Problem 4 - Physics for Scientists and Engineers with Modern Physics

Question

(I) The coefficient of static friction between hard rubber and normal street pavement is about \(0.90 .\) On how steep a hill (maximum angle) can you leave a car parked?

Step-by-Step Solution

Answer

The maximum angle is approximately 42 degrees.

Step 1: Identify the Forces

To solve this problem, we need to analyze the forces acting on the car on the slope of the hill. The car is stationary, so static friction is preventing it from sliding down. The forces involved are the gravitational force and the static friction force.

Step 2: Apply Static Friction Formula

The static friction force can be expressed as \( f_s \leq \, \mu_s \, N \), where \( \mu_s = 0.90 \) is the coefficient of static friction and \( N \) is the normal force. For a car on an inclined plane, \( f_s = m g \sin(\theta) \) and \( N = m g \cos(\theta) \), where \( m \) is the mass of the car, \( g \) is the acceleration due to gravity, and \( \theta \) is the angle of the hill.

Step 3: Set Static Friction to Maximum and Solve for Angle

Set the static friction force at its maximum, which means \( m g \sin(\theta) = 0.90 \, m g \cos(\theta) \). Simplifying this equation, we cancel out \( m g \) from both sides and get \( \tan(\theta) = 0.90 \).

Step 4: Calculate the Maximum Angle

To find the maximum angle \( \theta \), we calculate the arctangent: \( \theta = \tan^{-1}(0.90) \). Use a calculator to find \( \theta \).

Key Concepts

Coefficient of Friction

The coefficient of friction is a measure that represents how much frictional force exists between two surfaces. In our daily lives, it determines how easily one object will slide over another. Specifically, it includes two types: static and kinetic friction. Here, we're focusing on static friction, which is the force that prevents an object from moving when it's at rest. It's crucial in this exercise because our car is parked and it's the static friction keeping it from sliding down the hill.

• Static Coefficient: This is often larger than the kinetic coefficient because it takes more force to start moving an object than to keep it moving.
• Value: In this problem, the coefficient is given as 0.90, indicating a fairly high resistance to sliding.

The formula to calculate static friction force is given by: \( f_s \leq \, \mu_s \, N \) Where:

• \( f_s \) is the static friction force.
• \( \mu_s \) is the coefficient of static friction.
• \( N \) is the normal force, which we'll explore more in another section!

Understanding this coefficient helps us calculate how steep a hill can be before the car starts to slide.

Inclined Plane

An inclined plane is a flat surface tilted at an angle, different from horizontal. It's an essential concept in physics as it helps to understand forces in action when an object is on a slope. Imagine the hill as an inclined plane in this situation. When a car is on this plane, several forces act on it:

• Gravity: Pulls the car downward.
• Normal Force: Acts perpendicular to the surface.
• Static Friction: Acts to hold the car in place, preventing it from sliding down.

For our exercise, we set the gravitational force component along the incline equal to static friction, given by the formula:\( m g \sin(\theta) = \mu_s \, m g \cos(\theta) \). By solving this, we find the maximum angle (\( \theta \)) of the incline where the car remains parked. The inclined plane's role is vital in decomposing the gravitational force into two components: one parallel and one perpendicular to the plane. This helps in calculating necessary forces for balance and friction.

Normal Force

Normal force is a term for the force that a surface exerts to support the weight of an object resting on it. It acts perpendicular to the surface. In this case, it's a critical player in our problem. For inclined planes, the normal force is not just the object's weight but is lessened by the incline angle. The equation to calculate it is: \( N = m g \cos(\theta) \) Where:

• \( m \) is the mass of the car.
• \( g \) is the acceleration due to gravity, approximately \( 9.8 \, \text{m/s}^2 \).
• \( \theta \) is the incline angle.

Normal force is crucial because it directly affects the maximum static friction (recall our formula \( f_s \leq \, \mu_s \, N \)). When static friction reaches its peak, it equals \( \mu_s \, N \). By understanding normal force, we can better predict and calculate how forces will change in various scenarios, particularly inclined surfaces.