Kinetic energy of a spring-mass system

[peachy112girl]

Homework Statement

The total energy of a spring-mass system oscillating horizontally is 1600 J. What will be the kinetic energy (in joules) of the mass when its displacement is one-fourth of its amplitude?

Homework Equations

total energy = KE + PE = .5kA²
KE = .5mv²
PE = .5kx²

The Attempt at a Solution

All I have written down is-
1600 J = .5kA²

So, I have no idea how to start solving this problem. Can anyone help me get started? Any help is appreciated!

 

[denverdoc]

At the extreme ends of the displacement, 1/2kx^2=1600 where x is amplitude.

This leads to 1/2k (.25x)^2 = WHAT WE NEED TO KNOW. Set it up as a proportional problem.

AS the total energy is 1600J, the KE will be 1600-PE.

 

[kerimek]

I can provide an explanation and steps for solving this problem. First, it is important to understand the concept of kinetic energy in a spring-mass system. Kinetic energy is the energy an object possesses due to its motion. In this case, the mass is oscillating horizontally, which means it is moving back and forth. The kinetic energy of the mass is dependent on its velocity, which is constantly changing as it oscillates.

To solve this problem, we need to use the equation for total energy, which states that the total energy of a spring-mass system is equal to the sum of its kinetic energy and potential energy. In other words, the energy stored in the spring (potential energy) is converted into the energy of motion (kinetic energy) as the mass oscillates.

So, we can rewrite the given equation as:

1600 J = KE + PE

Next, we need to use the equations for kinetic energy and potential energy in a spring-mass system. These are:

KE = 1/2 * m * v^2

PE = 1/2 * k * x^2

Where m is the mass of the object, v is its velocity, k is the spring constant, and x is the displacement from equilibrium.

Now, we can substitute these equations into the first one:

1600 J = 1/2 * m * v^2 + 1/2 * k * x^2

Since we are given that the total energy is 1600 J, we can solve for the kinetic energy, which is what the question is asking for. To do this, we need to know the values of the mass (m), spring constant (k), and displacement (x).

We are given that the total energy is 1600 J, but we do not have enough information to determine the values of m and k. However, we are given that the displacement is one-fourth of the amplitude. This means that x = 1/4 * A, where A is the amplitude of the oscillation. We can rewrite this as:

1600 J = 1/2 * m * v^2 + 1/2 * k * (1/4 * A)^2

Now, we can simplify this equation and solve for the kinetic energy:

1600 J = 1/2 * m * v^2 + 1/32 * k * A^2

160