Problem 30 - College Physics

Question

\(\bullet\) An 80 \(\mathrm{N}\) box initially at rest is pulled by a horizontal rope on a horizontal table. The coefficients of kinetic and static fric- tion between the box and the table are \(\frac{1}{4}\) and \(\frac{1}{2},\) respectively. What is the friction force on this box if the pull is (a) 0 N, (b) \(25 \mathrm{N},(\mathrm{c}) 39 \mathrm{N},(\mathrm{d}) 41 \mathrm{N},(\mathrm{e}) 150 \mathrm{N} ?\)

Step-by-Step Solution

Answer

The friction forces are 0 N, 25 N, 39 N, 20 N, and 20 N for pulls of 0 N, 25 N, 39 N, 41 N, and 150 N, respectively.

Step 1: Identify the Forces

The box has a weight of 80 N, which means the normal force \( F_n \) will also be 80 N as the box is on a horizontal table. The pulling force \( F_p \) varies as specified.

Step 2: Calculate Maximum Static Friction

The maximum static friction \( F_{s_{max}} \) can be calculated using the coefficient of static friction \( \mu_s = \frac{1}{2} \). Thus, \( F_{s_{max}} = \mu_s \cdot F_n = \frac{1}{2} \times 80 = 40 \; \text{N}. \) If the pulling force is less than or equal to 40 N, the box will not move.

Step 3: Calculate Kinetic Friction

Once the box starts moving, we use the coefficient of kinetic friction \( \mu_k = \frac{1}{4} \). Therefore, the kinetic friction force \( F_k \) is \( F_k = \mu_k \cdot F_n = \frac{1}{4} \times 80 = 20 \; \text{N}.\)

Step 4: Determine the Friction Force for each Pulling Force

Evaluate the friction force for each pulling force: - (a) 0 N: Static friction is \( 0 \; \text{N} \) as no pull is applied.- (b) 25 N: Pull \( < \) Static friction max, \( F = 25 \; \text{N} \). - (c) 39 N: Pull \( < \) Static friction max, \( F = 39 \; \text{N} \). - (d) 41 N: Pull \( > \) Static friction max, box moves, \( F = 20 \; \text{N} \). - (e) 150 N: Box is moving, \( F = 20 \; \text{N} \).

Key Concepts

Static Friction

Static friction is the force that keeps an object at rest when an external force is applied until that force becomes large enough to cause movement. When you try to move a heavy box across the floor, the static friction resists your effort until you apply sufficient force. This force can be calculated using the formula for maximum static friction: \[ F_{s_{max}} = \mu_s \cdot F_n \]where \( \mu_s \) is the coefficient of static friction and \( F_n \) is the normal force, which is usually the weight of the object.

• In our exercise, the static friction helps the box resist movement up to 40 N (as calculated from the coefficient of 0.5 and a normal force of 80 N).
• So, if the pulling force is equal to or less than 40 N, the static friction matches this force and keeps the box stationary.

Static friction is crucial in everyday activities, from walking without slipping to driving and braking in vehicles. It provides stability until motion is desired, turning into kinetic friction once the object starts moving.

Kinetic Friction

After overcoming static friction, the box transitions into motion, and kinetic friction takes over. Kinetic friction is what you feel when an object slides over a surface, and it opposes the direction of the movement.The equation for kinetic friction is similar to that of static friction:\[ F_k = \mu_k \cdot F_n \]Here, \( \mu_k \) is the coefficient of kinetic friction, generally lower than static friction because it takes less force to keep an object moving than to start it moving.

• In this exercise, once the pulling force exceeds 40 N, the box begins to slide, and kinetic friction becomes constant at 20 N, calculated using a coefficient of 0.25 and a normal force of 80 N.
• Even with larger pulling forces, kinetic friction remains the same as long as the surface and speed do not change significantly.

Kinetic friction helps explain why continual force is required to keep an object in motion and how surfaces affect sliding objects differently.

Forces in Physics

Forces are fundamental interactions that bring about changes in motion and shape of objects. In physics, they are represented as vectors, having both magnitude and direction. Forces can be counted by summing up individual forces together, known as net force. Understanding these concepts is essential for solving problems involving motion and friction. Some key aspects of forces in this context include:

• Normal Force: A support force exerted by a surface perpendicular to the object resting on it. In this exercise, the box's weight ( 80 N ) equals the normal force.
• Applied Force: Any force that attempts to change the object's state, like the pulling force on the box.
• Friction Force: The resistance force that opposes motion between surfaces in contact, which can be static or kinetic as seen here.

When dealing with friction and forces, it's crucial to understand how these vectors interact to determine if an object remains at rest or starts to move. Using this knowledge helps to solve various real-world problems, like determining the best way to move objects or understanding the forces at play when driving a car.