For small displacements from the surface

PHYSICS · Multi-part · A Level · Physics · Fields and their consequences · Fields and their consequences

Problem

For small displacements from the surface, the expressions for

g

at height

h

and at depth

d

may be approximated. Note: The explicit instruction to use a binomial expansion has been removed. (i) Starting from the exact expression for

g

at altitude, obtain a simplified relation showing that the fractional change in

g

is approximately

-\frac{2h}{R}

when

h \ll R

. (ii) Repeat the procedure for depth

d

to show that the fractional change in

g

is approximately

-\frac{d}{R}

when

d \ll R

. (iii) Using results of (i) and (ii), evaluate the ratio

\frac{\text{fractional change per metre of depth}}{\text{fractional change per metre of altitude}}

near the surface and interpret its physical meaning.