Proving SHM using energy

In summary, the conversation discusses using the conservation of energy to show that a system undergoes simple harmonic motion. The equations used are 1/2mv^2 + mgh = A constant and 1/2mv^2 + mgcx^2 = A constant, with the goal of finding an equation of the form acc = - constant * position. The conversation also touches on finding the velocity and acceleration in this system.
  • #1
Mattofix
138
0

Homework Statement



i have a bead on a wire shaped by y=cx^2 where y is the height - use conservation of energy to show that the system unfergoes simple harmonic motion. - no friction!

Homework Equations



I think to prove its SHM i need to get an equation in the form acc= - w^2 x

The Attempt at a Solution



KE + PE = A constant
1/2mv^2 + mgh = A constant
1/2mv^2 + mgcx^2 = A constant

... is this right? i am pretty sure i need it in the above form them from that i know what w is and can work out the period of this moyion.
 
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  • #2
please help - with this i can't do the rest of the question :s
 
  • #3
Well as you said above, you need to show that the system satisfy an equation of the form acc = - constant * position.
More precisly something saying Z'' = - constant * Z. It doesn't matter what this constant exactly is, besides being a constant.
The v in your equation from the the kinetic energy is the SPEED of the bead so it's not the derivative of your x. What is v then? What happens if you plug that expression for v into your equation:

1/2mv^2 + mgcx^2 = constant

then taking the time derivative, so get something with dobbel derivative which is what you are seeking.
 
  • #4
what do i plug in for v? I am totally confused...
 
  • #5
It's the length of the velocity vector, so it's
v^2 = x'^2 + y'^2
where x' and y' are the velocities in the x- and y- direction respectivly (ie time derivatives of x and y resp.). What is y' in this case?
 
  • #6
y' = 2xcx' ?
 
  • #7
ahhh -= please help somebaody - its in for tomrrow
 
  • #8
Mattofix said:
y' = 2xcx' ?

This is correct.
 

1. What is SHM and how can energy be used to prove it?

Simple Harmonic Motion (SHM) is a type of periodic motion where an object oscillates back and forth around a central equilibrium point. Energy can be used to prove SHM by showing that the total mechanical energy of the system remains constant throughout the motion.

2. How is potential energy involved in SHM?

Potential energy is involved in SHM through the restoring force that brings the object back to its equilibrium position. As the object moves away from the equilibrium point, its potential energy increases, and as it returns, the potential energy is converted back into kinetic energy.

3. Can kinetic energy be used to prove SHM as well?

Yes, kinetic energy can also be used to prove SHM. As the object moves towards its equilibrium point, its kinetic energy increases, and as it moves away, the kinetic energy decreases. This cyclical pattern of energy conversion is a key characteristic of SHM.

4. What is the equation for total mechanical energy in SHM?

The equation for total mechanical energy in SHM is E = 1/2kx², where E is the total mechanical energy, k is the spring constant, and x is the displacement from the equilibrium point. This equation shows that the total mechanical energy remains constant throughout the motion.

5. How can the conservation of energy be used to prove SHM?

The conservation of energy states that energy cannot be created or destroyed, only transferred or converted. In SHM, the total mechanical energy remains constant, showing that energy is conserved throughout the motion. This is a key factor in proving SHM using energy.

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