Solving Energy of a Spring: Find Velocity at 2/3 E

In summary, an object with mass m attached to a spring with force constant k oscillates with simple harmonic motion. Its maximum displacement from equilibrium is A and the total mechanical energy of the system is E. To find the object's velocity when its potential energy is 2/3 of the total energy, we can use the equation E=KE+PE, and substitute in the known values to solve for v. However, this only gives us the maximum velocity, not the velocity when the potential energy is 2/3 of the total energy. Further calculations are needed to find this velocity.
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Homework Statement



An object of mass m attached to a spring of force constant oscillates with simple harmonic motion. The maximum displacement from equilibrium is A and the total mechanical energy of the system is E.

What is the object's velocity when its potential energy is 2/3 E ?


Homework Equations



E=KE+PE_s



The Attempt at a Solution



1/2kA^2 = 2/3(1/2kA^2) + 1/3(1/2mv^2)

solving for v...

1/6mv^2 = 1/2kA^2 - 1/3kA^2

1/6mv^2 = 1/6 kA^2

mv^2 = kA^2

v= sqrt(k/m) A

But it's wrong...any help? Thanks!
 
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  • #2
velocity when its potential energy is 2/3 E
when PE = 2/3*E then KE = 1/3*E
At maximum displacement, v = 0, KE = 0, so E = PE =1/2*k*A²

1/2kA^2 = 2/3(1/2kA^2) + 1/3(1/2mv^2)
This says E = 2/3*(maxPE) + 1/3*(maxKE) and is only true when v is the maximum velocity. Not the velocity you are looking for in the question.
 

1. How do you determine the energy of a spring?

To determine the energy of a spring, you need to know its spring constant (k) and the displacement (x) from its equilibrium position. The energy can be calculated using the formula E = 1/2 * k * x^2.

2. What is the significance of finding the velocity at 2/3 of the energy of a spring?

Finding the velocity at 2/3 of the energy of a spring can help us understand the behavior and motion of the spring. It can also give us information about the maximum potential energy and kinetic energy of the spring at that point.

3. Can the velocity at 2/3 of the energy of a spring be negative?

Yes, the velocity at 2/3 of the energy of a spring can be negative if the spring is moving in the opposite direction of its initial displacement. This indicates that the spring is slowing down and losing energy.

4. How does the mass of the spring affect the velocity at 2/3 of its energy?

The mass of the spring does not directly affect the velocity at 2/3 of its energy. However, a heavier spring will require more energy to reach that velocity compared to a lighter spring with the same spring constant and displacement.

5. Can the velocity at 2/3 of the energy of a spring be greater than the initial velocity?

Yes, it is possible for the velocity at 2/3 of the energy of a spring to be greater than the initial velocity. This indicates that the spring has gained additional kinetic energy and is moving faster than its initial state.

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