**Goal:** Reasoning with 2nd law.

**Source:** UMPERG

Consider the three situations shown below. In each case two small carts

are connected by a spring. A constant force F is applied to the

leftmost cart in each case. In each situation the springs are

compressed so that the distance between the two carts never changes.

Which of the following statements must be true regarding the compression

of the spring in each case? Assume the springs are identical.

- Compression A = Compression B = Compression C
- B = C < A
- A < B = C
- A < B < C
- B < A < C
- C < A < B
- A < C < B
- None of the above
- Cannot be determined

## Commentary:

## Answer

(5) The total mass is the same so the acceleration of the systems must

be the same. In each case the spring exerts the only horizontal force

on the cart to the right. The spring force must be largest for the 3M

cart and smallest for the M cart: B < A < C.

## Background

This item requires students to reason. It is difficult to resort to

equation manipulation to answer this question. One difficulty with the

problem is that it involves a complex system (two carts connected by a

spring).

## Questions to Reveal Student Reasoning

Is it really possible to compress the carts so that they stay a fixed

distance apart? What forces act on each cart? Will the carts

accelerate or move with a constant velocity? Compare the carts

acceleration.

## Suggestions

Draw a free-body diagram for each cart.

Define a new problem in terms of the carts on the right: Each cart is

given an applied force so that each has the same acceleration. How do

the applied forces compare?