Answers to the Exercises - 3D Math Primer for Graphics and Game Development - page 785

By Newton's third law, we assume that each force on the left is equal in magnitude but
opposite in direction to the corresponding force on the right. Next, since we assume that
the stretching of the rope is negligible, the tension at one end must be equal to the tension
at the other end, so all the T forces have equal magnitude. Since the children are both
accelerating, there must be a net force on each of them causing their displacement. Earth
is pushing the girl harder than the rope is pulling her, so she accelerates backwards. For
the boy, the opposite is true, and the directions of the forces cause him to move forwards.
So the reason that the children, as a system, accelerate relative to Earth is because the
girl's pushing force is larger than the boy's pushing force, resulting in a net force on the
children in the direction of the girl.
3. False. The acceleration due to gravity is constant, but the force due to gravity increases
proportionately with mass.
4. This is a straightforward application of Newton's law of universal gravitation with the
distance equal to the radius of Earth plus the orbit altitude.

d = 6, 371 km + 340 km = 6, 711 km.

Plugging this value and the mass of Earth into Equation (12.3), we have

−11 N m 2
kg 2

(5.98×10 24 kg)m 2
(6.711×10 6 m) 2

f = G m 1 m 2
d 2

= 6.673×10

= (8.86 N) m 2
kg

m
s 2

= 8.86 m 2

We observe a few things about this result. First, it most definitely is not zero; in fact, it is
only about 10% less than the acceleration due to gravity at Earth's surface. Although the
term “zero gravity” is often used to describe the environment of objects orbiting in space,
we see that this term is a bit of a misnomer, since gravity is quite alive and well, even at
340 km above Earth's surface. In fact, it is gravity that supplies the necessary centripetal
acceleration to maintain the orbit.
Second, we compare this answer to our results from Exercise 11.12, and we see that the
numbers are the same. (Well, almost exactly the same. The discrepancy of 0.1% is a
result of some slight simplifications to the problem and rounding.) This match leads us
to answer the second part of the problem. The apparent weightlessness exists because the
space station and all the objects in it are in free fall. Apparent weightlessness occurs in any
free-fall situation, no matter what the force of gravity and even if the object isn't orbiting
(for example, in a falling elevator or amusement park ride or in NASA's “vomit comet”
aircraft).
The difference between a falling elevator and an object orbiting Earth is that the free fall
in the space station continues indefinitely. The orbit speed and altitude are selected such
that the acceleration due to gravity is exactly the same as the centripetal acceleration, and
unlike a falling elevator, the space station never gets any closer to the ground. The space
station keeps “falling over the horizon” and never hits bottom.
5. At the critical angle, the force of static friction f s exactly balances the lateral component
of the force of gravity, g . The maximum friction is equal to the magnitude n of the
normal force times the coe cient of static friction s . The normal force is equal to g ⊥ , the