Dynamics: Max horizontal speed of a collar with three restricting springs

[jaredogden]

Homework Statement

A 1.2-kg collar C may slide without friction along a horizontal rod.
It is attached to three springs, each of constant k 5 400 N/m and
150-mm undeformed length. Knowing that the collar is released
from rest in the position shown, determine the maximum speed it
will reach in the ensuing motion.

The position shown shows the collar C on a horizontal rod. There are three springs attached to it at the same point on the collar. Spring 1 is attached from the collar straight down to a pin at 150 mm below the collar (undeformed). Spring 2 is attached to another pin connection 150 mm below the collar and 150 mm to the left of the collar (45-45-90 triangle). And finally Spring 3 is attached 150 mm below the collar and 300 mm to the left of it.

Homework Equations

E = KE + PE + U
PE of a spring = 1/2kΔx²
E_i = E_f

The Attempt at a Solution

First find the deformed length of the springs.
Spring 3: l = √((.15m)² + (.3m)²)
l = 0.3354 m
Δx₃ = 0.3354m - 0.15m
Δx₃ = 0.1854m

Spring 2: l = √((.15m)² + (.15m)²)
l = 0.21213 m
Δx₂ = 0.21213m - 0.15m
Δx₂ = 0.06213m

Spring 1: Undeformed Δx = 0m

now E_i = PE (from the deformed springs, no KE or U)
E_i = 1/2k₁(Δx₁)¹ + 1/2k₂(Δx₂)² + 1/2k₃(Δx₃)³
E_i = 1/2(400N/m)(0m)² + 1/2(400N/m)(0.06213m)² + 1/2(400N/m)(0.1854m)²
E_i = 7.6459J

From the Law of Conservation of Energy E_i = E_f and velocity will be max when there is no spring potential energy restricting the collar and only KE.

E_f = 1/2mv²
7.6459J = 1/2(1.2kg)v²
3.57 m/s = v

That is the answer I got however the answer in the book is v = 3.19 m/s
I'm guessing one of the springs will still no matter what restrict some movement? I'm not sure what else to do on this one.

 

[tiny-tim]

hi jaredogden!

From the Law of Conservation of Energy E_i = E_f and velocity will be max when there is no spring potential energy restricting the collar and only KE.

nope

velocity will be max when spring potential energy is min (not zero), won't it? …

and they've set up the question so that it's fairly obvious where the min is, so just calculate the PE there ​

 

[jaredogden]

That was my last question, I guess there will always be some sort of spring potential acting on it, but it is at its least when the collar has moved 150 mm to the left and the outer springs are barely deformed while the middle one isn't at all.

I added in a 2(1/2)(400N/m)(0.06213m)² component which would account for the two outer springs.

I got the right answer after that! Thanks so much for the little push in the right direction!