Advertisements
Advertisements
प्रश्न
If the radius of the earth decreases by 1% without changing its mass, will the acceleration due to gravity at the surface of the earth increase or decrease? If so, by what per cent?
उत्तर
If we consider the Earth to be a perfect sphere, then the acceleration due to gravity at its surface is given by \[g = G\frac{M}{R^2}\]
Here, M is the mass of Earth; R is the radius of the Earth and G is universal gravitational constant.
If the radius of the earth is decreased by 1%, then the new radius becomes
\[R' = R - \frac{R}{100} = \frac{99}{100}R\]
\[ \Rightarrow R' = 0 . 99R\]
New acceleration due to gravity will be given by
\[g' = G\frac{M}{R '^2} = G\frac{M}{(0 . 99R )^2}\]
\[ \Rightarrow g' = 1 . 02 \times \left( G\frac{M}{R^2} \right) = 1 . 02g\]
Hence, the value of the acceleration due to gravity increases when the radius is decreased.
Percentage increase in the acceleration due to gravity is given by
\[\frac{g' - g}{g} \times 100\]
\[ = \frac{0 . 02g}{g} \times 100\]
= 2 %
APPEARS IN
संबंधित प्रश्न
Derive an expression for critical velocity of a satellite revolving around the earth in a circular orbit.
A sonometer wire vibrates with frequency n1 in air under suitable load of specific gravity of . When the load is immersed in water, the frequency of vibration of wire n2 will be .........
What is the decrease in weight of a body of mass 600kg when it is taken in a mine of depth 5000m?
[ Radius of earth = 6400km, g = 9.8 m/s2 ]
A stone of mass 100 g attached to a string of length 50 cm is whirled in a vertical circle by giving velocity at lowest point as 7 m/s. Find the velocity at the highest point. [Acceleration due to gravity = 9.8 m/s2 ]
Derive an expression for acceleration due to gravity at depth ‘d’ below the earth’s surface.
Io, one of the satellites of Jupiter, has an orbital period of 1.769 days and the radius of the orbit is 4.22 × 108 m. Show that the mass of Jupiter is about one-thousandth that of the sun.
A hole is drilled half way to the centre of the Earth. A body is dropped into the hole. How much will it weigh at the bottom of the hole if the weight of the body on the Earth’s surface is 350 N?
Prove that `g_h=g(1-"2h"/R)` where gh is the acceleration due to gravity at altitude h and h << R (R is the radius of the earth).
‘g’ is the acceleration due to gravity on the surface of the Earth and ‘R’ is the radius of the
Earth.
Show that acceleration due to gravity at height ‘h’ above the surface of the Earth is
`gh = g (R/(R+H))^2`
What is the decrease in weight of a body of mass 500 kg when it is taken into a mine of depth 1000 km? (Radius of earth R = 6400 km, g = 9.8 m/s2)
