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.

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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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**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.
Transcribed Image Text:**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.
Transcribed Image Text:**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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