Perform the following steps for each scenario listed below

Perform the following steps for each scenario listed below:

- Try to make an intuitive physical argument for whether or not the total mechanical energy is conserved in the system

- If you think mechanical energy is conserved, try to prove it by mathematically verifying that the net work done by non-conservative forces is zero

- If you think mechanical energy is not conserved, try to prove it by computing the total mechanical energy at different points in time and showing that they\'re nor the same, and try to determine where the non-conservative work is coming from. (Hint: Are there types of energy other than mechanical energy?)

Scenarios:

4. The system: a ball and a massless spring. At the initial time, the ball is dropped above the spring. At the final time, the ball has come to a halt and the spring is compressed.

5. The system: a rocket taking off. At the initial time, the rocket is sitting on the ground, filled with fuel. At the final time, the fuel has ignited, and the rocket is moving upward.

Solution

4) Mechanical Energy is conserved in the system because there are no non-conservative forces acting on the system. The spring is massless as well.

let the ball dop from a height h on a spring with stiffness k, finally the ball comes to rest compressing the spring.

Therefore applying conservation of energy

mg*h = 0.5 * k * x^2    =>   x = (2mgh/k)^0.5     where x is the deformation in the spring.

Hence no non conservative forces act on this system.

5) A rocket taking will not have its total mechanical energy conserved because the drag forces acting on the high speed rocket are not negligible.

initally if the rocket is launched with speed vo then its KE = 0.5 *m * vo^2

finally if the rocket is at a height h and speed v then TE = m*g*h + 0.5*m*v^2

But 0.5mvo*2 is not equal to m*g*h + 0.5*m*v^2 because of the drag forces which do some non conservative work on the rocket.