# Connected bodies - Spring and 2 blocks

• zorro
In summary, a block of mass m is connected to another block of mass M by a massless spring of spring constant k on a smooth horizontal plane. Initially at rest, a constant force F starts acting on M causing it to move and the spring to extend. The maximum extension of the spring can be determined by working from the reference frame of the center of mass of the system, or by using the equation F1x1 + F2x2 = 0.5k(x2-x1)^2. However, the latter method may not yield accurate results as it ignores the intermediate stages of the movement.

## Homework Statement

A block of mass m is connected to another block of mass M by a massless spring of spring constant k. The blocks are kept on a smooth horizontal plane. Initially, the blocks are at rest and the spring is unstretched when a constant force F starts acting on the block of mass M to pull it. Find the max. extension of the spring.

## The Attempt at a Solution

I can find the answer working from the reference frame of COM of the system of 2 blocks and spring.

I am interested in finding it from ground frame.

Net force on M, F1 = F-kx, where x is the extension in the spring
Net force on m, F2 = kx

Let the displacements of the two blocks be x1 and x2.

F1x1 + F2x2 =0.5k(x2-x1)2 (x2-x1 is the extension produced)

I am unable to find another equation.

F1x1 + F2x2 =0.5k(x2-x1)2 (x2-x1 is the extension produced)

You should not include F2x2. Only include external forces!
But anyway this even won't give you result

I don't know if you can just subtract the forces or not.
I mean as soon as force F will act, body will start to move ... with even little expansion in spring, m2 will move with force kx'
this force will inc. with time till kxmax
I am not sure if you can just use the initial and final conditions and ignore the intermediate stages!
well I'm still trying and tell you my progress!

Did you come up with something?

## 1. What is a connected body system?

A connected body system refers to a group of objects that are connected to each other and move together as a whole. The motion of one object affects the motion of the other objects in the system.

## 2. What is a spring in a connected body system?

A spring is a type of elastic object that can stretch and compress, storing and releasing potential energy. In a connected body system, a spring is often used to connect two or more objects and provide a restoring force that brings them back to their original position.

## 3. How does a spring affect the motion of connected bodies?

A spring can affect the motion of connected bodies by providing a force that opposes the motion and causes the bodies to oscillate back and forth. The stiffness of the spring, as well as the mass and velocity of the bodies, will determine the frequency and amplitude of the oscillations.

## 4. What is the difference between a simple and compound connected body system?

A simple connected body system involves only one spring connecting two objects, while a compound connected body system has multiple springs connecting more than two objects. In a compound system, the motion of one object can affect the motion of multiple objects through the different springs.

## 5. How are forces and accelerations related in a connected body system?

In a connected body system, the forces acting on each object are related to its mass and acceleration through Newton's second law of motion. The sum of all the forces acting on an object is equal to its mass multiplied by its acceleration, which can help determine the motion of the connected bodies.