Problem 6 - College Physics

Question

A box rests on a frozen pond, which serves as a frictionless horizontal surface. If a fisherman applies a horizontal force with magnitude 48.0 \(\mathrm{N}\) to the box and produces an acceleration of magnitude \(3.00 \mathrm{m} / \mathrm{s}^{2},\) what is the mass of the box?

Step-by-Step Solution

Answer

The mass of the box is 16.0 kg.

Step 1: Identify the Known Variables

We are given the force \( F = 48.0 \, \mathrm{N} \) and the acceleration \( a = 3.00 \, \mathrm{m/s^2} \).

Step 2: Recall the Formula for Newton's Second Law of Motion

Newton's Second Law states that \( F = m \cdot a \) where \( F \) is the force applied, \( m \) is the mass of the object, and \( a \) is the acceleration.

Step 3: Rearrange the Formula to Solve for Mass

We need to find the mass, \( m \). Rearrange the formula: \( m = \frac{F}{a} \).

Step 4: Substitute the Known Values into the Equation

Plug in the values for force and acceleration: \( m = \frac{48.0 \, \mathrm{N}}{3.00 \, \mathrm{m/s^2}} \).

Step 5: Calculate the Mass

Divide the force by the acceleration to get the mass: \( m = \frac{48.0}{3.00} = 16.0 \, \mathrm{kg} \).

Key Concepts

Force

At the core of Newton's Second Law, force is a key concept. It is defined as the interaction that causes an object to change its velocity, effectively pushing or pulling it. Force is a vector quantity, which means it has both magnitude and direction. In the given problem, the force has a magnitude of 48.0 N and is applied horizontally.

• The unit of force in the International System of Units (SI) is the Newton (N).
• A force can cause an object to accelerate, decelerate, remain in motion, or change direction.

Understanding force is crucial because it explains how objects interact with one another. In many physics problems, calculating the force helps us understand the subsequent motion of objects. This problem simplifies things by assuming no friction, highlighting the direct relationship between force and motion through acceleration.

Acceleration

Acceleration measures how quickly an object's velocity changes over time. In simpler terms, acceleration tells us how fast an object is speeding up or slowing down. In our example, the acceleration is given as 3.00 m/s². This means that every second, the box's velocity on the frozen pond increases by 3.00 meters per second.

• Acceleration is a vector quantity, the same as force, meaning it has both a direction and magnitude.
• In Newton's Second Law, acceleration is directly proportional to the force exerted on the object.
• The unit of acceleration in SI is meters per second squared (m/s²).

In physics, understanding acceleration allows us to predict how quickly or slowly an object moves when a certain force is applied. This concept is crucial in problems related to motion, as it links force and mass together.

Mass Calculation

To find the mass of an object in scenarios governed by Newton's Second Law, we use the formula \( m = \frac{F}{a} \). This rearrangement of the law \( F = m \cdot a \) allows us to solve for mass when force and acceleration are known, as in our example.

• Mass is a measure of the amount of matter in an object.
• In the International System of Units, mass is measured in kilograms (kg).
• Mass is not affected by forces like gravity, making it distinct from weight.

In our problem, calculating mass involved dividing the given force (48.0 N) by the given acceleration (3.00 m/s²), resulting in a mass of 16.0 kg for the box. Mass calculation is foundational in physics because it helps us understand an object's resistance to acceleration when a force is applied.