Question

A man weighs \(0.80 \mathrm{kN}\) on Earth. What is his mass in kilograms?

Short answer

The man's mass is approximately 81.55 kg.

Step-by-step solution

Understand the relationship between weight and mass

Weight is the force exerted by gravity on an object. It is related to mass by the equation: \( W = m \cdot g \), where \( W \) is the weight in newtons (N), \( m \) is the mass in kilograms (kg), and \( g \) is the acceleration due to gravity. On Earth, \( g \) is approximately \( 9.81 \mathrm{m/s^2} \).

Write the equation for the situation

Given the weight of the man is \( 0.80 \mathrm{kN} \), which is the same as \( 800 \mathrm{N} \) (since \( 1 \mathrm{kN} = 1000 \mathrm{N} \)). Use the equation \( W = m \cdot g \) to relate the weight and mass: \( 800 = m \cdot 9.81 \).

Solve for mass \( m \)

Rearrange the equation \( 800 = m \cdot 9.81 \) to solve for \( m \): \( m = \frac{800}{9.81} \). Calculate this to find \( m \approx 81.55 \mathrm{kg} \).

Key concepts

Newton's Second Law

Newton's Second Law of Motion is a fundamental principle in physics that explains the relationship between the motion of an object and the forces acting upon it. According to this law, the force applied to an object is equal to the mass of the object multiplied by its acceleration: \[ F = m \cdot a \] Where \( F \) is the force measured in newtons (N), \( m \) is the mass in kilograms (kg), and \( a \) is the acceleration in meters per second squared (m/s²).

• When a force is applied to an object, it causes the object to accelerate, changing its speed or direction.
• The larger the mass of the object, the more force is needed to achieve the same acceleration.

For example, if we want to understand how gravity affects a person standing on Earth, we consider their weight (the force of gravity on them). This weight is also calculated using Newton’s second law, where the force of gravity \( W \) equals mass \( m \) times the acceleration due to gravity \( g \). This concept is crucial when converting weight into mass, especially in exercises like finding a man's mass based on his weight.

Acceleration due to Gravity

The acceleration due to gravity, often represented as \( g \), is the rate at which objects accelerate towards the Earth when falling freely in a vacuum with no air resistance. On Earth, the value of \( g \) is approximately \( 9.81 \text{ m/s}^2 \). This means any object in free fall will accelerate at this rate, making it a constant factor in many physics calculations.

• Gravity is a universal force of attraction that acts between all mass in the universe.
• The standardized value of \( 9.81 \text{ m/s}^2 \) is used for calculations involving gravitational force on Earth's surface.

Understanding this concept helps us relate weight and mass through the equation \( W = m \cdot g \). Once we know the weight of an object and the acceleration due to gravity, we can calculate the mass. This is especially useful for converting the weight of objects, like people, into mass units like kilograms.

Unit Conversion

Unit conversion is essential for solving physics problems as it ensures consistency in calculations. In physics, different quantities have specific units and often need to be translated into one another to apply formulas correctly. When dealing with forces, such as weight, converting units like kilonewtons (kN) to newtons (N) is necessary because equations usually utilize the base unit of newtons for force:

• 1 kilonewton equals 1,000 newtons.
• Knowing this conversion allows us to rephrase the weight from high-level units like kN to N for precise calculations.

For example, converting a man's weight from \(0.80 \text{kN}\) to \(800 \text{N}\) is crucial to properly apply the weight-mass relation formula \( W = m \cdot g \). This conversion allows us to compute mass accurately. Mastery of unit conversions supports solving complex physics tasks and ensures the accuracy of results across different scenarios.