Solving a 2 Mass Spring System: Forces, Equations, and Solutions"

[hurdler788]

Homework Statement

Consider two blocks of identical mass m. Block 1 can only move horizontally on a smooth table, and block 2 can only move vertically down the side of the table. Both blocks are connected by a spring (which is a direct path, so it should actually go through the table, but it doesn't) The spring has rest length L and spring constant k. No friction in this problem.

Homework Equations

Need the forces on each block to write Newton's equations

The Attempt at a Solution

I know block one only has the spring force and block 2 has both gravity and the spring force. The spring force is -kx where x is the distance from the equilibrium position of the spring. This is where i get stuck. I can't figure out how to write the change in the length of the spring.

Someone please help!

 

[Jhero]

Thank you for your post. You are on the right track with your understanding of the forces on each block. To write Newton's equations, we need to consider the forces acting on each block and the resulting motion.

For block 1, the only force acting on it is the spring force, which is given by -kx. Since the block can only move horizontally, we can write Newton's second law as ma = -kx, where a is the acceleration of the block. Since the block is moving horizontally, the acceleration is in the same direction as the displacement x, so we can write this equation as ma = -kx, where a is the acceleration and x is the displacement from the equilibrium position.

For block 2, there are two forces acting on it: gravity and the spring force. The gravitational force is given by mg, where g is the acceleration due to gravity. The spring force is still given by -kx, but now the displacement x is the change in length of the spring. Since the block can only move vertically, we can write Newton's second law as ma = -kx + mg, where a is the acceleration and x is the change in length of the spring.

To find the change in length of the spring, we can use the Pythagorean theorem. Since the blocks are connected by the spring, the change in length of the spring is equal to the hypotenuse of a right triangle with sides x and L. Therefore, we can write x^2 + L^2 = (L + x)^2, which simplifies to x = L√2.

Substituting this into our equations for block 1 and block 2, we get:

ma = -k(L√2) for block 1
ma = -k(L√2) + mg for block 2

Solving for a in each equation, we get:

a = -k(L√2)/m for block 1
a = (-k(L√2) + mg)/m for block 2

These are the equations of motion for each block. We can use these equations to solve for the acceleration and then the velocity and position of each block as a function of time.

I hope this helps. Let me know if you have any further questions.