# University of Wisconsin Green Bay

For no apparent reason, you decide to pull a heavy box of books across the floor using a spring. When you get the box moving at a constant velocity, the spring is stretched 0.10 m past its relaxed length. If the box has a mass of 35 kg and if the coefficient of friction between the box and the floor is 0.43, what is the spring constant of the spring? Assume that you pull the spring parallel to the floor.

Hint: The free body diagram and approach to solving the problem are exactly the same as if you push the box directly, or pull it with a rope. Don’t let the spring deter you from using this example with a simpler problem.

• In this problem, you are provided information about the motion of an object (the box moves at constant velocity) and about the forces causing that motion (the pull by the spring and friction.) Force and motion of a single object are always related through Newton’s Second Law, so this is a force or 2nd Law problem.

• The key equation for any problem that relates forces and motion is Newton’s Second Law. Regardless of what quantity you are asked to find, begin with the Second Law. If additional information is needed, it will become apparent as you proceed.

In this case, you are not asked for either a force or acceleration, but rather for a quantity that is related to a force. That does not affect how you solve the problem. The key interaction is force, and details of what you are asked to find don’t show up until the final algebraic steps.

• It doesn’t matter whether you start with the x- or the y- equation. If you begin with the x- equation, you will find that you need to solve the y- equation for normal force before you can complete the solution, so you are steered to the information you need regardless of where you begin.

Spring constant, or k, is the quantity that was requested in the problem and so no further mathematical steps are needed. Note that it isn’t until you fill in definitions for the forces (gravity = mg, friction = µn, spring force = kx) that you need to think at all about which information is given and what you are asked to find.

• In this problem, the box travels at a constant velocity in the x-direction. This means that the horizontal forces pulling to the right (the spring force) exactly balance the horizontal forces pulling to the left (friction.) Remember Newton’s First Law: when the net force on an object is zero, the velocity of that object doesn’t change. This seems counter-intuitive. In your experience, you always need to provide a force to keep an object moving—because you need to balance the force of friction that acts against it.

In the y-direction, the velocity of the box is also constant (the box remains at rest.) Therefore, the upward normal force from the floor exactly balances the downward pull of gravity.

The algebraic process that you use in this problem to arrive at a numerical solution is one that you will use over and over again: solve Newton’s Second Law in the y-direction to find normal force, and then use the normal force to calculate frictional force which acts in the x-direction.