Harmonic Oscillator kinetic and potential energies

It just gives me the same question, but with 1/2.In summary, a simple harmonic oscillator with a total energy of E has a kinetic energy of 7/16E and a potential energy of 9/16E when the displacement is three-fourths the amplitude. The value for the displacement when the kinetic energy equals one half the potential energy is approximately 0.8165 times the amplitude.
  • #1
lydster
7
0
A simple harmonic oscillator has a total energy of E.

(a) Determine the kinetic and potential energies when the displacement is three-fourths the amplitude. (Give your answer in terms of total energy E of the oscillator.)

Kinetic energy ______________ x E <----(times E)

Potential energy _____________ x E <-----(times E)(b) For what value of the displacement does the kinetic energy equal one half the potential energy? (Give your answer in terms of the amplitude A of the oscillator.)_________________ AI followed an example from the book, which was the same question, except for A and B it was one-half, and their answers are for (a) Kinetic is 3/4 E and potential is 1/4 E. and I have no clue on B

Thanks
 
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  • #2
lydster said:
A simple harmonic oscillator has a total energy of E.

(a) Determine the kinetic and potential energies when the displacement is three-fourths the amplitude. (Give your answer in terms of total energy E of the oscillator.)

Kinetic energy ______________ x E <----(times E)

Potential energy _____________ x E <-----(times E)(b) For what value of the displacement does the kinetic energy equal one half the potential energy? (Give your answer in terms of the amplitude A of the oscillator.)_________________ A

I followed an example from the book, which was the same question, except for A and B it was one-half, and their answers are for (a) Kinetic is 3/4 E and potential is 1/4 E. and I have no clue on B
I don't think those answers are right.

The potential energy for a harmonic oscillator is:

[tex]PE = \frac{1}{2}kx^2[/tex]

The total energy is the PE when KE=0 which occurs at maximum amplitude. ie total E is:

[tex]E = \frac{1}{2}kA^2[/tex]

So [tex]KE = E - PE = \frac{1}{2}k(A^2 - x^2) = \frac{1}{2}kA^2(1 - (\frac{x}{A})^2) = E(1 - (\frac{x}{A})^2)[/tex]

where x is the displacement and A is the maximum amplitude.

So for a), if displacement is 3/4 of A, then KE = 7/16 of E and PE is 9/16 of E

For b) if KE = .5PE, then [itex]PE = E/1.5 = 1/3kA^2[/itex]. You can work out the displacement from that.

AM
 
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  • #3
answers

Yeah I got a friend to try those numbers, and they didn't work out. He got the same answers as you, and they are wrong. Hmmmm...I dunno. Everything that you said makes sense
 
  • #4
lydster said:
Yeah I got a friend to try those numbers, and they didn't work out. He got the same answers as you, and they are wrong. Hmmmm...I dunno. Everything that you said makes sense
What makes you think the 7/16 , 9/16 answer is wrong?

The answer to b), if [itex]kA^2/3 = kx^2/2[/itex] then

[tex]x = \sqrt{\frac{2}{3}}A = .8165A[/tex]

What does your book say?

AM
 
  • #5
Because my online quiz thing automatically says if I'm right or wrong, and those answers were off. It said that I was within 10%-100% of the actualy answer. My book doesn't really say anything on that.
 

1. What is a harmonic oscillator?

A harmonic oscillator is a type of system that exhibits periodic motion, where the restoring force is directly proportional to the displacement of the system from its equilibrium position. It can be described by a mathematical model known as the harmonic oscillator equation.

2. What is kinetic energy in a harmonic oscillator?

Kinetic energy in a harmonic oscillator refers to the energy associated with the motion of the system. It is dependent on the mass and velocity of the system and is given by the equation KE = 1/2 * m * v^2, where m is the mass and v is the velocity of the oscillating object.

3. How is potential energy related to a harmonic oscillator?

Potential energy in a harmonic oscillator refers to the energy stored in the system due to its position relative to its equilibrium point. It is dependent on the spring constant and the displacement from equilibrium and is given by the equation PE = 1/2 * k * x^2, where k is the spring constant and x is the displacement from equilibrium.

4. How does the total energy of a harmonic oscillator change over time?

The total energy of a harmonic oscillator remains constant over time, as long as there are no external forces acting on the system. This is known as the principle of conservation of energy, where the sum of kinetic and potential energy remains constant.

5. What are some real-life examples of harmonic oscillators?

Some common examples of harmonic oscillators in real-life include pendulums, guitar strings, and mass-spring systems. These systems exhibit periodic motion and can be described by the harmonic oscillator equation.

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