A weight hanging on a vertical spring is set in motion with...

A weight hanging on a vertical spring is set in motion with a downward velocity of $6 \mathrm{cm} / \mathrm{sec}$ from its equilibrium position. Assume that the constant $\omega$ for this particular spring and weight combination is $2 .$ Write the formula that gives the location of the weight in centimeters as a function of the time $t$ in seconds. Find the amplitude and period of the function and sketch its graph for $t$ in the interval $[0,2 \pi] .$ (See Section 5.2 for the general formula that describes the motion of a spring.)

Key Concepts

Period

The period is the time it takes for one complete cycle of oscillation and is intimately connected to angular frequency by the formula T = 2?/?. Understanding the period allows one to predict the timing and repetition of the oscillatory motion.

Simple Harmonic Motion

Simple harmonic motion (SHM) describes the oscillatory movement of systems such as a weight on a spring, where the restoring force is proportional to the displacement from equilibrium. This fundamental concept in physics and differential equations explains how systems oscillate periodically over time, with the motion typically expressed using sine and cosine functions.

Angular Frequency

Angular frequency (denoted by ?) is a measure of how rapidly an object in oscillatory motion cycles through its states, defined as the rate of phase change in radians per second. It is crucial in relating the temporal evolution of the system to its oscillatory behavior and directly influences the period of the motion.

Initial Conditions in SHM

Initial conditions specify the starting displacement and velocity of the system, which are used to determine the specific constants (such as amplitude and phase shift) in the general solution of the SHM differential equation. These conditions ensure that the solution accurately reflects the observed motion from a given time, typically t = 0.

Amplitude

Amplitude is the maximum displacement of the oscillating system from its equilibrium position, representing the energy of the oscillation. It is a key parameter that defines the extent of the oscillation for a simple harmonic oscillator.

Transcript

00:01 The question tells us that a spring was set in motion downwards at 6 centimeters per second and the spring constant is 2.

00:08 And they ask us to write the formula that gives the weight's location as a function of time and find the amplitude and period and graph in the interval from 0 to 2 pi.

00:20 So i wrote the spring formula down over here.

00:24 It is this over this equation.

00:29 And since they give us the motion and the spring constant, we can really just plug in these numbers into this equation to find the formula.

00:40 So v0 is our original velocity.

00:45 The omega, or this variable that looks like a w, that is going to be our spring constant.

00:51 And x0 is at the original position.

00:54 So if i plug the numbers in, we get x, which is the location.

00:59 That's what we're solving for, is equal to v0, which is 6, over the spring constant 2 times sign of 2t.

01:11 And then since x0 is, well, zero, because it doesn't say like how far the spring was compressed or stretched, we can put in zero and basically everything back here cancels out.

01:26 So we don't have to write this at all.

01:29 So this will be our formula.

01:32 And obviously i can simplify it.

01:35 So i'm going to do that.

01:37 So x is equal to 3 sine of 2t.

01:41 And this will be the equation that we're going to be graphing.

01:46 This is the formula of the weights location.

01:49 When it says function of time, it just means that the variable on the other side is going to be t.

01:54 So in this case, we don't have a y value.

01:56 We have a t value instead.

01:58 So now i can move on to finding the amplitude and period.

02:08 The amplitude is going to be this value in front of the sign, which is 3.