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 \(\mathrm{N}\) 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 Known Values

The problem provides the horizontal force applied to the box and the resulting acceleration produced. We know:- Force \( F = 48.0 \text{ N} \)- Acceleration \( a = 3.00 \text{ m/s}^2 \)

Step 2: Recall the Formula for Force

The force applied on an object is calculated using Newton's second law of motion, which is:\( F = m \, a \)where \( F \) is the force, \( m \) is the mass, and \( a \) is the acceleration.

Step 3: Rearrange the Formula for Mass

We need to find the mass \( m \), so we rearrange the formula:\( m = \frac{F}{a} \)

Step 4: Substitute Known Values into the Formula

Using the known values, substitute them into the rearranged formula:\( m = \frac{48.0 \text{ N}}{3.00 \text{ m/s}^2} \)

Step 5: Calculate the Mass

Perform the division to find the mass:\( m = 16.0 \text{ kg} \)

Step 6: Verify Units

The units for mass are in kilograms (kg), which is appropriate as Force (N) divided by Acceleration (m/s²) results in kg.

Key Concepts

Force and Acceleration

In the context of Newton's second law of motion, force and acceleration have a direct relationship. Newton's second law states that the force exerted on an object is proportional to the acceleration of that object multiplied by its mass, which is mathematically represented as:\[ F = m \cdot a \]- **Force (F)**: Measured in newtons (N), it is the push or pull applied to an object. In our exercise, a force of 48.0 N is applied.- **Acceleration (a)**: This is the rate at which an object's velocity changes. It is measured in meters per second squared (m/s²). Here, the acceleration is 3.00 m/s².The relationship indicates that for a constant mass, any increase in force will result in an increase in acceleration. Conversely, for a given force, increasing the mass will decrease the acceleration. This intuitive interaction is foundational for understanding how objects move in response to applied forces.

Frictionless Surface

A frictionless surface is an idealized concept, often used in physics problems to simplify calculations. In reality, most surfaces will have some degree of friction that opposes motion. However, describing the surface as frictionless allows us to ignore frictional forces that would otherwise complicate the calculation. - **No Opposing Forces**: Since the surface is frictionless, there are no forces opposing the movement of the box, aside from the force applied. - **Ideal Conditions**: This means that all the force applied to the box is used solely for acceleration, making it easier to apply Newton's second law without adjustments for friction. By removing friction from the equation, the problem becomes a straightforward application of the force to achieve the acceleration. This assumption helps break down the larger concepts in a physics problem into simpler, more manageable components.

Mass Calculation

Calculating mass in problems like this one involves rearranging Newton's second law. Here, we are given both the force and acceleration, and we need to find the mass of the object.The formula by Newton's second law is \( F = m \cdot a \). By rearranging this formula, we solve for mass \( m \):\[ m = \frac{F}{a} \]- **Substitution of Values**: Substitute the given values: force (48.0 N) and acceleration (3.00 m/s²).Using these values, calculate the mass of the box:\[ m = \frac{48.0 \text{ N}}{3.00 \text{ m/s}^2} = 16.0 \text{ kg} \]- **Verification of Units**: This calculation confirms that the units are consistent with mass being measured in kilograms (kg) since newtons divided by meters per second squared result in kilograms.This approach ensures accuracy in determining the mass and helps students practice manipulating equations to solve for a desired variable.