A weight on the end of a spring is oscillating in harmonic...

A weight on the end of a spring is oscillating in harmonic motion. The equation model for the oscillations is $d(t)=6 \sin \left(\frac{\pi}{2} t\right),$ where $d$ is the distance (in centimeters) from the equilibrium point in $t$ sec. a. What is the period of the motion? What is the frequency of the motion? b. What is the displacement from equilibrium at $t=2.5 ?$ Is the weight moving toward the equilibrium point or away from equilibrium at this time? c. What is the displacement from equilibrium at $t=3.5 ?$ Is the weight moving toward the equilibrium point or away from equilibrium at this time? d. How far does the weight move between $t=1$ and $t=1.5$ sec? What is the average velocity for this interval? Do you expect a greater or lesser velocity for $t=1.75$ to $t=2 ?$ Explain why (FIGURE CAN NOT COPY)

A weight on the end of a spring is oscillating in harmonic motion. The equation model for the oscillations is $d(t)=6 \sin \left(\frac{\pi}{2} t\right),$ where $d$ is the distance (in centimeters) from the equilibrium point in $t$ sec.
a. What is the period of the motion? What is the frequency of the motion?
b. What is the displacement from equilibrium at $t=2.5 ?$ Is the weight moving toward the equilibrium point or away from equilibrium at this time?
c. What is the displacement from equilibrium at $t=3.5 ?$ Is the weight moving toward the equilibrium point or away from equilibrium at this time?
d. How far does the weight move between $t=1$ and $t=1.5$ sec? What is the average velocity for this interval? Do you expect a greater or lesser velocity for $t=1.75$ to $t=2 ?$ Explain why
(FIGURE CAN NOT COPY)

Key Concepts

Instantaneous and Average Velocity

Instantaneous velocity in oscillatory systems is determined by the time derivative of the displacement function and tells us the speed and direction of motion at a precise moment. Average velocity, on the other hand, is calculated over a time interval by dividing the net change in displacement by the time elapsed, providing an overall rate of motion during that period.

Displacement from Equilibrium

Displacement in oscillatory motion describes the distance and direction from the equilibrium point at any given time. It is a critical measure that informs whether an object is at rest, at its maximum excursion, or passing through the equilibrium position.

Period and Frequency

The period is the time taken for one complete cycle of the oscillation, and it is inversely related to the frequency, which represents the number of cycles completed per unit time. These concepts are fundamental in understanding the timing characteristics of any oscillatory system.

Amplitude

Amplitude is the maximum displacement from the equilibrium position in a periodic motion. It indicates the extent or intensity of the oscillation but does not affect the period or frequency of the motion.

Sinusoidal Function Modeling

Sinusoidal functions, like sine and cosine, are used to represent periodic phenomena because they naturally encapsulate the cyclic variation of displacement, velocity, and acceleration. They are essential for modeling oscillatory motions in various contexts ranging from mechanical vibrations to electrical signals.

Harmonic Motion

Harmonic motion refers to a type of periodic oscillatory movement where the restoring force is directly proportional to the displacement and acts in the opposite direction. This principle underlies many physical systems, such as springs and pendulums, and is described mathematically by sinusoidal functions.

Transcript

00:01 The function of the harmonic motion is d of t is equal to 6 sine of pi over 2.

00:16 This means that the weight is going to follow a sinusorial path.

00:23 We can graph this, we know that the highest point will be 6 because the amplitude is 6 and the lowest point will be negative 6.

00:35 The weight is going to start at the equilibrium point, which is here, is going to reach the highest point, then it's going to go back to the equilibrium point, reach the lowest point, and then it's going to go back to the equilibrium point again.

01:03 For part a, they're asking us to find a period, which we can use our formula to pi divided by b.

01:12 We know that b is equal to pi over 2 which means that the period is going to be 4 seconds they're asking us a real life problem so we have to give them a real life solution which would be 4 7 the frequency is just the inverse of the period which means 1 over 4 again we give them a real life solution so we have to put units in this case the frequency is measured in cycles per second.

01:52 Now for part b they're asking us what is the distance at time 2 .5.

02:03 So we can plug this in here.

02:05 D of 2 .5 is equal to 6 sine of pi divided by 2 times t which t in this case is 2 .5.

02:18 So we get that the distance is negative 4 .24.

02:27 Just like in part a, we have to give them units.

02:30 In this case, we're talking about distances, so we would give them centimeter.

02:36 To know if it's moving away or towards the center, we can use our graph...

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