Spring oscillation question HARD

[WINSTEW]

Homework Statement

Two objects are placed in a 1 dimensional box with springs of different constants at each end of the box. The box is made of wood and has a length of L. The first object has a mass of M and is initially compressed a distance of xinitial onto spring 1 of spring constant k. When M is released it collides elastically with a second object of 6M with is initially at rest. Both objects recoil in opposite directions and ignoring friction how far will the initial spring recompress when M strikes it. How far will the second spring compress when 6M hits it knowing that the spring constant is 2k. The total length of each spring is 1.5m at equilibrium.

Homework Equations

1/2kx^2
1/2mv^2
energy balance
momentum balance

The Attempt at a Solution

This is a problem I am having trouble getting started. I know the concept but I can't get it to the paper. The 1st spring has a potential energy that will be converted into a kinetic energy of the object M. This will give M momentum equal to its velocity X mass. This will collide with 6M transferring some energy. Since the momentum is conserved the the 6M will have a positive velocity and the m will now have a negative velocity and less momentum. This velocity can then be used to find out how much spring potential energy is needed to obtain this, and then find the compression needed.

 

[LowlyPion]

Homework Statement

Two objects are placed in a 1 dimensional box with springs of different constants at each end of the box. The box is made of wood and has a length of L. The first object has a mass of M and is initially compressed a distance of xinitial onto spring 1 of spring constant k. When M is released it collides elastically with a second object of 6M with is initially at rest. Both objects recoil in opposite directions and ignoring friction how far will the initial spring recompress when M strikes it. How far will the second spring compress when 6M hits it knowing that the spring constant is 2k. The total length of each spring is 1.5m at equilibrium.

Homework Equations

1/2kx^2
1/2mv^2
energy balance
momentum balance

The Attempt at a Solution

This is a problem I am having trouble getting started. I know the concept but I can't get it to the paper. The 1st spring has a potential energy that will be converted into a kinetic energy of the object M. This will give M momentum equal to its velocity X mass. This will collide with 6M transferring some energy. Since the momentum is conserved the the 6M will have a positive velocity and the m will now have a negative velocity and less momentum. This velocity can then be used to find out how much spring potential energy is needed to obtain this, and then find the compression needed.

Looks like you have the sense of the solution.

Since you know V_i in terms of the givens
1/2*k*X_i² = 1/2*M*V_i²

And you can figure the resulting velocities in terms of V_i, you should be able to work the effect of the resulting v1 back to spring1 and v2 to spring2. Looks like some things will cancel out.

 

[nlightner]

The same approach can be used for the second spring using the final velocities and the spring constant 2k.

I would start by first identifying the key concepts and equations that are relevant to this problem. These include conservation of energy and conservation of momentum. From the given information, it is clear that the initial energy of the system is stored in the compressed spring and is converted into kinetic energy when the first object is released. This kinetic energy is then transferred to the second object upon collision, and the remaining energy is converted back into potential energy in the second spring.

To solve this problem, we can use the equation for potential energy stored in a spring, 1/2kx^2, to find the initial compression of the first spring. This energy will then be transferred to the second object, which will have a kinetic energy of 1/2mv^2. Using the conservation of momentum, we can equate the initial momentum of the first object (M) to the final momentum of both objects (M and 6M) after the collision. This will allow us to solve for the final velocity of the two objects.

Once we have the final velocities, we can use the equation for potential energy in a spring again to find the compression of the second spring. Since the spring constant is given as 2k, we can simply multiply the initial compression by 2 to find the final compression of the second spring.

It is also important to note that the total length of each spring at equilibrium is given as 1.5m, which can be used to check our final answers to ensure they are physically reasonable.

In summary, to solve this problem we would use the equations for potential energy and kinetic energy, along with the conservation of momentum, to find the initial compression of the first spring and the final compression of the second spring. We can also use the given information to check our final answers and ensure they are physically valid.