Game Development Reference
In-Depth Information
■
Tidbit
In 1672, Sir Isaac Newton was the first to note in writing that the flight of a tennis ball was affected
by its spin. Apparently even when he was exercising, Newton's mind was on physics.
The size of the Magnus force exerted on an object depends on the characteristics of the
fluid, the geometry of the object, and the speed at which the object is traveling. To characterize
Magnus force, or any other lifting force for that matter, an equation very similar to that used for
drag in Equation (5.12) is typically used. The Magnus force,
F
M
, is expressed as a function of the
fluid density,
r
, the velocity magnitude,
v
, a characteristic area,
A
, and a nondimensional
number known as a lift coefficient,
C
L
.
1
---
C
L
rv
2
A
F
M
=
(5.28)
Determining the Magnus force lift coefficient for an arbitrarily shaped object is a difficult
problem. If certain simplifying assumptions are made, analytical relations are possible for
some simple shapes. For example, according to Bernoulli's equation, the Magnus force lift
coefficient for a sphere is the product of the sphere radius,
r
, and angular velocity,
w
, divided by
the translational velocity magnitude,
v
.
rw
v
C
L
=
------
(5.29)
The ratio in Equation (5.29) is also known as the
rotational spin ratio
. The Magnus lift
coefficient equation for a cylinder rotating about the long axis of the cylinder is given in
Equation (5.30).
2
prw
v
C
L
=
-------------
(5.30)
The direction of Magnus force will always be perpendicular to both the velocity vector and
the spin axis. One way to determine the direction of Magnus force is to take the vector cross
product of the velocity and spin axis vectors. An equation for the cross product of two vectors
was presented in Chapter 2 in Equation (2.11).
The force diagram for a projectile subject to gravity, aerodynamic drag, and Magnus force
is shown in Figure 5-15. Also shown in Figure 5-15 is the direction that the object is spinning. If
the object has backspin, as it does in Figure 5-15, the Magnus force will be directed upward and
will lift the object. In this case, the Magnus force will counteract the downward force of gravity
and allow a projectile to stay in the air longer than it normally would. If the rotation of the
object was in the opposite direction (that is, topspin), the direction of the Magnus force vector
would be directed downward and would cause the object to fall faster.
