Chapter 13, Problem 55A

The symbol g can mean acceleration due to gravity or gravitational field strength. Show that the units of g can be expressed as either $m/s^2$ or N/kg

Solution

To Show:The unit of g can be expressed as either ${\text{m/s}}^{\text{2}}\text{or N/kg}$

Introduction:

Acceleration of body is the rate of change of velocity with respect to time. According to the Newton’s law of gravitation, every massive body is attracted towards the earth with constant acceleration g . The value of constant acceleration due to gravity is $\text{9}{\text{.8 m/s}}^{\text{2}}$ .

Explanation:

According to basic kinematics, when force is acted on body, it changes its velocity with respect to time. In this case body is said to be accelerated.

  $\begin{array}{l}a=g=\frac{dv}{dt}\\ g=\frac{dv}{dt}\\ \text{Dimensional analysis gives,}\\ g=\frac{\left[\frac{\text{m}}{\text{s}}\right]}{\left[\text{s}\right]}\\ g=\left[\frac{\text{m}}{{\text{s}}^{\text{2}}}\right]\end{array}$

Instead of external force, when body is moving under the action of gravitational force, acceleration can be given by,

Dimensional analysis of above equation gives the unit of acceleration,

  $g=\frac{Gm}{r^2}$

By using dimensional analysis,

  $\begin{array}{l}g=\frac{\left[\frac{{\text{N×m}}^{\text{2}}}{{\text{kg}}^{\text{2}}}\right]\left[\text{kg}\right]}{\left[{\text{m}}^{\text{2}}\right]}\\ g=\left[\frac{\text{N}}{\text{kg}}\right]\end{array}$

Conclusion:

Therefore, using the basic equation for acceleration in kinematics as well as Newton’s second law of motion, g can be obtained with different units.