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Old Apr 19th 2015, 12:35 AM   #1
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Kinetic Enery of a Spring?

A cart attached to a spring vibrates with amplitude A. What fraction of the total energy of the cart-spring system is kinetic energy when the cart is at position x=A/2.0?

K/(Utotal) = ?

Express your answer using two significant figures.

I have figured that the maximum kinetic energy of a spring will be 0.5mv^2, and the maximum potential energy will be 0.5kx^2. Therefore, the fraction of kinetic energy out of the total energy will be 0.5mv^2/((0.5mv^2)+(0.5kx^2)), correct?

The second part of the question:

At what position is the cart when its kinetic energy equals its elastic potential energy?
Express your answer in terms of A.

I figure that 0.5mv^2 = 0.5kx^2 but get confused trying to solve in terms of A.
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Old Apr 19th 2015, 05:52 AM   #2
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Start by recognizing that the total mechanical energy of the spring/mass at any point is:

PE + KE = 0.5 kx^2 + 0.5 mv^2 = U = constanr.

Here v (velocity) is a function of position (x). In you attempt at a solution you have a correct equation, but since you don;t know what v is at the point in question you got stuck. But consider that at maximum displacement where x is maximum and equal to A the velocity v = 0. Hence the constant U in the above equation is:

0.5kA^2 = U

So now for part 1: the PE at any point is 0.5kx^2, and you are asked to find the ratio of PE/U at x = 0.5A. Can you take it from here?

As for part 2: If PE = KE, then both are equal to U/2. Hence:

PE = 0.5 kx^2 = U/2 = 0.5 (0.5 kA^2)

Solve for x.
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Old Sep 28th 2015, 04:24 AM   #3
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Kinetic energy of a spring

I have added some points here which is related to your question. I think this will be useful for you.

In physics, you can examine how much potential and kinetic energy is stored in a spring when you compress or stretch it. The work you do compressing or stretching the spring must go into the energy stored in the spring. That energy is called elastic potential energy and is equal to the force, F, times the distance, s:
W = Fs
As you stretch or compress a spring, the force varies, but it varies in a linear way (because in Hooke’s law, force is proportional to the displacement).

therefore W = Fs

The distance (or displacement), s, is just the difference in position, xfxi, and the average force is (1/2)(Ff + Fi). Therefore, you can rewrite the equation as follows:

For example, suppose a spring is elastic and has a spring constant, k, of

and you compress the spring by 10.0 centimeters. You store the following amount of energy in it:

You can also note that when you let the spring go with a mass on the end of it, the mechanical energy (the sum of potential and kinetic energy) is conserved:
PE1 + KE1 = PE2 + KE2
When the moving mass reaches the equilibrium point and no force from the spring is acting on the mass, you have maximum velocity and therefore maximum kinetic energy — at that point, the kinetic energy is
by the conservation of mechanical energy.

Last edited by topsquark; Sep 28th 2015 at 08:10 AM. Reason: Removed ad link
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