Question

An object weighs \(25 \mathrm{kN}\) at a location where the acceleration of gravity is \(9.8 \mathrm{~m} / \mathrm{s}^{2}\). Determine its mass, in \(\mathrm{kg}\).

Short answer

The mass is approximately 2551.02 kg.

Step-by-step solution

Understand the relationship between weight and mass

The weight of an object is given by the formula: \[ W = m \times g \] where:- \(W\) is the weight,- \(m\) is the mass, and- \(g\) is the acceleration due to gravity.

Identify the given values

From the problem statement, the given values are:- Weight, \(W = 25 \text{kN} = 25000 \text{N} \)- Acceleration due to gravity, \( g = 9.8 \text{ m/s}^2 \)

Rearrange the formula to solve for mass

Rearrange the weight formula to solve for mass, \( m \): \[ m = \frac{W}{g} \]

Substitute the given values into the rearranged formula

Substitute \( W = 25000 \text{N} \) and \( g = 9.8 \text{m/s}^2 \) into the formula: \[ m = \frac{25000 \text{N}}{9.8 \text{m/s}^2} \]

Perform the calculation

Calculate the mass: \[ m = \frac{25000}{9.8} \approx 2551.02 \text{kg} \]

Key concepts

Weight and Mass Relationship

Weight and mass are commonly confused, but they are distinct concepts. Mass is the amount of matter in an object and is measured in kilograms (kg).
Weight, on the other hand, is the force exerted by gravity on that mass and is measured in newtons (N).
The relationship between weight and mass is given by the formula: \[ W = m \times g \] where

• \(W\) is the weight,
• \(m\) is the mass, and
• \(g\) is the acceleration due to gravity.

This formula shows that weight depends on both mass and gravity.
For instance, the same object will weigh less on the Moon than on Earth because the Moon's gravitational pull is weaker.

Acceleration Due to Gravity

Acceleration due to gravity (\(g\)) is the rate at which objects accelerate downwards due to Earth's gravitational pull. On Earth, this value is approximately \(9.8 \text{ m/s}^2\).
It's an essential constant in the weight equation and can vary slightly depending on location (e.g., altitude and latitude).
Understanding \(g\) allows us to determine how massive objects are by measuring their weight. Therefore, if you know an object's weight and the local gravity, you can calculate its mass using the formula we derived earlier: \[ m = \frac{W}{g} \]

Unit Conversion in Physics

Unit conversion is crucial in physics to ensure that calculations are accurate and consistent. In this exercise, we converted the given weight from kilonewtons (kN) to newtons (N).
1 kN equals 1000 N, which simplifies the calculation.
That’s why the initial weight of \(25 \text{kN}\) became \(25000 \text{N}\).
Another example is length conversions, such as 1 kilometer (km) equals 1000 meters (m).
Always ensure consistent units before performing calculations to avoid errors.