Question

A car of mass \(M\) travels in a straight line at constant speed along a level road with a coefficient of friction between the tires and the road of \(\mu\) and a drag force of \(D\). The magnitude of the net force on the car is a) \(\mu M g\). b) \(\mu M g+D\). c) \(\sqrt{(\mu M g)^{2}+D^{2}}\) d) zero.

Short answer

Answer: a) \(\mu M g\).

Step-by-step solution

Identify the forces acting on the car

The car is subject to two forces: friction (F_f) and drag (D). Friction is given by the formula: F_f = μMg, where μ is the coefficient of friction, M is the mass of the car, and g is the acceleration due to gravity. Drag force is already given by D.

Apply Newton's Second Law

As the car is moving at a constant speed in a straight line, we can say that the sum of the forces acting on it in the horizontal direction is zero. Therefore, we can apply Newton's second law: F_net = 0.

Sum the forces in the horizontal direction

The sum of the forces acting on the car is F_net = Friction force - Drag force = F_f - D.

Use the formula for friction force

Now we can substitute the formula for friction force (F_f = μMg) into the equation for the net force: F_net = μMg - D.

Solve for the net force

Since the net force is zero, we have 0 = μMg - D. Adding D to both sides, we get D = μMg.

Choose the correct answer

Based on our analysis, the magnitude of the net force on the car is equal to the drag force, which in turn is equal to μMg. Therefore, the correct answer is: a) \(\mu M g\).

Key concepts

Friction Force

When an object moves across a surface, the opposing force that resists this motion is called the friction force. This force is dependent on the nature of the contact between the surface and the object. For a car moving on a road, the friction force is what enables the tires to grip the road surface and propel the car forward. In the provided exercise, the friction force is calculated using the formula F_f = \(\mu M g\). Here, \(\mu\) represents the coefficient of friction between the car’s tires and the road, M is the mass of the car, and g is the acceleration due to gravity.

Understanding the role of friction in motion is essential for solving problems related to mechanics. In the context of our exercise, the friction force points opposite to the direction of the drag force exerted by air resistance and is essential to maintain constant speed.

Drag Force

The force exerted by fluid-like air or water that opposes the motion of an object is known as the drag force. This force increases with the object’s speed and depends on various factors, including the shape and size of the object and the characteristics of the fluid. For a car traveling at constant speed, the drag force represents the air resistance working against the motion of the vehicle.

In our exercise scenario, the drag force is denoted by D, and it directly impedes the forward motion of the car. To maintain a constant speed, the car needs to output a force that counters the drag force. That countering force, in this case, is provided by the engine, which is directly related to the friction force between the tires and the road.

Net Force

The net force acting on an object is the vector sum of all the forces acting on that object. According to Newtons's Second Law, the net force is directly related to the object's acceleration, described by the formula F_net = m*a, where m is the mass and a is the acceleration.

In the context of the exercise, the car maintains a constant speed, indicating that its acceleration (a) is zero. Therefore, by Newton's Second Law, the net force must also be zero. This implies that the friction force that pushes the car forward is exactly balanced by the drag force, resulting in no net acceleration of the car.

Constant Speed Motion

Motion at a constant speed means that an object is moving at a steady rate over time, without speeding up or slowing down. When we say a car is moving at a constant speed, this implies that the acceleration of the car is zero.

For an object to continue moving at constant speed, especially in the presence of opposing forces like friction and drag, there must be a balance of forces. In the case of our exercise, the propulsive force due to the engine’s work (related to friction) must equal the retarding drag force for the car to sustain constant speed motion on the level road.

Coefficient of Friction

The coefficient of friction is a dimensionless scalar value that describes the ratio of the force of friction between two bodies and the force pressing them together. Notated commonly as \(\mu\), it depends on the materials of the surfaces in contact. A higher coefficient indicates that the surfaces have greater resistance to sliding relative to each other.

In our textbook exercise, \(\mu\) represents the coefficient of friction between the car tires and the road. Together with the normal force—which in this scenario is the product of the car's mass (M) and gravitational acceleration (g)—it determines the maximum frictional force that can be exerted without slipping. This coefficient is a crucial parameter for evaluating the interaction between tires and road surface and for ensuring the proper balance of forces for constant speed motion.