Problem 77 - Physics for Scientists and Engineers with Modern Physics

Question

A \(2.0-\mathrm{kg}\) silverware drawer does not slide readily. The owner gradually pulls with more and more force, and when the applied force reaches \(9.0 \mathrm{~N}\), the drawer suddenly opens, throwing all the utensils to the floor. What is the coefficient of static friction between the drawer and the cabinet?

Step-by-Step Solution

Answer

The coefficient of static friction is approximately 0.459.

Step 1: Understand the Problem

We need to find the coefficient of static friction between a silverware drawer and the cabinet. The drawer starts moving when a force of \(9.0 \mathrm{~N}\) is applied, and the mass of the drawer is \(2.0 \mathrm{~kg}\).

Step 2: Identify the Relevant Formula

The force of static friction can be calculated using the formula: \(f_s = \mu_s \cdot N\), where \(f_s\) is the static frictional force, \(\mu_s\) is the coefficient of static friction, and \(N\) is the normal force. The normal force for the drawer is the same as its weight due to gravity.

Step 3: Calculate the Normal Force

The normal force \(N\) is equal to the gravitational force on the drawer, which is calculated by \(N = m \cdot g\), where \(m = 2.0 \mathrm{~kg}\) and \(g = 9.8 \mathrm{~m/s^2}\). Hence, \(N = 2.0 \mathrm{~kg} \times 9.8 \mathrm{~m/s^2} = 19.6 \mathrm{~N}\).

Step 4: Solve for the Coefficient of Static Friction

Since the drawer begins to move at a force of \(9.0 \mathrm{~N}\), this force is equal to the maximum static frictional force (\(f_s\)). We can now solve for \(\mu_s\) using the equation \(9.0 \mathrm{~N} = \mu_s \cdot 19.6 \mathrm{~N}\).

Step 5: Calculation

Rearrange the equation from Step 4 to find \(\mu_s\): \(\mu_s = \frac{9.0 \mathrm{~N}}{19.6 \mathrm{~N}} = 0.459\).

Key Concepts

Static Friction

Static friction is the force that keeps two objects at rest relative to each other from starting to slide. It acts parallel to the surface of contact and opposes the initiation of motion. In the context of our exercise, static friction is what keeps the silverware drawer from sliding at first. This frictional force must be overcome by the applied force in order to start the movement of the drawer.

Static friction is not a constant value but can range from zero up to a maximum value. This maximum static frictional force is described by the formula:

• \( f_s = \mu_s \cdot N \)

Where \( f_s \) is the maximum static frictional force, \( \mu_s \) is the coefficient of static friction, and \( N \) is the normal force.

In the problem, the maximum applied force before the drawer moves is 9.0 N. Hence, this is the maximum static frictional force holding the drawer in place.

Normal Force

The normal force is an essential concept in understanding the interaction between surfaces. It acts perpendicular to the surface that an object is resting on and balances out the force of gravity acting on that object.

To calculate the normal force, we use the equation:

• \( N = m \cdot g \)

Where \( N \) is the normal force, \( m \) is the mass of the object, and \( g \) is the acceleration due to gravity (approximated as 9.8 m/s² on Earth).

In the silverware drawer scenario, the normal force is calculated as \( N = 2.0 \text{ kg} \times 9.8 \text{ m/s}^2 = 19.6 \text{ N} \). This force is acting vertically upwards, counterbalancing the weight of the drawer. The normal force is crucial because it is used to determine the static frictional force.

Force of Gravity

The force of gravity is a fundamental force that pulls objects towards each other. On Earth, it gives objects weight and is the reason why they stay grounded. It acts downwards from the center of mass of the object.

The force of gravity for an object near the surface of the Earth is calculated using the formula:

• \( F_g = m \cdot g \)

Where \( F_g \) is the force of gravity, \( m \) is the mass, and \( g \) is the gravitational acceleration (9.8 m/s²).

For the drawer in our exercise, \( F_g = 2.0 \text{ kg} \times 9.8 \text{ m/s}^2 \), which equals 19.6 N. This weight of the drawer is exactly balanced by the normal force when it is stationary, which is why it doesn’t fall or rise.

Physics Problem Solving

Physics problem solving often involves understanding and connecting various concepts through equations and logical steps. In this specific exercise, the process of finding the coefficient of static friction required:

• Comprehending what is asked: the resistance to initial movement, specifically the coefficient of static friction.
• Identifying related formulas: Specifically, \( f_s = \mu_s \cdot N \).
• Calculating essential forces: First, determine the normal force through the formula \( N = m \cdot g \) to be 19.6 N.
• Applying logic and calculations: Identify the maximum static frictional force equal to the applied 9.0 N and rearrange the formula to solve for the coefficient of static friction.

By breaking down each step, students learn to solve physics problems systematically and thoroughly, avoiding the intimidation of complex-looking formulas. With this systematic approach, finding the solution becomes an achievable task.