The Earth’s gravitational field strength, g = 9.81 N kg-1, is also sometimes given as the acceleration due to gravity, g = 9.81 m s-2. Show that these units are equivalent.
The Correct Answer and Explanation is :
The correct answer is: ( g = 9.81 \, \text{N kg}^{-1} = 9.81 \, \text{m s}^{-2} )
To demonstrate that the units of gravitational field strength ( g = 9.81 \, \text{N kg}^{-1} ) and acceleration due to gravity ( g = 9.81 \, \text{m s}^{-2} ) are equivalent, we need to analyze the units involved in both expressions.
- Understanding the units of gravitational field strength (N/kg):
- The unit of force is the Newton (N), which is defined as ( 1 \, \text{N} = 1 \, \text{kg} \cdot \text{m s}^{-2} ).
- This means that a Newton is the force required to accelerate a mass of 1 kilogram at a rate of 1 meter per second squared.
- Therefore, when we express gravitational field strength in terms of Newtons per kilogram, we have:
[
\text{N/kg} = \frac{\text{N}}{\text{kg}} = \frac{\text{kg} \cdot \text{m s}^{-2}}{\text{kg}}.
] - The kilograms (kg) in the numerator and denominator cancel out:
[
\text{N/kg} = \text{m s}^{-2}.
] - Thus, we find that ( 1 \, \text{N kg}^{-1} = 1 \, \text{m s}^{-2} ).
- Understanding the units of acceleration due to gravity (m/s²):
- The unit ( \text{m s}^{-2} ) directly represents acceleration. It indicates that for every second, the velocity of an object increases by a certain number of meters per second.
- The value ( g = 9.81 \, \text{m s}^{-2} ) specifies that an object in free fall will accelerate towards the Earth at ( 9.81 \, \text{m/s}^2 ).
From our analysis, we see that both representations of gravitational field strength essentially describe the same physical phenomenon but use different unit conventions. The equivalence of the units ( \text{N kg}^{-1} ) and ( \text{m s}^{-2} ) underlines the fundamental relationship between force, mass, and acceleration as captured by Newton’s second law of motion. Thus, we can confidently conclude that the two expressions for gravitational strength are indeed equivalent, with ( g = 9.81 \, \text{N kg}^{-1} = 9.81 \, \text{m s}^{-2} ).