### Understanding Oscillatory Motion of a Mass on a SpringThe following explanation is aimed at helping students understand the oscillatory motion of a mass attached to a spring through a position-time graph. This material can be used for educational purposes in physics courses dealing with harmonic motion.#### The Position-Time Graph DescriptionThe graph (Figure 10-9) illustrates the position-time relationship for a mass oscillating vertically on the end of a spring. In the graph, the mass is shown at three different positions (1, 2, and 3) during one complete cycle of its motion.The vertical axis represents the position of the mass with points labeled A and -A, where 'A' is the maximum displacement from the equilibrium position (O). The horizontal axis represents time.#### Tasks**a. Draw the Forces**- Draw the forces acting on the mass at each of the positions (1, 2, and 3).- Ensure the length of the force vectors reflects the relative magnitudes of the forces at these positions.**b. Draw the Velocity Vectors**- Draw the velocity vectors next to each block (1, 2, and 3).- Indicate the direction and relative magnitude of the velocity at each position.**c. Draw the Acceleration Vectors**- Draw the acceleration vectors next to each block (1, 2, and 3).- Show the direction and relative magnitude of the acceleration at each position.**d. Direction of Net Force and Acceleration at Position 1**- Determine the direction of the net force at position 1.- Identify the direction of the acceleration at position 1.- Provide an explanation by comparing the size of the spring force and the weight at position 1.#### Detailed Explanation for Position Analysis1. **Position 1 (Top of the Arc)**: - At this point, the spring force (upward) is at its maximum, opposing the weight (downward). - The net force direction will be upwards as the spring force is greater than the weight. - The acceleration direction is therefore upwards.2. **Position 2 (Equilibrium Position)**: - The spring force equals the weight, resulting in a net force of zero. - The velocity is at its maximum while the acceleration is zero.3. **Position 3 (Bottom of the Arc)**: - Here, the spring force is the smallest compared to the weight

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The diagram below shows the position-time graph for a mass that is oscil- lating vertically on the end of a spring. The mass is shown superimposed on the graph at three different instants during a single cycle. Check page one of the lab for a similar graph that might be helpful. a. Draw the forces acting on the mass at each of those positions. Make sure the length of the force vectors give the relative magnitude of the forces at those positions. b. Draw the velocity vectors next to each block to show the direction and relative magnitude of the velocity at each position. C. Draw the acceleration vectors next to each block to show the direction and relative magnitude of the accel- eration at each position. A +X TIME-> What is the direction of the net force at position 1? What direc- tion is the acceleration? Give an explanation by comparing the size of the spring force and weight at position 1. d. 0 -X- -A Figure 10-9

The diagram below shows the position-time graph for a mass that is oscil- lating vertically on the end of a spring. The mass is shown superimposed on the graph at three different instants during a single cycle. Check page one of the lab for a similar graph that might be helpful. a. Draw the forces acting on the mass at each of those positions. Make sure the length of the force vectors give the relative magnitude of the forces at those positions. b. Draw the velocity vectors next to each block to show the direction and relative magnitude of the velocity at each position. C. Draw the acceleration vectors next to each block to show the direction and relative magnitude of the accel- eration at each position. A +X TIME-> What is the direction of the net force at position 1? What direc- tion is the acceleration? Give an explanation by comparing the size of the spring force and weight at position 1. d. 0 -X- -A Figure 10-9

Principles of Physics: A Calculus-Based Text

5th Edition

ISBN:9781133104261

Author:Raymond A. Serway, John W. Jewett

Publisher:Raymond A. Serway, John W. Jewett

Chapter12: Oscillatory Motion

Section: Chapter Questions

Problem 52P

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Concept explainers

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.

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Question

### Understanding Oscillatory Motion of a Mass on a Spring

The following explanation is aimed at helping students understand the oscillatory motion of a mass attached to a spring through a position-time graph. This material can be used for educational purposes in physics courses dealing with harmonic motion.

#### The Position-Time Graph Description
The graph (Figure 10-9) illustrates the position-time relationship for a mass oscillating vertically on the end of a spring. In the graph, the mass is shown at three different positions (1, 2, and 3) during one complete cycle of its motion.

The vertical axis represents the position of the mass with points labeled A and -A, where 'A' is the maximum displacement from the equilibrium position (O). The horizontal axis represents time.

#### Tasks

**a. Draw the Forces**

- Draw the forces acting on the mass at each of the positions (1, 2, and 3).
- Ensure the length of the force vectors reflects the relative magnitudes of the forces at these positions.

**b. Draw the Velocity Vectors**

- Draw the velocity vectors next to each block (1, 2, and 3).
- Indicate the direction and relative magnitude of the velocity at each position.

**c. Draw the Acceleration Vectors**

- Draw the acceleration vectors next to each block (1, 2, and 3).
- Show the direction and relative magnitude of the acceleration at each position.

**d. Direction of Net Force and Acceleration at Position 1**

- Determine the direction of the net force at position 1.
- Identify the direction of the acceleration at position 1.
- Provide an explanation by comparing the size of the spring force and the weight at position 1.

#### Detailed Explanation for Position Analysis

1. **Position 1 (Top of the Arc)**:
- At this point, the spring force (upward) is at its maximum, opposing the weight (downward).
- The net force direction will be upwards as the spring force is greater than the weight.
- The acceleration direction is therefore upwards.

2. **Position 2 (Equilibrium Position)**:
- The spring force equals the weight, resulting in a net force of zero.
- The velocity is at its maximum while the acceleration is zero.

3. **Position 3 (Bottom of the Arc)**:
- Here, the spring force is the smallest compared to the weight

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