Question

A large box containing your new computer sits on the bed of your pickup truck. You are stopped at a red light. When the light turns green, you stomp on the gas and the truck accelerates. To your horror, the box starts to slide toward the back of the truck. Draw clearly labeled free-body diagrams for the truck and for the box. Indicate pairs of forces, if any, that are third-law action- reaction pairs. (The horizontal truck bed is \(not\) frictionless.)

Short answer

Draw forces on free-body diagrams and identify action-reaction force pairs using Newton's third law.

Step-by-step solution

Understand the Forces Acting on the Box

First, identify the forces acting on the box. These include the gravitational force acting downward, the normal force acting upward from the truck bed, and the force of friction opposing the motion of the box relative to the truck. Label these forces on a free-body diagram of the box.

Identify the Interaction with the Truck

Recognize that when the truck accelerates, the box tends to stay in place (relative to its state of rest) due to inertia. This implies the truck applies a forward frictional force on the box, whereas the box applies an equal and opposite frictional force backward on the truck according to Newton's third law.

Draw the Free-Body Diagram for the Box

Draw the box on the truck bed and include arrows representing the gravitational force acting downwards, the normal force acting upwards, and the static friction force acting forwards (relative to the box's motion). Label these forces appropriately.

Determine the Forces Acting on the Truck

Identify that the truck is subjected to forces including the normal force from the ground, gravitational force, applied force from the engine (driving the truck forward), and the frictional force exerted backwards by the box.

Draw the Free-Body Diagram for the Truck

Draw the truck with arrows for the forces mentioned: gravitational force acting downward, normal force from the ground upward, friction from the box backward, and engine force forward. Label these forces correspondingly.

Indicate Action-Reaction Pairs

Link the action-reaction pairs: the forward frictional force by the truck on the box and the backward frictional force by the box on the truck. Label these as third-law force pairs.

Key concepts

Free-Body Diagrams

Free-body diagrams are visual representations used to illustrate the forces acting on an object. In our scenario with the box and the pickup truck, we have to draw two diagrams: one for the box and another for the truck. These diagrams help to analyze the different forces and predict the motion of the objects involved.
To start with the box, we have:

• A gravitational force pulling it downwards, often represented as an arrow pointing down and labeled "weight" or "gravity."
• A normal force from the truck bed pushing upwards against the box, balancing the gravitational force. This is labeled as "normal force."
• A static frictional force that acts forward, opposing the box's tendency to slide backward when the truck accelerates. This force is labeled as "friction" and acts parallel to the truck bed.

For the truck, the free-body diagram should include:

• The gravitational force acting down on it and a corresponding normal force from the ground acting up.
• The engine force driving the truck forward is labeled as "engine force."
• The backward frictional force exerted by the box, depicted as an arrow pointing backward, this opposes the engine force.

Labeling each force on these diagrams helps to clearly understand the physical interactions between the box and the truck.

Frictional Force

The frictional force plays a crucial role in the interaction between the box and the truck's surface. It is a force that resists the relative motion of two surfaces in contact. In this exercise, friction prevents the box from immediately sliding off the truck.
There are two main types of friction to consider: static and kinetic friction.

• Static friction: When the truck starts accelerating, the box resists motion due to static friction. This force operates until it reaches a maximum threshold defined by the materials in contact and the normal force.
• Kinetic friction: If the box begins to slide, kinetic friction takes over. This is typically less than static friction and acts to slow down the sliding box.

The frictional force here is vital because it not only acts to keep the box steady initially but also determines how much force is needed from the truck's engine to overcome it. Within the context of Newton's Third Law, the truck exerts a forward frictional force on the box as it accelerates, preventing it from sliding off.

Inertia

Inertia is a property of matter that describes its resistance to any change in its state of motion. In simple terms, an object at rest will stay at rest, and an object in motion will stay in motion unless acted upon by a force. This is highlighted by Newton's First Law, often called the law of inertia.
In our given problem, when the truck accelerates from a stop, the box inside wants to remain at rest due to inertia. This natural tendency is why the box initially stays put relative to the ground as the truck moves forward, leading to the illusion that it is sliding backwards when it is actually the truck moving forward. This opposing motion relative to the truck causes friction between the box and the truck bed to act, trying to move the box along with the truck.
The concept of inertia is central to understanding why the box moves or doesn't move and why friction is necessary to counteract its effects. Without friction, the box would not accelerate with the truck but would instead stay still, eventually falling off as the truck continues to gain speed.

Action-Reaction Pairs

Newton's Third Law of motion, often stated as "For every action, there is an equal and opposite reaction," introduces the concept of action-reaction pairs. These pairs occur whenever two objects interact. Each object exerts a force on the other, and these forces are equal in magnitude but opposite in direction.
In the context of the truck and the box:

• The truck exerts a forward frictional force on the box. This is the action force aiming to accelerate the box along with the truck.
• The box, in turn, exerts an equal and opposite frictional force back on the truck, serving as the reaction force.

This example vividly illustrates action-reaction pairs. Understanding this concept helps us predict how forces affect motion. For instance, if the frictional force is too small due to a slippery surface, the action-reaction relationship might not effectively keep the box from sliding backward. Recognizing these pairs is essential for problem-solving in physics, especially in scenarios involving friction and inertia.