MAT 17A - DISCUSSION #6
Problem 1. Optimal pitching Since the World Series just started, let’s think
about baseball. When a pitcher throws a ball, he steps forward, rotates his torso,
rotates his shoulder with respect to his torso, rotates his forearm with respect to his
upper arm, snaps his wrist and extends his fingers. The muscles controlling each of
these body parts contract, and the timing is such that muscles controlling proximal
(near the trunk) segments are activated before distal (near the finger tips) ones.
The multi-segment nature of this motion is extremely important for maximal velocity.
In fact, the primary difference between a change-up and a fast-ball is that the changeup is gripped in the palm. This reduces the ability of the fingers to contract, which
removes one segment from the motion, and ball velocity is about ten miles per hour
slower than a fast-ball, where the fingers contribute to ball velocity.
In this problem, we’ll try to understand why multiple links help you throw faster.
Here are two simplified models of throwing:
θ
ball
φ
θ
y
x
hinge
In the model on the left, a single link pivots about a hinge. Suppose that we want to
maximize the ball’s velocity in the x direction (note the axes in the lower-left corner
of the figure). Let’s write an expression for the ball’s velocity in the x direction as a
dθ
function of `, the segment length, θ the angle of the “arm,” and
the rate that θ
dt
changes with time, called the angular velocity.
Here’s how we’ll do it:
a) Write an expression for the ball’s x-position in terms of ` and θ.
b) Differentiate your expression from a) with respect to time. (Recall that θ(t) is a
function of time, and recall that
dθ
is the angular velocity.)
dt
dx
dθ
, find the value of θ that maximizes your expression for
.
dt
dt
To do this you can graph your expression using R/R-studio or any other method.
How is the “arm” positioned in order to get maximal ball velocity in the x direction?
Does that make sense?
dθ
d) Suppose that the “arm” has length 1m, and the angular velocity
= 10 radidt
ans/second. What’s the maximum speed the ball can go? Write your answer in m/s,
and also use the conversion factor 1m/s = 3.6mph to convert your answer to miles
per hour (mph). Recall that radians are defined from the relationship between the
arc length, s, of a “wedge” of a circle with angle θ and radius r, given by s = rθ.
Therefore when the units of radians are used in the expression for angular velocity,
1
dθ
are
.
the units of
dt
time
c) For a fixed value of
In the model in the right figure, one link pivots about a hinge and a second link
pivots about the first. Again, suppose that we want to maximize the ball’s velocity
in the x direction (note the axes in the lower-left corner of the figure). Let’s write
an expression for the ball’s velocity in the x direction as a function of `1 and `2 , the
dθ
and φ the
segment lengths, θ the angle of the “upper arm,” its angular velocity
dt
dφ
angle of the “forearm” with respect to the upper arm, and its angular velocity
.
dt
e) repeat steps a–c for the model at the right. Your final answer will include the
dθ
dφ
variables, `1 , θ, `2 , φ,
and
.
dt
dt
f) Now suppose that the “arm” has same length as before (1m), and the joint occurs
in the middle, giving `1 = 0.5m, and `2 = 0.5m. Again, suppose that the angular
dθ
velocity
= 10 radians/second, but now the distal link also has angular velocity
dt
dφ
= 10 radians/second. What’s the maximum speed the ball can go? Write your
dt
answer in m/s, and also use the conversion factor 1m/s = 3.6mph to convert your
answer to miles per hour (mph).
g) Suppose that the model on the left (parts a-d) represents a change-up, while the
model on the right (parts e and f) represents a fast-ball. Does that make sense? How
does that second link contribute to ball speed?
Problem 2. Population Density
Shenzhen, which is a major city in southern China’s Guangdong Province, is one of
the fastest growing cities in the world. In 1996, the area of Shenzhen was 600 km2
and it was expanding at a rate of 40 km2 /year. The population of Shenzhen in 1996
was 5,000,000 people, and it was growing at a rate of 400,000 people/year.
a) Determine the population density of Shenzhen in 1996.
b) Use the data above to determine how fast the population density of Shenzhen was
growing in 1996.
(Include the appropriate units).