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Problem 36 Hard Difficulty

An elastic band is hung on a hook and a mass is hung on the lower end of the band. When the mass is pulled downward and then released, it vibrates vertically. The equation of motion is $ s = 2 \cos t + 3 \sin t, t \ge 0, $ where s is measured in centimeters and $ t $ in seconds. (Take the positive direction to be downward.)
(a) Find the velocity and acceleration at time $ t. $
(b) Graph the velocity and acceleration functions.
(c) When does the mass pass through the equilibrium position for the first time?
(d) How far from its equilibrium position does the mass travel?
(e) When is the speed the greatest?

Answer

a)$$\begin{array}{l}{v=-2 \sin t+3 \cos t} \\ {a=-2 \cos t-3 \sin t}\end{array}$$
b) See solution
c) The mas will be at equilibrium for the first time when $t \approx 2.55$
d) 3.6056 centimeters
e) $\approx 2.5536 \quad\text{seconds}$

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Video Transcript

it's clear. So it name right here. So velocity is the derivative of the position we have us is equal to to co sign was three sign When we derive this, we've got negative to sign less three co sign To get acceleration, we have to derive velocity Getting negative to co sign minus three. Sign per per p. We're gonna draw a graph of velocity and acceleration. So let's start from here. This is gonna be velocity an acceleration, something like this report. See, you know that the mass in equilibrium is when the acceleration is zero and we got acceleration in part A. So you get zero is equal to negative to co sign minus three sign when we get a T value. Well, arken of negative 2/3 which is around negative 0.59 We know that the tan function has a period of Hi. So it's gonna be a equilibrium when t is equal to negative 0.59 plus and pie. And we could just plug in one for and and you get 2.55 So the mass ISS equilibrium when t is equal to 2.55 party, we're gonna use the velocity function, which is the derivative of our position function. We're gonna make a equal to zero when we get a value for tea to be our 10 off three three house, which is around 0.98 to eight. We plug this in to our equation. Her distance equation for us. Really good. Around 3.6 0567 years. Known for part E. We know that speed is the absolute value of velocity. So we're gonna take us of tea. It is equal to zero, and we get our 10 of the negative 2/3 for a tea. And we know this tangerine inverse of negative two birds gives a negative number. We know that we can have a negative. So we need to write a specific equation. Just he is equal to an inverse of 2/3 waas ti pi. Since it repeats every rotation off pie and we're just gonna plug in one for Kay, we get two 0.5536 seconds