Problem 23 - College Physics

Question

When a horizontal force of \(300 \mathrm{~N}\) is applied to a 75.0 \(\mathrm{kg}\) box, the box slides on a level floor, opposed by a force of kinetic friction of \(120 \mathrm{~N}\). What is the magnitude of the acceleration of the box?

Step-by-Step Solution

Answer

The acceleration of the box is \(2.4 \, \text{m/s}^2\).

Step 1: Identify the Forces

First, identify the forces acting on the box. There is a horizontal force of 300 N being applied, and there is a force of kinetic friction of 120 N opposing the motion.

Step 2: Calculate the Net Force

To find the net force acting on the box, subtract the force of friction from the applied force: \[ F_{net} = F_{applied} - F_{friction} = 300 \, \text{N} - 120 \, \text{N} = 180 \, \text{N} \]

Step 3: Apply Newton's Second Law

Use Newton's second law to find the acceleration. The law states \( F = ma \), where \( F \) is the net force, \( m \) is the mass, and \( a \) is the acceleration. Rearrange to find \( a \): \[ a = \frac{F_{net}}{m} = \frac{180 \, \text{N}}{75.0 \, \text{kg}} \]

Step 4: Calculate the Acceleration

Perform the division to find the acceleration:\[ a = \frac{180}{75} = 2.4 \, \text{m/s}^2 \]

Key Concepts

Understanding Kinetic Friction

Kinetic friction is a force that acts between moving surfaces. When two objects slide over each other, kinetic friction opposes the direction of motion. This force is caused by the microscopic interactions between the surfaces. In our exercise, the kinetic friction force is given as 120 N. This means that as the box moves, this force tries to slow it down, acting in the opposite direction of the applied force.
It is important to remember that kinetic friction is generally less than static friction, which is the force needed to start the motion. Once the object is moving, overcoming kinetic friction becomes the primary concern.

• Kinetic friction depends on the materials in contact and the normal force (perpendicular force between the surfaces).
• The coefficient of kinetic friction often denoted by \( \mu_k \), quantifies this force when combined with the normal force.

Understanding these dynamics helps in predicting how much force is needed to keep an object in motion. In many physics problems like ours, you are given the frictional force directly, making calculations easier.

Exploring Net Force

Net force is the total force acting on an object after all opposing forces are considered. It is simply the difference between all forces in opposite directions. In our example, net force is the result of subtracting the kinetic friction from the applied force.
Net force determines how much an object will be accelerated. According to Newton's Second Law, motion is only affected by the presence of a net force.

• If the net force is zero, the object remains at a constant velocity or at rest, illustrating balanced forces.
• If the net force is non-zero, it means the forces are unbalanced, causing acceleration.

With our net force equation: \( F_{net} = F_{applied} - F_{friction} = 300 \, \text{N} - 120 \, \text{N} = 180 \, \text{N} \,\) you can see we've accounted for both the driving force and the opposing frictional force, resulting in a positive net force which dictates the motion of the box.

Acceleration Calculation Through Newton's Second Law

Calculating acceleration involves Newton's Second Law of Motion, which states that the force exerted on an object equals the mass of the object times its acceleration \( F = ma \). To find acceleration, simply rearrange the formula to \( a = \frac{F_{net}}{m} \). In our given problem:
The net force is 180 N and the mass of the box is 75.0 kg.
Plug these into the formula:
\[ a = \frac{180 \, \text{N}}{75.0 \, \text{kg}} \]
Perform the division to calculate acceleration:
\[ a = 2.4 \, \text{m/s}^2 \]
Through this calculation, you can see how effectively the basic principles of physics help predict motion. This practical use of Newton's laws allows for the intricacies of forces and motion to be understood and predicted, providing a logical approach to real-world problems.