# A spring with mass, KE of spring

• makeAwish
In summary, the conversation discusses the kinetic energy of a spring and how it is usually ignored. The work done to stretch or compress the spring is given by 0.5kx^2, where x is the distance from equilibrium. The conversation then asks to calculate the kinetic energy of a spring with one end fixed and the other end moving with a linearly varying speed. The solution involves dividing the spring into pieces and finding the speed and mass of each piece, and then integrating to find the total kinetic energy.
makeAwish

## Homework Statement

We usually ignore the kinetic energy of the moving coils of a spring, but let's try to get a reasonable approximation to this. Consider a spring of mass M, equilibrium length L0, and spring constant k. The work done to stretch or compress the spring by a distance L is 0.5kx^2, where x = L – L0.
(a) Consider a spring, as described above that has one end fixed and the other end moving with speed v. Assume that the speed of points along the length of the spring varies linearly with distance l from the fixed end. Assume also that the mass M of the spring is distributed uniformly along the length of the spring. Calculate the kinetic energy of the spring in terms of M and v.
(Hint: Divide the spring into pieces of length dl; find the speed of each piece in terms of l, v, and L; find the mass of each piece in terms of dl, M, and L; and integrate from 0 to L. The result is not 0.5Mv^2, since not all of the spring moves with the same speed.)

The attempt at a solution

v = (qL^2)/2 where q is the constant proportionality of v and l
m = (M^2)/(2 landa) where landa is the linear mass density

I'm not sure if my current workings are correct. And how to get rid of these constants?

this in an integral question

you need to show you got your equations, try writing down an equation for the energy of an infinetesimal piece of the spring ie. dE in terms of v(x) and dm, then think how dm is related to dx

## 1. What is the formula for calculating the kinetic energy of a spring?

The formula for calculating the kinetic energy of a spring is KE = 1/2 * m * v^2, where m is the mass attached to the spring and v is the velocity of the mass.

## 2. How does the mass of the object attached to a spring affect its kinetic energy?

The mass of the object directly affects the kinetic energy of the spring. As the mass increases, the kinetic energy also increases because there is more mass to move and thus a higher velocity is required.

## 3. How is the kinetic energy of a spring related to its potential energy?

The kinetic energy of a spring is directly related to its potential energy. When a spring is compressed or stretched, it has potential energy stored in it. As the spring releases this potential energy, it converts into kinetic energy.

## 4. How does the velocity of the mass affect the kinetic energy of a spring?

The velocity of the mass has a direct effect on the kinetic energy of a spring. As the velocity increases, so does the kinetic energy. This is because the formula for kinetic energy includes the velocity term squared.

## 5. Can the kinetic energy of a spring ever be negative?

No, the kinetic energy of a spring cannot be negative. This is because kinetic energy is a measure of the energy an object has due to its motion, and motion cannot have a negative value. However, the velocity of the mass attached to the spring can be negative, resulting in a negative value for the kinetic energy formula. This simply means that the mass is moving in the opposite direction of its initial position.

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