# Oscillation of a particle

1. Apr 24, 2013

### steve2510

1. The problem statement, all variables and given/known data

A particle oscillates about a fixed point. Its distance, x(m) from the origin is given by the equation x=3sin(2t) + 2cos(2t) -2.

Find

i) its velocity,
ii) where it first comes to rest,
iii) its maximum velocity.

2. Relevant equations

3. The attempt at a solution

Well firstly dx/dt = v
∴ v = 6cos(2t) - 4sin(2t)

For i) The velocity is just found out by placing a value of T into the above equation. I'm confused as to why no value of t has been stated as the velocity is changing depending on what value of t is used.

For ii) It comes to rest when v=0
So 6cos(2t) - 4cos(2t) = 0 A=2T

Rcos(A-B) = RcosAcosB - RsinAsinB

Comparing coefficents of cosA and SinA,

RcosB= 6 RsinB= 4

RsinB/RcosB = TanB
B = tan^-1 4/6 = 0.588 rads = B

If RsinB = 4 .........R(sin 0.588) = 4
R= 7.21

7.21(cos(A-0.588) = 6cos(2t) - 4sin(2t) = 0
7.21(cos(A-0.588)) = 0 A=2t
Cos = 0 at pi/2 and 3pi/2 so Cos(2t-0.588) = 0
2t-0.588 = pi/2
2t= 2.159
t= 1.080

If I put that back into the equation, it doesn't equal 0 which means I must have gone wrong somewhere.

iii) max velocity, I assume I need to take the derivative again and plug in values of t to see which produces a maximum i.e <0

Last edited by a moderator: Apr 24, 2013
2. Apr 24, 2013

### ehild

It is 6cos(2t) - 4sin(2t)=0 instead. Which means 4sin(2t)=6cos(2t), that is tan(2t)=1.5

ehild

3. Apr 24, 2013

### steve2510

Okay, I can't believe I didn't see that, so at that time, it comes to rest. To find where it comes to rest, I guess I plug that t back into the orignal equation.

Then to find the maximum value of velocity hmm? I'm reading maximum, so I'm thinking second derivative, but I'm not sure how that helps.

Last edited by a moderator: Apr 24, 2013
4. Apr 24, 2013

### milesyoung

Yeah, use your value of t to find x(t), the position of the particle at time t.

You've probably done it a lot of times before. Find the roots of dv/dt (solutions to dv/dt = 0) to find values of t that gives maxima/minima of v.