## Illustration 4.5: Pull Your Wagons

Two toy wagons, attached by a lightweight rope (of negligible mass), are pulled with a constant force using another lightweight rope (again of negligible mass) as shown in the animation (position is given in centimeters and time is given in seconds). Restart. The mass of the red wagon is 2.0 kg and the mass of the blue wagon is 1.2 kg. What is the force of the hand on the rope, and what is the tension in the rope joining the two wagons? To answer these questions, you must apply Newton's second law. However, when applying Newton's second law, you must first define the system that you are considering. Let's answer each question separately.

What is the force of the hand on the rope? Begin by defining the system to which we will apply Newton's second law. Since we want to determine the force of the hand on the rope, we start by choosing the rope to be our system. What forces act on the rope? It may help to draw a free-body diagram. View a free-body diagram of the rope along with the animation.

Note that there are two forces on the rope, the force of the hand on the rope in the +x direction and the force of the red wagon on the rope in the -x direction. These two forces are equal in magnitude; therefore, the net force on the rope is zero. But how can it be zero if the rope's acceleration is NOT zero? Since the mass of the rope is negligible, we set the mass of the rope equal to zero, and because of Newton's second law, the net force on the rope is zero. Of course, the rope's mass, in reality, is not zero; however, it's close enough to zero that we can say that it's approximately zero. Ultimately, this means that the tension in the rope is constant.

What is the force of the red wagon on the rope? What system should we now consider? We have two choices: (1) Consider the red wagon as the system or (2) consider the blue wagon, the red wagon, and the rope between them as the system. Either choice can lead you to the answer, but choice (2) is the most direct and best choice in order to solve the problem most quickly.

Consider the two wagons and rope as the system as depicted in this animation. The gray box represents the system. Now, draw a free-body diagram for the system and then view the animation again in order to check your answer. Once you have drawn the free-body diagram and identified the forces, you can apply Newton's second law, determine the force of the red wagon on the rope, and solve for the force of the hand on the rope.

That answers the first question. Now for the second question: What is the tension in the rope joining the two wagons? You can answer this question by following a similar procedure. Identify your system, draw a free-body diagram, and apply Newton's second law.

Illustration authored by and placed into the public domain by Aaron Titus.

Physlets were developed at Davidson College and converted from Java to JavaScript using the SwingJS system developed at St. Olaf College.