ERTH3020 Tutorial 2:
Gravity Method - Units and Practice Calculations
This tutorial is aimed at:
• Revision of basic theoretical ideas relating to the Gravity method
• Practice at handling units, manipulating equations, performing calculations
• Practice at scientific comprehension
Q1. Force and Newton’s 2nd Law
(a)
Write down Newtons 2nd Law (F = ma) and show that if a mass of 1 kg is accelerated
at 1 ms−2 , the force required is 1 kg ms−2 .
(b)
The SI unit of force is the Newton, which is defined as the force required to give a mass
of 1 kg an acceleration of 1 ms−2 . Convince yourself that 1 Newton is an alternative
unit for 1 kg ms−2 .
(c)
A 70kg person jumps off a diving board. Show that the earth’s gravity is exerting a
force of approximately 686 N on the person.
(d)
The same 70kg person is standing on the side of the pool. What is force being exerted
on the person by the earth’s gravity? If acceleration is a = F/m why is the person not
accelerating downwards?
(e)
Suppose that a bike with rider weighs 100 kg. The bike is accelerated from rest and
reaches a speed of 20 km/hr in 10 sec. What is the acceleration in SI units? Show that
the force required is approximately 55.5N.
Tutorial: Gravity Exercises
ERTH3020
Page 1
Q2. Newton’s Law of Universal Gravitation
This law describes the gravitational force between any two particles:
F =
Gm1 m2
r2
where the symbols have the usual meanings.
(a)
Calculate the gravitational force acting between two 90 kg people separated by 10m.
(Assume that the particle requirement can be ignored for the purpose of this estimate.)
(b)
Show that this force is about 0.0000000001 times the force exerted by the cyclist in Q1
(e).
(c)
Rearrange Newton’s Law to express G in terms of the other parameters.
(d)
Hence show that the units of G must be N m2 kg−2
(e)
By combining Newton’s 2nd Law, and the Law of Universal Gravitation, write an expression for the acceleration caused by a body of mass m, on another body at a distance
r.
(f)
Using the known values of G, and the mass and radius of the earth, estimate the approximate acceleration of a body at the surface of the earth. (Assume the earth acts as
if all mass is at its centre.)
(g)
In theory G could be estimated by accurately measuring the force or acceleration between bodies at a known separation. A spacecraft of mass 10000 kg is returning to
earth and at a distance of 1000 km above the surface its gravitational acceleration is
measured to be approximately 7.5 ms−2 . Based on this experiment, estimate the value
of G. Show that the error in this experimental estimate is around 2.3 %.
Q3. Work and Gravitational Potential
(a)
The work (W ) done by a force (F ) acting over a distance s is given by W = F ∗ s. A
Joule of work is defined as the work done by a force of 1N moving over a distance of
1m. Convince yourself that 1 J is the same as 1 Nm, orequivalently 1 kg m2 s−2 .
(b)
Consider the bike rider in Q1(e). Show that in the first 10 s he travels a distance 27.75m.
(Hint: use s = 0.5at2 .)
Tutorial: Gravity Exercises
ERTH3020
Page 2
(c)
Hence show that the amount of work he does in covering this distance is approximately 1540 J.
(d)
In geophysics the definition of Gravitational Potential relates to the work done moving
a 1 kg mass. By substituting units in the expression U = GM/r confirm that the units
of U are J kg−1 .
(e)
Calculate the average gravitational potential at the surface of the earth (under the approximation that all mass is acting at the centre),
(f)
Now calculate the gravitational potential 1 m above the surface. Use this to estimate
the gravitational acceleration at the surface of the earth. Hint: use a numerical approach to estimate g = − dU
.
dr
(g)
Show that the process of numerical differentiation used in Q3(f) results in the correct
units of acceleration.
Q4. Gravity Anomalies
In practical geophysics, we are interested in the gravity anomaly produced by the socalled density contrast of the target relative to the country rocks.
A gravity meter is being used to locate an old buried pipeline. It is of known to be
of sufficient size that the it can be assumed to have a density of either 0 or 1000 kg
m−3 (depending whether it is dry or water filled). The country rocks have a density of
approximately 2500 kg m−3 . In one location a number of similar anomalies are found
having an average peak value of -0.85 gu. In another location anomalies are smaller
with average peak of -0.35 gu. All anomalies have a half width of order 5m.
(a)
Estimate the depth to the centre of the pipe
(b)
As studied in Practical 1, the peak anomaly of an infinite horizontal cylinder is:
2πGR2 σ
Z
Based on the possibility that the different anomalies relate to dry and water-filled pipe,
deduce the size of the pipe.
Tutorial: Gravity Exercises
ERTH3020
Page 3
Q5. Analytical vs Numerical Approaches
A spaceship is travelling from earth to the moon. At this particular time of the month
the distance between the centres of the earth and moon is 400000 km. There is a particular point where the ship will effectively become weightless, when the gravitational
force exerted by the earth is equal to that exerted by the moon.
(a)
Show that at this point in the journey the following relationship holds:
2
2
ME /RE
= MM /RM
where ME and MM are the masses of earth and moon and RE and RM are the distances
of the ship from the centres of earth and moon respectively.
(b)
The point in space at which this happens can be found analytically. Start with the
relationship in (a) and substitute RM = R−RE , where R is the known distance between
the earth and moon (400000 km). Hence show that the required distance RE satisfies
the quadratic equation:
2
(1 − ME /MM )RE
+ 2R(ME /MM )RE − R2 (ME /MM ) = 0
(c)
By inserting known values for ME , MM andR, show that the coefficients in this quadratic
are:
a = −80.22
b = 6.50 ∗ 1010
c = −1.3 ∗ 1019
(d)
Use the well known solution
√
b2 − 4ac
2a
to solve for RE , and prove that the point at which the ship is weightless is approximately 360,050 km from the earth’s centre (i.e. about 39,950 km from the moon).
−b ±
(e)
Now solve the same problem using a numerical approach. Using Excel, Python or
similar, calculate and plot the acceleration caused by the earth, as a function of the distance from the earth’s centre. On the same graph plot the corresponding acceleration
due to the moon. The point at which these two curves cross is the required ’weightless’ location. Obviously you should get the same result for analytical and numerical
approaches.
Tutorial: Gravity Exercises
ERTH3020
Page 4