# Hooke's Law derivation problem involving two blocks

• Emethyst
In summary, the conversation discusses how to find the maximum amplitude of oscillation for a block on a frictionless surface connected to a wall by a spring, with a second block resting on top. The coefficient of static friction between the blocks is given as u. The attempt at a solution involves setting up Fnet equations, but there are errors in the equations and a different approach is suggested.
Emethyst

## Homework Statement

A block with mass M rests on a frictionless surface and is connected to a horizontal spring of force constant k, the other end of which is attached to a wall. A second block with mass m rests on top of the first block. The coefficient of static friction between the blocks is u. Find the maximum amplitude of oscilliation such that the top block will not slip on the bottom block.

## Homework Equations

Hooke's Law (F = -kx), F = ma

## The Attempt at a Solution

So far what I managed to do is get two Fnet equations, setting the lefthand direction as the negative direction (this being the direction of the spring force and acceleration). The first Fnet equation is (m + M)(a) = -Fspring and the second 0 = umg - Fspring, or umg = Fspring. After this I start encountering problems. I am supposed to end up with x =(ug(m + M))/k as the equation to find the maximum amplitude, but I end up with x = umg/k for in Hooke's Law I first found acceleration: a = -kx/m, then made m = m + M, and finally subbed all three equations together and solved for x. Can someone point out where I'm going wrong here and show me the correct way to derive the correct formula? Any help would be greatly appreciated.

Well, you said that you made m = m + M.. if you put that into your final equation then it is equivalent to what you said you should be getting.

There are problems with your Fnet equations. For example, the first one is missing the force due to friction from the top block. And Fspring shouldn't be acting on the second block at all. Draw a picture and be very careful in labeling it!

I think you're going about this the wrong way. Start by figuring out what the condition is that makes the blocks slip. When they slip, that means one block is accelrating with respect to the other. So Fnet_topblock DOES NOT EQUAL Fnet_bottomblock.

Also, remember that Ffric doesn't have to equal u*Fnormal. It be less than that, just not more than that.

This is a great problem! Thanks for bringing it to my attention.

Ok, so there is then going to be only one Fnet equation?

Sort of. There are two Fnet equations (because there are two moving bodies) but the idea is to combine them into a single equation, then figure out what you have to do to break the equality.

## 1. What is Hooke's Law?

Hooke's Law is a principle in physics that states the force applied to an elastic object is directly proportional to the amount of stretch or compression of the object, as long as the elastic limit is not exceeded.

## 2. What is the derivation problem in Hooke's Law involving two blocks?

The derivation problem in Hooke's Law involving two blocks refers to the process of finding the mathematical relationship between the force applied to two connected blocks and the resulting displacement of the blocks.

## 3. How do you solve the Hooke's Law derivation problem involving two blocks?

To solve the Hooke's Law derivation problem involving two blocks, you need to use the principles of Newton's Second Law and Hooke's Law to create an equation that relates the force applied to the blocks and the displacement of the blocks. This can be done by considering the forces acting on each block and using the fact that the elastic force is equal for both blocks.

## 4. What is the significance of solving the Hooke's Law derivation problem involving two blocks?

Solving the Hooke's Law derivation problem involving two blocks allows us to understand and predict the behavior of elastic objects under different forces. It also helps us to determine the spring constant, which is a measure of the stiffness of the object.

## 5. Are there any assumptions made in the Hooke's Law derivation problem involving two blocks?

Yes, there are some assumptions made in the Hooke's Law derivation problem involving two blocks. These include assuming that the blocks are in a state of equilibrium, that the blocks are connected by an ideal spring with no mass, and that there is no external forces acting on the system apart from the applied force.

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