Problem 12 - College Physics

Question

(a) What is the mass of a book that weighs 3.20 \(\mathrm{N}\) in the laboratory? (b) In the same lab, what is the weight of a dog whose mass is 14.0 \(\mathrm{kg} ?\)

Step-by-Step Solution

Answer

(a) 0.326 kg (b) 137.34 N

Step 1: Understanding Weight and Mass

Weight is the force with which an object is pulled towards the Earth and is given by the equation \( W = m \times g \), where \( W \) is the weight, \( m \) is the mass, and \( g \) is the acceleration due to gravity. On Earth, \( g \) is approximately \( 9.81 \, \mathrm{m/s^2} \).

Step 2: Calculation of Mass from Weight

To find the mass of the book, rearrange the formula for weight: \( m = \frac{W}{g} \). Substitute the given values, \( W = 3.20 \, \mathrm{N} \) and \( g = 9.81 \, \mathrm{m/s^2} \), to find \( m \).\[m = \frac{3.20}{9.81} \approx 0.326 \, \mathrm{kg}\]

Step 3: Calculation of Weight from Mass

To find the weight of the dog, use the same weight formula. You are given \( m = 14.0 \, \mathrm{kg} \) and \( g = 9.81 \, \mathrm{m/s^2} \). Calculate the weight \( W \).\[W = 14.0 \times 9.81 = 137.34 \, \mathrm{N}\]

Key Concepts

Force and Weight

In the realm of physics, understanding the concept of force and weight is crucial. Weight is a type of force that acts on every object with mass. Specifically, it's the gravitational force exerted on an object due to Earth's gravitational pull.

When we speak of weight, we use the formula:

• \( W = m \times g \)

where \( W \) represents the weight, \( m \) is the mass of the object, and \( g \) is the acceleration due to gravity, which on Earth is approximately \( 9.81 \, \mathrm{m/s^2} \).

This formula shows that weight is directly proportional to both mass and gravity, meaning a greater mass or a larger gravitational pull results in a greater weight. In physics problems, determining either mass from weight or vice-versa involves rearranging this basic equation.

Mass and Gravity

Mass is a fundamental property of an object, representing the amount of matter it contains, and unlike weight, mass does not change regardless of location. Mass is often measured in kilograms in the metric system.

Gravity, on the other hand, is a natural force that draws objects towards each other. On Earth, this force is quantified by the acceleration due to gravity, denoted by \( g \).

Understanding how mass and gravity relate is essential for exploring how objects behave under gravitational forces. For instance, if you know the weight of an object and want to find its mass, you rearrange the weight formula:

• \( m = \frac{W}{g} \)

This allows you to find mass by dividing the weight by the gravitational acceleration, making it helpful in practical problems, just like the given exercise of determining the mass of a book given its weight.

Physics Calculations

Physics involves calculations that utilize established formulas to solve real-world problems. These calculations help you move between concepts like force, weight, mass, and gravity effectively.

For example, if you know an object's mass and wish to find its weight, you multiply the mass by the gravitational acceleration:

• \( W = m \times g \)

In the exercise provided, such calculation allows us to find out the weight of a dog by applying this formula.

Conversely, if given weight, you can determine mass through the formula:

• \( m = \frac{W}{g} \)

These processes exemplify core physics calculations that take place regularly in problem-solving scenarios.