Problem 109 - University Physics with Modern Physics

Question

You observe a 1350 -kg sports car rolling along flat pavement in a straight line. The only horizontal forces acting on it are a constant rolling friction and air resistance (proportional to the square of its speed). You take the following data during a time interval of \(25 \mathrm{s} :\) When its speed is \(32 \mathrm{m} / \mathrm{s},\) the car slows down at a rate of \(-0.42 \mathrm{m} / \mathrm{s}^{2},\) and when its speed is decreased to \(24 \mathrm{m} / \mathrm{s},\) it slows down at \(-0.30 \mathrm{m} / \mathrm{s}^{2} .\) (a) Find the coefficient of rolling friction and the air drag constant \(D\) . (b) At what constant speed will this car move down an incline that makes a \(2.2^{\circ}\) angle with the horizontal? (c) How is the constant speed for an incline of angle \(\beta\) related to the terminal speed of this sports car if the car drops off a high cliff? Assume that in both cases the air resistance force is proportional to the square of the speed, and the air drag constant is the same.

Step-by-Step Solution

Answer

Find coefficients with equations; use incline equilibrium for speed; relate speeds through force balance.

Step 1: Establish the Forces in Play

Identify the forces acting on the car: the force of rolling friction, which depends on the weight of the car, and the drag force, which depends on the speed of the car squared. The total decelerating force is given by \(F = ma = - ext{friction} - ext{drag}\). The rolling friction can be expressed as \( f = ext{coefficient of friction} \times ext{normal force} = \mu mg\) while the drag force is \( F_d = Dv^2 \) where \(v\) is the speed and \(D\) is the drag constant.

Step 2: Set Up Equations at Given Speeds

Use the given deceleration rates to set up equations. At 32 m/s, \(ma = -f - F_d\) becomes \(-0.42 \times 1350 = -\mu \times 1350 \times 9.8 - D \times (32)^2\), and at 24 m/s, it becomes \(-0.30 \times 1350 = -\mu \times 1350 \times 9.8 - D \times (24)^2\).

Step 3: Solve the System of Equations for \(\mu\) and \(D\)

Solve the two equations generated in Step 2 to find \(\mu\) and \(D\). This can be accomplished by isolating one variable in one of the equations and substituting back into the other or using matrix algebra for linear systems.

Step 4: Calculate Constant Speed Down Incline

Introduce the component of gravitational force down the incline planes as \(mg\sin(\theta)\), where \(\theta = 2.2^\circ\). Set up the equilibrium equation \(mg\sin(\theta) = \mu mg\cos(\theta) + Dv^2\) at constant speed, solve for \(v\).

Step 5: Relate Constant Speed on Incline to Terminal Speed

At terminal velocity (speed when all forces balance during free fall), the net force is zero, so \(-mg + Dv_t^2 = 0\) yields \(v_t = \sqrt{\frac{mg}{D}}\). Comparing this with the constant incline speed, you'll find they both depend on \(mg\) and \(D\) but differ by the sine and cosine of \(\theta\) in the incline case.

Key Concepts

Understanding Rolling Friction

Rolling friction is a type of force that opposes the motion of a car as it moves along a surface. Unlike sliding friction which can be quite high, rolling friction is relatively small and occurs because of the deformations of the surfaces in contact. For a car, this friction mostly happens between the tires and the road. The force due to rolling friction depends on the normal force acting on the car, which is essentially its weight.

• The formula for rolling friction is given as: \( f = \mu mg \)
• Here, \( \mu \) is the coefficient of rolling friction.
• \( m \) is the mass of the car and \( g \) is the acceleration due to gravity.

When calculating the coefficient of rolling friction in problems, this small force is crucial in understanding how easily a car moves and stops. Rolling friction allows vehicles to maintain traction and aids in more controlled driving.

Effect of Air Resistance

Air resistance, often known as drag force, is an opposing force that resists the motion of an object moving through air. For the sports car in the problem, air resistance works against its forward motion and becomes more significant as the speed increases.

• The drag force is given by the formula: \( F_d = Dv^2 \)
• \( D \) is the drag constant which depends on factors like shape and surface area of the car.
• \( v \) is the velocity or speed of the car.

This quadratic relationship means that as the car goes faster, the effect of air resistance increases more significantly. Understanding air resistance is essential for calculating how fast a car can go before external forces, such as wind, substantially slow it down. It is this resistance that needs to be balanced with other forces to achieve terminal velocity.

Reaching Terminal Velocity

Terminal velocity is achieved when the net force on an object falling or moving through a medium like air becomes zero, resulting in constant speed. For the sports car, when it reaches its terminal speed, all the forces acting upon it, such as gravity and drag force, balance each other out, and the car stops accelerating.

• The formula for terminal velocity is: \( v_t = \sqrt{\frac{mg}{D}} \)
• At terminal velocity, the force due to gravity \( mg \) is balanced by the drag force \( Dv_t^2 \).

Understanding terminal velocity helps in predicting the behavior of moving objects under various resistive forces, such as in cases of free-fall scenarios or rolling down an inclined plane.

Newton's Laws and Motion

Newton's laws of motion are the foundation for understanding movements of objects under various forces. In the context of this exercise, Newton's laws help in setting up the equations needed to find the coefficients of rolling friction and air resistance.

• Newton's First Law implies that a body in motion remains in motion unless acted upon by an external force, such as rolling friction in this case.
• Newton’s Second Law (\( F = ma \)) allows us to calculate forces based on the mass of the car and its acceleration or deceleration.
• Newton’s Third Law states that for every action, there is an equal and opposite reaction, which is essential in understanding interaction forces like friction and air resistance.

These laws provide the framework to analyze and describe the car's motion and response to forces it encounters.

Exploring Inclined Plane Physics

Inclined plane physics deals with the movement of objects on a slope, which is a fundamental study in physics. When the sports car moves down an inclined surface, it experiences a component of gravitational force pulling it downhill.

• The force acting down the plane is \( mg\sin(\theta) \).
• The normal force perpendicular to the incline is \( mg\cos(\theta) \).

This setup results in the equation of motion, balancing gravity, rolling friction, and air resistance to find the speed of the car.The motion of the car on an inclined plane directly relates to Newton's laws and the effects of forces discussed earlier, offering insight on how slopes affect objects' movement and speed. Understanding inclined plane physics is crucial for calculating the equilibrium speed and predicting dynamic scenarios in transportation.