How Does Spring Compression Affect Mass Velocity on a Frictionless Table?

[pdubb]

A massless spring is between a 1-kilogram mass and a 3-kilogram mass, but it is not attached to either mass. Both masses are on a horizontal frictionless table. In an experiment, the 1-kilogram mass is held in place, and the spring is compressed by pushing on the 3-kilogram mass. The 3-kilogram mass is then released and moves off with a speed of 10 meters per second.

(a) Determine the minimum work needed to compress the spring in this experiment.
(b) Determine the final velocity of each mass relative to the cable after the masses are released.

I am totally lost on this problem. I mean, I understand what it is asking for, but I don't know which equation to use and what variables to solve for. I know the equation for work is W= 1/2kx^2.
I know that when the spring is compressed with mass of 3 kg, then there's only spring potential energy, when the mass is released with a speed of 10m/s then there's only kinetic energy, but i don't know what to do to solve this problem.

 

[CaptainZappo]

Assuming both masses are released, I believe you should also consider what the 1 kg mass is doing. What does conservation of momentum tell you? Being that kinetic energy is a scalar quantity, how can you relate the movement of these two masses to the minimum input work required?

 

[ogb p]

I would suggest approaching this problem by first identifying the relevant physical principles and equations that can be applied. In this case, the key principles involved are conservation of energy and Hooke's law for springs.

For part (a), we can use the equation for the potential energy stored in a spring, which is given as 1/2kx^2, where k is the spring constant and x is the displacement of the spring from its equilibrium position. In this case, since the spring is being compressed, x will be negative. We also know that the 3-kilogram mass is being used to compress the spring, so we can use its weight (mg) as the force applied to compress the spring. Therefore, the equation for the work done to compress the spring would be W = -1/2kx^2 = -1/2(mg)x.

For part (b), we can use the principle of conservation of energy, which states that the total energy of a system remains constant. In this case, the initial energy of the system is the potential energy stored in the compressed spring, and the final energy is the kinetic energy of the masses. We can set these two energies equal to each other and solve for the final velocities. The equation would be 1/2kx^2 = 1/2mv^2, where v is the final velocity of each mass. Since we know the mass (1 kg and 3 kg) and we can calculate the spring constant (k) using the given information, we can solve for the final velocities.

In summary, to solve this problem, we need to use the equations for potential energy stored in a spring, work done by a force, and conservation of energy. By identifying the relevant principles and equations, we can solve for the minimum work needed to compress the spring and the final velocities of the masses.