Problem 76 - Physics for Scientists and Engineers with Modern Physics

Question

A coffee cup on the horizontal dashboard of a car slides forward when the driver decelerates from \(45 \mathrm{~km} / \mathrm{h}\) to rest in \(3.5 \mathrm{~s}\) or less, but not if she decelerates in a longer time. What is the coefficient of static friction between the cup and the dash? Assume the road and the dashboard are level (horizontal).

Step-by-Step Solution

Answer

The coefficient of static friction is approximately 0.364.

Step 1: Understand the problem

We need to find the coefficient of static friction that prevents a coffee cup from sliding forward on a car's dashboard when the driver decelerates. The car's initial speed is given as 45 km/h, and the deceleration time is 3.5 seconds or less.

Step 2: Convert speed to meters per second

Convert the initial speed from km/h to m/s. \[45 \text{ km/h} = \frac{45 \times 1000}{3600} = 12.5 \text{ m/s}\]

Step 3: Calculate deceleration

Use the formula for constant acceleration to calculate deceleration (a): \[v = u + at \]where \( v = 0 \) m/s (final velocity), \( u = 12.5 \) m/s (initial velocity), and \( t = 3.5 \) s.Rearrange for \( a \):\[0 = 12.5 + a \times 3.5 \Rightarrow a = -\frac{12.5}{3.5} \approx -3.57 \text{ m/s}^2\]

Step 4: Relate deceleration to static friction

The maximum static friction force matches the deceleration force if the cup doesn't slide. Using the equation \( f_s = m \cdot a \), where \( f_s = \mu_s \cdot m \cdot g \), set the forces equal:\[\mu_s \cdot m \cdot g = m \cdot |a|\] Simplify to find \( \mu_s \):\[\mu_s = \frac{|a|}{g}\]where \( g \approx 9.8 \text{ m/s}^2 \).

Step 5: Calculate the coefficient of static friction

Substitute the known values:\[\mu_s = \frac{3.57}{9.8} \approx 0.364\]Thus, the coefficient of static friction is approximately 0.364.

Key Concepts

Coefficient of Static Friction

Static friction is a force that keeps an object at rest when it is acted upon by external forces. It prevents surfaces from sliding past each other. Imagine a coffee cup on a car dashboard. When the car slows down, static friction keeps the cup from sliding. However, once this force is overcome, the cup will start to slip. This concept is crucial in determining whether objects remain stable or move.

The coefficient of static friction is a value that tells us how large the static frictional force can be between two surfaces. It is represented by the symbol \( \mu_s \). The coefficient depends on the types of materials in contact and their surface textures.

To calculate it, we use the equation:

• \( f_s = \mu_s \cdot N \)
• \( f_s \) is the static frictional force
• \( N \) is the normal force (often equals gravitational force on flat surfaces)

Each material pair has a unique \( \mu_s \) value, showcasing how sticky or slippery the interface might be. In our problem, finding \( \mu_s \) involves knowing the deceleration of the car and the gravitational force, resulting in the cup's skid prevention mechanism.

Deceleration

Deceleration, a type of acceleration, occurs when an object slows down. This was crucial for solving our coffee cup problem. When the car decreases its speed from 45 km/h to a stop, it experiences deceleration.

Using the equation of motion:

• \( v = u + at \)
• \( v \) is the final velocity (0 m/s here since the car stops)
• \( u \) is the initial velocity (12.5 m/s, converted from km/h)
• \( t \) is the time over which the deceleration occurs

We are given that this occurs in 3.5 seconds or less. Deceleration can be calculated by rearranging the formula to find \( a \) (acceleration):
\[ a = \frac{v - u}{t} = -\frac{12.5}{3.5} \approx -3.57 \text{ m/s}^2 \]
This result indicates how fast the car reduces its speed in terms of meter per second squared. The negative sign confirms that it's a deceleration rather than an acceleration.

Constant Acceleration

When we mention constant acceleration, we're talking about a steady rate of change of velocity. Whether increasing or decreasing (in the case of deceleration), this concept implies the rate doesn't change over time.

In the coffee cup scenario, the car decelerates at a constant rate, which means its velocity decrease is uniform. A useful equation for such constant changes in velocity is:

• \( v = u + at \)

Where:

• \( v \) is the final velocity
• \( u \) is the initial velocity
• \( a \) is the acceleration (or deceleration here)
• \( t \) is the time

By assuming constant acceleration, calculations simplify considerably, which helps in determining the coefficient of static friction since we consider steady force interactions between the cup and the dashboard.

This understanding helps in predicting how different aspects of the movement and forces involved relate to one another in a simple and predictable manner.