Question

The mass of a book is $500\;g$ on the surface of the earth.

1. What will be its mass at a height equal to the radius of the earth?
2. Find the weight of the book at the surface of the earth.

Answer 1

Recall the key differences between mass of a body and weight of a body before proceeding with the calculations. The first question asks us to find the mass of the book at a height, so think about whether mass depends on the strength of the gravitational force acting on it or not. Similarly, in the second question, we are asked to find the weight of the book at the surface of the Earth. Account for the acceleration due to gravity in this case since weight is a measure of the influencing force acting on a body. To this end, arrive at the appropriate solution.

Formula Used:
Weight $W=mg$

Complete Solution:
It is essential to first understand the distinction between mass and weight before proceeding to evaluating the problem.

Mass is simply the measure of the quantity of matter contained in a body. It is always constant for a body irrespective of influencing forces acting on it. It is a fundamental quantity, and since it only has a magnitude it is a scalar quantity.

Weight, on the other hand, is a measure of the amount of force acting on a mass under the action of gravity. It is thus a measure of the attractive gravitational force acting on the mass. It is a derived quantity, and since it has both magnitude and direction it is a vector quantity. It is always directed towards the centre of gravity.

Now, let us get into the context of our question:
We are given a book that has a mass $m=500\;g= 500 \times 10^{-3}\;kg = 0.5\;kg$ on the surface of the Earth.

We are required to deduce the mass of the book at a height equal to the Radius of the Earth.

The difference is that the book is now at a height, where the strength of gravity is lesser than at the surface of the earth. However, we know that the mass of any body does not depend upon gravity and remains the same everywhere.

Therefore, the mass of the book at a height equal to the radius of the Earth will still be $500\;g$

We are now required to find the weight of the book at the surface of the earth.

We know that weight is influenced by the acceleration due to gravity and measures the amount of gravitational force acting an a body of certain mass. This is given as:

Weight $W=mg$, where m is the mass of the body and g is the acceleration due to gravity.
Now, the acceleration due to gravity on the surface of the earth has a value of about $g=9.8\;ms^{-2}$.

Therefore, the weight of the book at the surface of the earth will be:
$W = mg = 0.5 \times 9,8 \Rightarrow W= 4.9\;N$

Note:
Remember that in classical physics, mass of a body can never be zero. However, the weight of a body can be zero where there is no gravity. The weight of the body thus depends on the gravitational force, or the acceleration due to gravity acting on it.
Note that the SI unit of mass is Kilogram (kg), whereas the SI unit of weight is newton (N). However, sometimes, kilogram is interchangeably used to denote both mass and weight, so it is important to read the question carefully for further clarity.