Correct Answer - Option 2 : 0. 67 M
Concept:
Mass and Weight
Mass
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Weight
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Mass is the amount or measure of matter present in a body. Every matter has a mass.
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It is a measure of the Normal force we apply to any object. In normal conditions, it is equal to the force of gravity acting on our body.
W = mg
m is mass, g is the acceleration due to gravity
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It is fixed for a matter and does not change with the place.
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It varies from place to place. It is different on different planets as they have different values of g.
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It is having SI unit of Kg. Other units are gram, pound, etc.
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It is having SI unit of Newton. Usually, it is also denoted as Kg Wt
A mass having a mass 60 kg will weigh 60 Kg wt. on the earth's surface.
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Acceleration Due to Gravity:
- The force of attraction exerted by the earth on a body is called gravitational pull or gravity.
- We know that when a force acts on a body, it produces acceleration. Therefore, a body under the effect of gravitational pull must accelerate.
- The acceleration produced in the motion of a body under the effect of gravity is called acceleration due to gravity, it is denoted by g.
- The acceleration due to gravity on the surface is given as
\(g = \frac{{GM}}{{{R^2}}} \) ---(1)
G is universal constant, M is mass, R is radius of earth.
Calculation:
Given Weight of astronaut on earth W1 = mg = 75 kg wt.
m is the mass of the object, g is the acceleration due to gravity on earth,
g' is the acceleration due to gravity on the other planet X
Given Weight of astronaut on Planet X mg' = 50 W2 = kg wt.
\(\frac{W_2}{W_1} = \frac{mg'}{mg} = \frac{50}{75}\)
⇒ \( \frac{W_2}{W_1} =\frac{g'}{g} = \frac{2}{3}\)
⇒ \(g' = \frac{2}{3} \ g\) ---(2)
Given the radius of both planet is the same R,
Mass of earth is M, the mass of planet X is M'
The acceleration due to gravity of another planet is
\(g' = \frac{{GM'}}{{{R^2}}} \) ---(3)
Using (1) and (3) in (2) we get
⇒ \(\frac{GM'}{R^2} = \frac{2}{3} \frac{GM}{R^2} \)
⇒ \(M' = \frac{2}{3} M\)
⇒ M' = 0. 67 M
So, the correct option is 0. 67 M.