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Gravity : Notes/W.S. - 30
The Gravitational Potential Energy was previously defined as :
G.P.E. = mgh. This assumes that the G.P.E. on the Earth's surface for
a body of mass m is equal to zero and that the G.P.E. of that body at a
height h above the Earth's surface is equal to the work required to
move the object to a height h. (work = mgh, assumes mg is constant)
This definition of the G.P.E. is works only if h is small. The reason for
this is that the force of gravity decreases as the distance from the
Earth's center increases.
Force of Gravity
(R E = Earth Radius)
d (distance)
The force of gravity (shown above) is F = {GMm}/d2 . F decreases as
the distance d from the Earth's center increases. The work done in
moving an object a distance (R - RE ) above the Earth's surface equals
the area under the curve shown above. Calculating this area requires
using the calculus.
A more general definition of the G.P.E. is given as : G.P.E. = - GMm/d. G
is the gravitational constant, M is the mass of the Earth, m is the
mass of the object, and d is the distance from the Earth's center. The
zero energy is at infinity (d very large). The work done (area under the
above curve) on the object equals the ∆G.P.E., or G.P.E.(at R) G.P.E.(at RE ). This assumes that there is no change in the kinetic
energy. The G.P.E. for a body is shown below.
In general, the total energy of a body is constant. The total energy
equals the kinetic energy plus the gravitational potential energy. This
assumes that the only force acting on the body is gravity.
EK1 + EP1 = EK2 + EP2
Exercises :
1) Using the general formula, does the G.P.E. of a body increase or
decrease as it is moved farther from the Earth's center?
2)a) Find the G.P.E. of a 1.00 kg mass on the Earth's surface.
b) Find the G.P.E. of the 1.00 kg mass if it is moved to a height of 100.
km above the Earth's surface.
c) Find the ∆G.P.E..
d) Find the G.P.E. of the above mass if the formula G.P.E. = mgh is
used. Compare with the answer to question c. (assume g = 9.81)
e) How much work would be required to move the 1.00 kg object to
"infinity", from the Earth's surface.
3) Find the G.P.E. of the moon with respect to the Earth.
4) Find the minimum velocity that a rocket must have to escape the
Earth's gravity.
5) Find the initial speed that a rocket must have in order to reach a
height of 2.00 Earth radii above the Earth's surface.
6) A 1.00 kg object is dropped to the surface of the Earth from a
height of five Earth radii. Find the speed with which it hits the surface.
Assume that there is no air resistance.
Answers : 1) It increases, 2)a) -6.25x107 , J, b) -6.16x107 J, c)
+9.65x10 5 J, d) 9.81x105 J, e) +6.25x107 J, 3) -7.63x1028 J, 4)
1.12x10 4 m/s, 5) 9.13x103 m/s, 6) 1.02x104 m/s.