Spring System with Masses: Force and Vertical Oscillations

[Lewis]

I'm having lots of trouble with this one, help would be much appreciated!

A spring of stiffness k supports a box of mass M, in which is placed a block of mass m. If the system is pulled downward a distance d from the equilibrium position and then released, find the force of reaction between the block and the bottom of the box as a function of time. For what value of d will the block just begin to leave the bottom of the box at the top of the verticle oscillations? Neglect any air resistance.

 

[Hootenanny]

Please read this thread

https://www.physicsforums.com/showthread.php?t=4825

 

[kyle8921]

First of all, it is important to understand the physical principles at play in this system. The spring stiffness, k, is a measure of how much force is required to stretch or compress the spring by a certain distance. In this case, the spring is supporting a box with a mass M, which in turn contains a block of mass m. When the system is pulled down a distance d, the spring will exert a force on the block in the opposite direction, trying to restore the system to its equilibrium position.

As the block is released, it will begin to oscillate up and down due to the restoring force of the spring. This oscillation is a result of the interplay between the gravitational force acting on the block and the spring force. The block will reach its maximum height when the spring force is equal to the gravitational force, and it will then begin to fall back down.

To find the force of reaction between the block and the bottom of the box as a function of time, we can use the equation F = -kx, where F is the force, k is the spring stiffness, and x is the displacement from the equilibrium position. In this case, x will vary with time as the block oscillates up and down. Therefore, the force of reaction will also vary with time.

To find the value of d at which the block just begins to leave the bottom of the box at the top of the vertical oscillations, we can use the conservation of energy principle. At the top of the oscillations, the block will have potential energy due to its height and kinetic energy due to its motion. At this point, the spring will be fully compressed, and all of the initial potential energy will have been converted to kinetic energy. Using the equations for potential and kinetic energy and setting them equal to each other, we can solve for d.

It is important to note that this analysis neglects any air resistance, which would affect the motion of the block and the force of reaction. In a real-world scenario, air resistance would need to be taken into account to get a more accurate understanding of the system's behavior.

In conclusion, understanding the physical principles of spring systems and using equations such as F = -kx and conservation of energy can help us analyze and predict the behavior of this system. It is also crucial to consider any external factors, such as air resistance, that may affect the results. I hope this helps with your understanding of the problem and I am available