A mass hanging from a spring is set in motion, and its ensuing velocity is given by v(t)=2 πcos

step 1 So we have a mat hanging from a spring. So let's say this is our spring. And I have a mat. M han

A mass hanging from a spring is set in motion, and its...

Key Concepts

Integration to Obtain Position

In the context of motion, the position function of an object is obtained by integrating its velocity function over time. This process requires the careful application of antiderivatives along with any given initial conditions to uniquely determine the constant of integration. It is a fundamental method in calculus for relating rates of change to accumulated quantities.

Simple Harmonic Motion (SHM)

Simple harmonic motion describes a type of periodic oscillation where the restoring force is directly proportional to the displacement and acts in the opposite direction. In a mass-spring system, the motion is modeled using sine or cosine functions, capturing key characteristics such as amplitude, period, and phase shift, which are crucial for understanding the oscillatory behavior.

Trigonometric Functions in Oscillation

Trigonometric functions such as sine and cosine are essential in modeling periodic motions, especially in systems undergoing simple harmonic motion. They allow for the representation of the cyclical behavior of the system, including the determination of positions at any given time, the identification of maximum and minimum points, and overall graphing of the motion.

Identifying Extrema in Periodic Motion

Finding the maximum (high point) and minimum (low point) of a periodic function involves analyzing the points where the function attains its highest and lowest values respectively. This is typically done by either differentiating the function to find its critical points or by scrutinizing the inherent properties of the sine and cosine functions that naturally arise in periodic systems like a mass-spring oscillator.

Transcript

00:01 So we have a mat hanging from a spring.

00:04 So let's say this is our spring.

00:08 And i have a mat.

00:09 M hanging from this spring.

00:13 And it sets in motion, insuring its velocity.

00:19 So i have my velocity, v of t, giving us two pi, called pi, p, t.

00:29 So this is the velocity we have.

00:31 And we are saying that the velocity, velocity is a function it's a function which is what's positive such that i have see from zero towards high so this is the interval we have and we are told that's the motion is an upward one so it is moving in an upward one so for a, we would want to determine the position function.

01:18 So we want to look at what the position, the position function.

01:26 For t, for t greater than or equal to zero.

01:34 So how do we go about it? so the position function, let's say s of t is given to be s of nuts.

01:46 S not plus the integral from 0 to t v of t d t so to find the position s of t what we are doing is to add the displacement over the interval 0 towards t to the initial words position so this is our initial position so this is my initial position initial position position and this is what my displacement this is the displacement i'm looking at so there's the velocity v prime of what t so that if you add the two then you get what the position function we are trying to solve so what is s not the initial position we know it was equal to what zero.

03:08 So we know i have zero plus the integral to what? the displacement, which is what v of x, the words the x.

03:20 So wherever i see t, i substitute s.

03:23 And this is going to give me the integral to t.

03:26 I have words 2 pi cost pi words x the words the x which is equal to this is going to be equal to 2 pi over pi because of the trick function you differentiate the function here with respect to x you get pi you divide by that then you integrate the trick function so if we integrate course we are getting sine pi x the interval is from what zero to what t okay so for the lower interval this is going to go to zero so we have the upper interval so our position for our position function s of word t is therefore equal towards two sign pi words c.

04:40 So this is what the position was function.

04:46 Then we would also want to look at for the b part we want to find what sketch the position graph on the interval 0 towards 4.

05:01 So for b we are looking as graph the so we want to grab the position interval s of t on the interval 0 towards 4.

05:15 Okay, so let's see how it looks like since it's a sign function...

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