Problem 9 - University Physics with Modern 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 N to the box and produces an acceleration of magnitude 2.20 m/s\(^2\), what is the mass of the box?

Step-by-Step Solution

Answer

The mass of the box is approximately 21.82 kg.

Step 1: Identify Given Values

First, extract and identify all the given values from the problem statement. We know the force applied (\(F = 48.0 \text{ N}\)) and the acceleration produced (\(a = 2.20 \, \text{m/s}^2\)).

Step 2: Apply Newton's Second Law

According to Newton's Second Law, the force applied to an object equals the mass of the object times its acceleration, which can be described with the equation: \( F = m \times a \). We will use this to solve for the mass (\(m\)).

Step 3: Rearrange the Equation for Mass

Rearrange the equation \( F = m \times a \) to solve for mass (\(m\)). This gives us: \( m = \frac{F}{a} \).

Step 4: Substitute Values into the Equation

Substitute the known values into the equation: \( m = \frac{48.0 \, \text{N}}{2.20 \, \text{m/s}^2} \).

Step 5: Perform the Calculation

Calculate the mass by dividing the force by the acceleration: \( m = \frac{48.0}{2.20} \approx 21.82 \, \text{kg} \).

Key Concepts

Force

In the context of physics, force is an external influence that can change the state of rest or motion of an object. Essentially, a force can make an object start moving, stop moving, or change its speed and direction (known as acceleration).
When you think about forces, envision pushing a swing, pulling a wagon, or even gravity pulling you to the ground.
All these actions are examples of force in action.
Key points about force include:

• A force has both magnitude and direction, making it a vector quantity.
• In our example, the fisherman applies a horizontal force of 48.0 N to the box, which means the box starts to accelerate across the ice.
• For simplicity, in many physics problems, surfaces are considered frictionless to focus only on the forces at play, much like our frozen pond example.

Understanding force and its characteristics is crucial because it's one of the fundamental principles that govern motion and interaction in our universe.

Acceleration

Acceleration is the rate of change of velocity of an object. It tells us how quickly an object is speeding up or slowing down.
Think of a car at a traffic signal: when the light turns green and the driver steps on the accelerator, the car starts to move forward faster and faster; this change in speed over time is acceleration.
In our problem, the acceleration is given as 2.20 m/s², meaning that the box's velocity changes by 2.20 meters per second every second as a result of the applied force.
Important aspects of acceleration include:

• Like force, acceleration is a vector quantity, it has both magnitude and direction.
• It can occur due to various external forces such as gravity, friction, or applied forces as in our example.
• Newton’s Second Law provides a relationship between force and acceleration: more force means more acceleration, assuming the mass remains constant.

By grasping the concept of acceleration, you'll be better equipped to analyze how objects move and the forces that cause these changes.

Mass Calculation

Calculating the mass of an object is a fundamental task in physics, especially when understanding motion using Newton's Second Law.
The formula used is derived from this law, which states that the force applied to an object is equal to the product of its mass and acceleration. Mathematically, it's written as: \[ F = m \times a \]To find the object's mass, you rearrange the equation, solving for mass \(m\) by dividing the force \(F\) by the acceleration \(a\):\[ m = \frac{F}{a} \]In our example:

• The force is 48.0 N, and the acceleration is 2.20 m/s².
• Substitute these values into the equation to find: \( m = \frac{48.0}{2.20} \approx 21.82 \text{ kg} \).

This calculation will tell you the mass of the box being accelerated by the applied force. Understanding mass calculation enables you to connect measurable physical quantities to real-world scenarios, providing a clear picture of how forces affect motion.