Question

A machine part is undergoing SHM with a frequency of 4.00 Hz and amplitude 1.80 cm. How long does it take the part to go from $\chi =$ 0 to $\chi =-1.80$ cm?

Short answer

0.0625 seconds.

Step-by-step solution

Determine the Total Period of SHM

The period $T$ of a simple harmonic motion is the reciprocal of the frequency $f$. Given the frequency $f=4.00\text{ Hz}$, we can find the period using the formula: $$T=\frac{1}{f}=\frac{1}{4.00}=0.25\text{ seconds}$$

Identify Relevant Positions and Phases

In simple harmonic motion, the displacement $\chi$ goes from $0$ to the amplitude $A$ and back in half a period. The given positions are between $\chi =0$ and $\chi =-1.80\text{ cm}$, which indicates a motion from the center to extreme negative displacement. This corresponds to moving from phase $0$ to $-\pi /2$.

Calculate the Time Taken

The motion from $\chi =0$ to $\chi =-1.80\text{ cm}$ covers a quarter of the full cycle (as it moves from the mean position to the negative extreme, which is a quarter of oscillation). The total period is 0.25 seconds, hence the journey from $0$ to the negative displacement takes: $$\text{Time taken}=\frac{T}{4}=\frac{0.25}{4}=0.0625\text{ seconds}$$

Key concepts

Frequency

In simple harmonic motion (SHM), frequency tells us how quickly the system is oscillating. It is typically measured in Hertz (Hz), which is the number of complete cycles per second. For example, if the frequency is 4.00 Hz, this means the system completes 4 full cycles every second.

The frequency can be used to understand how fast the object is moving back and forth. If you have a high frequency, like 10 Hz, the object moves rapidly. Conversely, a lower frequency of 2 Hz signifies slower movement. If you know the time period of SHM, you can also find the frequency using the formula:

• Formula: Frequency, $f=\frac{1}{T}$

where $T$ is the period of the oscillation. This relationship is vital in comprehending how these properties interact with each other in harmonic motion.

Amplitude

Amplitude is a fundamental aspect of simple harmonic motion, indicating the maximum distance an object moves from its central position. Measured in meters (m) or centimeters (cm), it reflects the "size" of the motion. For instance, an amplitude of 1.80 cm means the object swings or moves 1.80 cm from its equilibrium point at its peak.

Amplitude does not affect the frequency or period of the oscillation; instead, it provides insight into the energy of the system. Larger amplitudes often suggest more energy within the system:

• A higher amplitude in a swinging pendulum means it swings wider.
• For a spring, a larger amplitude indicates further stretch or compression with each oscillation.

Recognizing the amplitude is vital for visualizing the reach of the oscillating object and understanding its movement extremes.

Period

The period in simple harmonic motion refers to the time it takes to complete one full cycle of motion. It is the duration from one point in the cycle back to that same point in the next cycle. In the provided exercise, the period is found using the inverse of the frequency:

• Formula: $T=\frac{1}{f}$

Given that the frequency is 4.00 Hz, the period becomes 0.25 seconds.

Period helps in understanding how long each oscillation lasts. Whether it's a pendulum or a vibrating string, a smaller period means quicker oscillations. Period is a critical parameter in timing aspects of oscillations, letting us plan and calculate other phases of motion accurately.

Phase

Phase in the context of simple harmonic motion helps pinpoint the specific stage or position of the system within its cycle at a given time. It provides an understanding of the position of the moving object at any point in time, usually expressed in radians or degrees.

Think of phase as a snapshot of the oscillation's progress. In the given example, when the part moves from the center to the negative extreme, it corresponds to a phase change from 0 to $-\frac{\pi }{2}$. This quarter-cycle shift is particularly useful for determining the timing and motion between various points in the cycle.

Phase differences can also represent the lead or lag between different oscillating systems or parts, clearing the path for complex motion analysis. Understanding phase helps in predicting future positions and syncing systems that rely on harmonic motion.