Gravitational and elastic energy

[pdrjuarez]

Homework Statement

A spring with spring constant k is hanging from the ceiling, at equilibrium point. The length of the spring in equilibrium is h₂. Then you hang a mass less string from the end of the spring, holding a mass m. The length of the string and the mass together equal y.
The height of the floor to the ceiling is h₁

Using elastic and gravitational energy equations, what is the necessary length of the spring and the mass combined (y), so that when you hang the mass, it just touches the floor? (in other words, solve for y?)

Homework Equations

E_e=kx²/2

E_g=mgh

 

[rl.bhat]

If x is the extension in the spring, then
h1 = h2 + x + y.
Substitute the value of x, in the relevant equations and solve for y.

 

[tomcue]

I would approach this problem by first analyzing the forces at play. The spring is initially at equilibrium, meaning that the force of gravity pulling down on it is balanced by the elastic force pulling upwards. When the mass is added to the end of the spring, the force of gravity increases, causing the spring to stretch further. The key to solving this problem is finding the point at which the force of gravity is equal to the elastic force, resulting in a stable equilibrium.

Using the equations provided, we can set Ee (elastic energy) equal to Eg (gravitational energy) and solve for y. This will give us the necessary length of the spring and mass combined for the mass to just touch the floor.

Ee = Eg

kx^2/2 = mgh

Since the spring is initially at equilibrium, the length of the spring is given as h2. We can substitute this value for x in the equation.

ky^2/2 = mgh

Solving for y, we get:

y = √(2mgh/k)

This is the necessary length of the spring and mass combined for the mass to just touch the floor. It is important to note that this equation assumes a massless string and a point mass at the end of the string. If the string has a mass and the mass is distributed along the string, the equation would need to be modified accordingly.

In conclusion, by using the concepts of elastic and gravitational energy, we can determine the necessary length of the spring and mass combined to achieve a stable equilibrium with the mass just touching the floor. This problem highlights the interplay between different forms of energy and the importance of understanding the forces at play in a system.