<< Chapter < Page Chapter >> Page >

This is the maximum reduction possible due to rotation. Indeed, we can neglect this variation for all practical purposes except where very high accuracy is required.

Vertical position

In this section, we shall discuss the effect of the vertical position of the point of measurement. For this, we shall consider Earth as a perfect sphere of radius “R” and uniform density, “ρ”. Further, we shall first consider a point at a vertical height “h” from the surface and then a point at a vertical depth “d” from the surface.

Gravitational acceleration at a height

Gravitational acceleration due to Earth on its surface is equal to gravitational force per unit mass and is given by :

g 0 = F m = G M R 2

Gravity at an altitude

Distance between center of Earth and particle changes at an altitude.

where “M” and “R” are the mass and radius of Earth. It is clear that gravitational acceleration will decrease if measured at a height “h” from the Earth’s surface. The mass of Earth remains constant, but the linear distance between particle and the center of Earth increases. The net result is that gravitational acceleration decreases to a value “g’” as given by the equation,

g = F m = G M R + h 2

We can simplify this equation as,

g = G M R 2 1 + h R 2

Substituting for the gravitational acceleration at the surface, we have :

g = g 0 1 + h R 2

This relation represents the effect of height on gravitational acceleration. We can approximate the expression for situation where h<<R.

g = g 0 1 + h R 2 = g 0 1 + h R - 2

As h<<R, we can neglect higher powers of “h/R” in the binomial expansion of the power term,

g = g 0 1 2 h R

We should always keep in mind that this simplified expression holds for the condition, h<<R. For small vertical altitude, gravitational acceleration decreases linearly with a slope of “-2/R”. If the altitude is large as in the case of a communication satellite, then we should resort to the original expression,

g = G M R + h 2

If we plot gravitational acceleration .vs. altitude, the plot will be about linear for some distance.

Acceleration .vs. linear distance

The plot shows variations in gravitational acceleration as we move vertically upwards from center of Earth.

Gravitational acceleration at a depth

In order to calculate gravitational acceleration at a depth “d”, we consider a concentric sphere of radius “R-d” as shown in the figure. Here, we shall make use of the fact that gravitational force inside a spherical shell is zero. It means that gravitational force due to the spherical shell above the point is zero. On the other hand, gravitational force due to smaller sphere can be calculated by treating it as point mass. As such, net gravitational acceleration at point “P” is :

Gravity at a depth

Distance between center of Earth and particle changes at an altitude.

g = F m = G M R - d 2

where “M’” is the mass of the smaller sphere. If we consider Earth as a sphere of uniform density, then :

M = V ρ

ρ = M 4 3 π R 3

Hence, mass of smaller sphere is equal to the product :

M = ρ V

M = 4 3 π R d 3 X M 4 3 π R 3 = R d 3 X M R 3

Substituting in the expression of gravitational acceleration, we have :

Questions & Answers

How can we take advantage of our knowledge about motion?
Kenneth Reply
pls explain what is dimension of 1in length and -1 in time ,what's is there difference between them
Mercy Reply
what are scalars
Abdhool Reply
show that 1w= 10^7ergs^-1
Lawrence Reply
what's lamin's theorems and it's mathematics representative
Yusuf Reply
if the wavelength is double,what is the frequency of the wave
Ekanem Reply
What are the system of units
Jonah Reply
A stone propelled from a catapult with a speed of 50ms-1 attains a height of 100m. Calculate the time of flight, calculate the angle of projection, calculate the range attained
Samson Reply
58asagravitasnal firce
water boil at 100 and why
isaac Reply
what is upper limit of speed
Riya Reply
what temperature is 0 k
0k is the lower limit of the themordynamic scale which is equalt to -273 In celcius scale
How MKS system is the subset of SI system?
Clash Reply
which colour has the shortest wavelength in the white light spectrum
Mustapha Reply
how do we add
Jennifer Reply
if x=a-b, a=5.8cm b=3.22 cm find percentage error in x
Abhyanshu Reply
x=5.8-3.22 x=2.58
what is the definition of resolution of forces
Atinuke Reply

Get the best Physics for k-12 course in your pocket!

Source:  OpenStax, Physics for k-12. OpenStax CNX. Sep 07, 2009 Download for free at http://cnx.org/content/col10322/1.175
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Physics for k-12' conversation and receive update notifications?