Problem 44 - College Physics

Question

A truck is pulling a car on a horizontal highway using a horizontal rope. The car is in neutral gear, so we can assume that there is no appreciable friction between its tires and the highway. As the truck is accelerating to highway speeds, draw a free-body diagram of (a) the car and (b) the truck. (c) What force accelerates this system forward? Explain how this force originates.

Step-by-Step Solution

Answer

(a) Draw forces: tension, gravity, normal on the car. (b) On the truck, draw forces: friction, tension, gravity, normal. (c) The driving force is the static friction acting on the truck's tires.

Step 1: Drawing Free-Body Diagram of the Car

To draw the free-body diagram of the car, identify the forces acting on it. Since there is negligible friction, the focus is on the tension in the rope pulling the car forward. The forces include: - Tension ( T ) in the rope acting horizontally forward. - Gravitational force ( mg ) acting downward, where m is the mass of the car and g is the acceleration due to gravity. - Normal force ( N ) acting upward, balancing the gravitational force. Thus, the diagram should show: 1. An arrow pointing right labeled T (tension in the rope). 2. An arrow pointing downward labeled mg (gravitational force). 3. An arrow pointing upward labeled N (normal force).

Step 2: Drawing Free-Body Diagram of the Truck

To draw the free-body diagram of the truck, identify the forces acting on it. Apart from gravitational and normal forces, consider the force exerted by the road on the truck to move it forward (friction) and the tension force exerted by the car: - Frictional force ( F_f ) between the truck's wheels and road acts forward. - Tension ( T ) in the rope acts backward, pulling the truck. - Gravitational force ( Mg ) acts downward, where M is the mass of the truck. - Normal force ( N_t ) acts upward, balancing the gravitational force. The diagram includes: 1. An arrow pointing right labeled F_f (frictional force). 2. An arrow pointing left labeled T (tension in the rope). 3. An arrow pointing downward labeled Mg (gravitational force). 4. An arrow pointing upward labeled N_t (normal force).

Step 3: Identifying the Force that Accelerates the System

The force that accelerates the truck-car system forward is the frictional force exerted by the highway on the truck's tires. This static frictional force arises because the truck's engine propels the wheels to push backward against the road, and by Newton's third law, the road pushes forward on the wheels with an equal force. This forward force is responsible for accelerating the truck, and through the rope tension, it also accelerates the car.

Key Concepts

Tension

In the context of a truck pulling a car via a rope, tension refers to the pulling force conducted along the rope. The tension force is always directed along the length of the rope and is the same at every point in an ideal (massless and inextensible) rope.

In this scenario, the tension is responsible for pulling the car forward. It originates from the truck's engine, which propels the vehicle forward, causing the rope to be pulled taut. This in turn pulls the car, creating a tension force.

• Tension acts in opposite directions on the truck and the car along the rope.
• For the car, tension acts forward, helping it move.
• For the truck, tension acts backward as it pulls the car, providing resistance.

Visualizing this on a free-body diagram of the car, you would see an arrow labeled T pointing to the right, representing tension's force direction.

Frictional Force

Despite its seemingly undermined presence in exercises focused on horizontal motion, the frictional force plays a crucial role, particularly between the truck's tires and the highway road. The friction here is actually a static friction that allows the truck to achieve motion.

Static friction is stronger than kinetic friction and prevents the tires from slipping. It arises because when the truck's wheels push backward against the road surface, the friction force from the road pushes the wheels forward, propelling the truck. This forward force is what moves the truck and, through the tension in the rope, the car as well.

• Frictional force can be seen on the truck's free-body diagram as an arrow pointing forward (in the direction of motion).
• This force is essential in translating the truck's engine output into forward movement.
• Without sufficient friction, the wheels would spin without generating forward movement.

Normal Force

Normal force is the perpendicular force exerted by a surface to support the weight of an object resting on it. For both the car and the truck in this highway scenario, the normal force acts vertically upward, balancing the downward gravitational force.

The normal force ensures that the vehicles do not sink into or rise off the highway. This force is critical in maintaining balance and facilitating the friction needed for movement.

• The normal force on the car is depicted as an arrow pointing upward, countering gravity.
• For the truck, there is a similar representation as an upward arrow that balances the truck's gravitational pull.

The normal force varies depending on the mass of the vehicle, directly counteracting the gravitational force (i.e., if mass increases, so does the normal force).

Gravitational Force

Gravitational force is the pull exerted by the Earth on any object with mass, directed toward the center of the Earth. For the truck and the car, this force is depicted in the diagrams as downward arrows.

• For the car: It is shown as mg, where m is the car's mass and g is the acceleration due to gravity.
• For the truck: It appears as Mg, with M being the truck's mass.

Gravitational force influences both vehicles by providing the necessary force that the normal force counters. While it does not directly contribute to horizontal motion, it's indispensable in ensuring stability and enabling the friction necessary for the truck's movement. In essence, gravity keeps both vehicles grounded on the highway.