How to use the energy equation for SHM to find the position at a given velocity?

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AI Thread Summary
The discussion focuses on solving a problem involving a mass-spring system undergoing simple harmonic motion (SHM). A 7kg weight on a spring with a constant of 327 N/m is analyzed to find the position at a velocity of 4.73 m/s, starting from an initial position of -1.22 m and velocity of 3.54 m/s. The key equation used is the conservation of mechanical energy, which relates initial and final kinetic and potential energies. The user initially struggles with setting up the equation but ultimately realizes that they can solve for the unknown position using the known variables. The correct final position is determined to be -1.13 m.
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Homework Statement



A 7kg weight is affixed to a spring with a constant of 327 N/m. System undergoing SHM.
Position x is -1.22m and velocity of positive 3.54 m/s. The system then moves to a position where the velocity is positive 4.73. Find the position where the velocity is 4.73 m/s

Homework Equations



I know that velocity increases in a + direction to the left of equilibrium. The initial position I am given,-1.22m is to the left of the equilibrium point, and I know the weight is speeding up toward the equilibrium point. I believe the relevant equation here is the conservation of mechanical energy for a mass-spring system.

The Attempt at a Solution



1/2mvf^2 + 1/2kxf^2 = 1/2mvi^2 + 1/2kxi^2
The 1/2 cancels

And I am stuck on how to set up the equation. I know if mvf^2 = kxi^2, then vf = √(kxi^2)/m, but that is for velocity, where is a known... ugh! I need a bump in the right direction so I can do this myself. I need to find Xf, and I think I need to take a negative square root somewhere... The book solution tells me the answer is -1.13m, but I am working on trying to figure out how To get there.

Please kick me in the right direction, so I can solve and understand this problem. Thank you!
 
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So you have $$ m v_f^2 + k x_f^2 = m v_i^2 + k x_i^2 .$$ Of all the symbols in the equation, you know ## k, \ m, \ v_i, \ v_f, \ x_i ##. The only unknown is ## x_f ##. Solve for it.
 
Ah! Thank you! It was looking me right in the face! Got it!
 
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