# Calculus - Equillibrium of band with mass

1. Oct 6, 2013

### JonGarces

1. The problem statement, all variables and given/known data
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≥0, where s is measured in cm and t in seconds. (Take the positive direction to be downward.)

a) Find the Velocity and acceleration at time t. *Easy and done*
b) Graph velocity and acceleration functions. *Easy and done*
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?

2. Relevant equations
s(t) = 2 cos t+3 sin t
v(t)= - 2 sin t+3 cos t
a(t)= - 2 cos t - 3 sin t

3. The attempt at a solution

I'm stumped at c. What I'm thinking is when the velocity function equals 0, but when i think more into it it starts to not make sense. Any advice is appreciated.

2. Oct 7, 2013

### MuIotaTau

Are there any forces acting on the mass in the instant it's at its equilibrium position? What does that tell us about the acceleration at that instant?

3. Oct 7, 2013

### HallsofIvy

Staff Emeritus
Spring motion will go equal distances on either side of the "equilibrium" point so one way to find the equilibrium is to find the maximum and minimum value and take half way between. Since this s is given by sine and cosine, another way to find the equilibrium is to write it as a single sine function.

4. Oct 7, 2013

### JonGarces

Alright thank you, basically I'm looking for the point of inflection. Aka where the 2nd derivative equals zero.