**Problem:**c. At what times does the mass reach its lowest point the first three times?d. At what times does the mass reach its highest point the first three times?---In-depth explanation and solution to these problems can be found in the following sections. **Detailed Instructions:**1. **Analyzing Oscillation Patterns:** - When a mass undergoes periodic motion, such as harmonic oscillation, it reaches various points of its path repeatedly. - The lowest point in an oscillation is typically referred to as the trough. - The highest point in an oscillation is generally known as the crest.2. **Determining Specific Times:** - To find the specific times at which these points are reached, one needs to understand the period of oscillation (T) and its initial phase. - Using the period, the time intervals at which the mass reaches these points can be calculated.Graphs and diagrams that illustrate oscillation over time, such as those showcasing the displacement vs. time for one cycle, will significantly aid in visualizing and understanding these concepts.**Note:**Students are encouraged to refer to their textbook or course materials for formulas related to periodic motion, such as those for a simple harmonic oscillator. **Motion of a Mass on a Spring:**In this problem, a mass hanging from a spring is set in motion. The velocity of the mass as a function of time \(v(t)\) is given by:\[ v(t) = -4\pi \sin \pi t \]where \( t \geq 0 \).**Assumptions:**- The positive direction is considered to be upward.- The initial position of the mass \( s(0) \) is given as 4.**Analysis:**1. **Understanding the Velocity Function:**The velocity function describes the rate of change of the position of the mass with respect to time. In this case, the velocity is a sinusoidal function indicating oscillatory motion, characteristic of a spring-mass system.2. **Direction and Initial Condition:**Since the positive direction is upward, the initial position \( s(0) = 4 \) implies that the mass starts from a position 4 units above a defined reference point.**Expected Outcomes:**By integrating the velocity function, the position function \( s(t) \) can be determined, providing insight into the displacement of the mass over time. The sine component in the velocity function suggests periodic motion, typical behavior for objects attached to a spring.**Application:**Understanding the motion of masses on springs is crucial in various fields such as mechanical engineering, physics, and even biological systems where similar oscillatory motions are observed.

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Description: **Problem:**c. At what times does the mass reach its lowest point the first three times?d. At what times does the mass reach its highest point the first three times?---In-depth explanation and solution to these problems can be found in the following

The equation of velocity :v(t) = - 4πsin πt s(0) = 4

Answered: A mass hanging from a spring is set in…

Science

Physics

A mass hanging from a spring is set in motion and its ensuing velocity is given by v(t) = - 4t sin at for t20. Assume that the positive direction is upward and s(0) = 4.

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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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.

Topic Video

Question

**Problem:**
c. At what times does the mass reach its lowest point the first three times?

d. At what times does the mass reach its highest point the first three times?

---

In-depth explanation and solution to these problems can be found in the following sections.

**Detailed Instructions:**

1. **Analyzing Oscillation Patterns:**
- When a mass undergoes periodic motion, such as harmonic oscillation, it reaches various points of its path repeatedly.
- The lowest point in an oscillation is typically referred to as the trough.
- The highest point in an oscillation is generally known as the crest.

2. **Determining Specific Times:**
- To find the specific times at which these points are reached, one needs to understand the period of oscillation (T) and its initial phase.
- Using the period, the time intervals at which the mass reaches these points can be calculated.

Graphs and diagrams that illustrate oscillation over time, such as those showcasing the displacement vs. time for one cycle, will significantly aid in visualizing and understanding these concepts.

**Note:**
Students are encouraged to refer to their textbook or course materials for formulas related to periodic motion, such as those for a simple harmonic oscillator.

**Motion of a Mass on a Spring:**

In this problem, a mass hanging from a spring is set in motion. The velocity of the mass as a function of time \(v(t)\) is given by:

\[ v(t) = -4\pi \sin \pi t \]

where \( t \geq 0 \).

**Assumptions:**
- The positive direction is considered to be upward.
- The initial position of the mass \( s(0) \) is given as 4.

**Analysis:**

1. **Understanding the Velocity Function:**

The velocity function describes the rate of change of the position of the mass with respect to time. In this case, the velocity is a sinusoidal function indicating oscillatory motion, characteristic of a spring-mass system.

2. **Direction and Initial Condition:**

Since the positive direction is upward, the initial position \( s(0) = 4 \) implies that the mass starts from a position 4 units above a defined reference point.

**Expected Outcomes:**

By integrating the velocity function, the position function \( s(t) \) can be determined, providing insight into the displacement of the mass over time. The sine component in the velocity function suggests periodic motion, typical behavior for objects attached to a spring.

**Application:**

Understanding the motion of masses on springs is crucial in various fields such as mechanical engineering, physics, and even biological systems where similar oscillatory motions are observed.

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