced, and then released, it oscillates. Suppose it is displaced 0.120 \(\mathrm{m}\) from its equilibrium position and released with zero initial speed. After \(0.800 \mathrm{s},\) its displacement is found to be 0.120 \(\mathrm{m}\) on the opposite side and it has passed the equilibrium position once during this interval. Find (a) the amplitude, (b) the period, and (c) the frequency of the motion.

Step-by-Step Solution

Answer

(a) Amplitude is 0.120 m; (b) Period is 1.600 s; (c) Frequency is 0.625 Hz.

Step 1: Understanding the Amplitude

The amplitude of an oscillating object on a spring is the maximum distance it moves from its equilibrium position. Here, the object is displaced 0.120 m from the equilibrium position initially. Since the oscillations occur symmetrically on either side of the equilibrium, the amplitude \( A \) is simply the initial displacement: \( A = 0.120 \text{ m} \).

Step 2: Calculating the Period with Key Observations

The object takes 0.800 s to move from one side to the opposite side while crossing the equilibrium once. This means it completes half of a full oscillation cycle (from maximum displacement on one side to the same displacement on the opposite side). Therefore, the time for one full period \( T \) is twice this duration: \( T = 2 \times 0.800 \text{ s} = 1.600 \text{ s} \).

Step 3: Determining the Frequency

Frequency \( f \) is defined as the number of oscillations completed per second. It is the reciprocal of the period \( T \). Using the formula \( f = \frac{1}{T} \), we find \( f = \frac{1}{1.600 \text{ s}} = 0.625 \text{ Hz} \). Therefore, the frequency of the oscillation is 0.625 Hz.

Key Concepts

Amplitude

Amplitude in simple harmonic motion refers to the maximum extent of the oscillation from its equilibrium position. It measures how far the object moves from the central point of balance and is a crucial parameter for understanding the dynamics of oscillation.
For instance, consider an object attached to a spring that is initially displaced by 0.120 meters. When the object is released, it moves symmetrically back and forth across the equilibrium point. The amplitude here is exactly 0.120 meters, which is precisely this initial displacement.
Understanding amplitude is important because it tells us about the energy in the system. A larger amplitude implies that the system has more energy, as it moves further from the equilibrium. Here are a few key points about amplitude:

• Amplitude is always measured in the same units as displacement, typically meters in these problems.
• In the absence of damping forces like friction, amplitude remains constant.
• It does not depend on other factors like mass or frequency for an ideal simple harmonic system.

Visualize amplitude as the height of the wave if the motion were plotted on a graph of position versus time. It's the peak that the object reaches on either side of its equilibrium state.

Period of Oscillation

The period of oscillation signifies the amount of time it takes for the object to complete a full cycle of motion. In the example given, it involves the time taken to move from one side, through the equilibrium, to the maximum displacement on the opposite side, and back.
In the described situation, the object takes 0.800 seconds to get from one anticlimax to the other. However, it's important to note that this is only half of a complete oscillation, as the full journey includes returning to the starting point. Therefore, the full period is doubled: \[ T = 2 \times 0.800 \text{ s} = 1.600 \text{ s} \]

This period provides insights into the speed and energy of the system. Here are several important aspects to understand about the period:

• The period is independent of the amplitude in simple harmonic motion without damping.
• It is influenced by the properties of the spring and the mass connected to it. Specifically, a stiffer spring or a lighter mass often leads to a shorter period.
• Period is measured in units of time, typically seconds.

The period is like the ticking of a clock — it tells us the rhythm of the oscillation.

Frequency

Frequency describes how often the oscillation occurs over a specified period, typically a second. It allows us to comprehend how many cycles the motion completes in a given timeframe.
Calculated as the reciprocal of the period, frequency provides a different perspective on the same motion. From the completed step about period, we determined the full period to be 1.600 seconds.
Using the formula for frequency:
\[ f = \frac{1}{T} = \frac{1}{1.600 \text{ s}} = 0.625 \text{ Hz} \]

This calculation shows the frequency of the object’s oscillation is 0.625 Hertz, meaning the object makes 0.625 complete cycles every second.
Here are some critical things to consider about frequency:

• It is inversely related to the period: as frequency increases, the period decreases, and vice versa.
• Frequency is measured in Hertz (Hz), where 1 Hz represents one cycle per second.
• In a mechanical context, a higher frequency often denotes a more rapid repetitive motion.

Understanding frequency helps paint a picture of the pace of oscillations. It's like the beats of a drum — fast beats (high frequency) or slow periodic thuds (low frequency).