In the image is a diagram showing the position-time graph for a mass that is oscillating vertically on the end of a spring. The mass is shown superimposed on the graph at three different instants during a single cycle. a. What is the net force and acceleration at position 2? b. What are the directions of the net force and acceleration at position 3? c. At which position, 1, 2, or 3, does the mass have the greatest velocity, and at which position does the mass have the greatest acceleration?
Simple harmonic motion
Simple harmonic motion is a type of periodic motion in which an object undergoes oscillatory motion. The restoring force exerted by the object exhibiting SHM is proportional to the displacement from the equilibrium position. The force is directed towards the mean position. We see many examples of SHM around us, common ones are the motion of a pendulum, spring and vibration of strings in musical instruments, and so on.
Simple Pendulum
A simple pendulum comprises a heavy mass (called bob) attached to one end of the weightless and flexible string.
Oscillation
In Physics, oscillation means a repetitive motion that happens in a variation with respect to time. There is usually a central value, where the object would be at rest. Additionally, there are two or more positions between which the repetitive motion takes place. In mathematics, oscillations can also be described as vibrations. The most common examples of oscillation that is seen in daily lives include the alternating current (AC) or the motion of a moving pendulum.
In the image is a diagram showing the position-time graph for a mass that is oscillating vertically on the end of a spring. The mass is shown superimposed on the graph at three different instants during a single cycle.
a. What is the net force and acceleration at position 2?
b. What are the directions of the net force and acceleration at position 3?
c. At which position, 1, 2, or 3, does the mass have the greatest velocity, and at which position does the mass have the greatest acceleration?
![**Figure 10-9: Representation of a Waveform Over Time**
The diagram in Figure 10-9 illustrates a waveform progressing through time. Let's break down the key elements of this graph:
- **Y-Axis (Vertical)**: The vertical axis represents the amplitude of the waveform. The points are marked as:
- \( A \) (maximum positive amplitude)
- \( +x \) (positive peak)
- \( 0 \) (zero or equilibrium point)
- \( -x \) (negative peak)
- \( -A \) (maximum negative amplitude)
- **X-Axis (Horizontal)**: The horizontal axis represents time.
- **Waveform**: It is depicted as a dashed green line showing periodic oscillations.
### Key Points in the Waveform:
1. **Point 1**:
- This point is marked at approximately \( x \) on the positive side of the vertical axis, indicating a peak in the waveform.
2. **Point 2**:
- This point is situated on the equilibrium line (0 amplitude) on the graph, corresponding to a point where the waveform crosses the time axis.
3. **Point 3**:
- This point is marked at approximately \( -x \) on the negative side of the amplitude axis, indicating the negative peak of the waveform.
### Analysis:
- **Wave Behavior**:
The graph shows a periodic wave that oscillates between the maximum positive amplitude (A) and the maximum negative amplitude (-A) over a continuous time period.
- **Crossing Points**:
The waveform crosses the equilibrium line at regular intervals, shown at the zero points for time.
### Educational Purpose:
Understanding this waveform graph can help students grasp concepts related to oscillations, waves, and periodic motion. The visual representation is fundamental in subjects like physics, engineering, and signal processing where wave behaviors and properties are studied.
**(Source: Figure 10-9 from the referenced textbook)**](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1a15759c-8b82-438f-9d6a-36f728df4f17%2Fbd770d7d-31c1-407f-92c7-ad8eace9a81c%2F4cboakl_processed.jpeg&w=3840&q=75)
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