6.3. DENSITY, MASS, AND CENTER OF MASS
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(b) For the 30g rod, will the center of mass lie at its midpoint, to the left of the
midpoint, or to the right of the midpoint? Why?
(c) For the 30g rod, find the center of mass, and compare your prediction in (b).
(d) At what value of x should the 30g rod be cut in order to form two pieces of
equal mass?
2. Consider two thin bars of constant cross-sectional area, each of length 10 cm, with
respective mass density functions ρ(x) =
1
1+x 2 and p(x) = e −0.1x .
(a) Find the mass of each bar.
(b) Find the center of mass of each bar.
(c) Now consider a new 10 cm bar whose mass density function is f (x) = ρ(x) +
p(x).
i. Explain how you can easily find the mass of this new bar with little to no
additional work.
ii. Similarly, compute
10
0
x f (x) dx as simply as possible, in light of earlier
computations.
iii. True or false: the center of mass of this new bar is the average of the
centers of mass of the two earlier bars. Write at least one sentence to say
why your conclusion makes sense.
3. Consider the curve given by y = f (x) = 2xe −1.25x + (30 − x)e −0.25(30−x) .
(a) Plot this curve in the window x = 0 . . . 30, y = 0 . . . 3 (with constrained scaling
so the units on the x and y axis are equal), and use it to generate a solid of
revolution about the x-axis. Explain why this curve could generate a reasonable
model of a baseball bat.
(b) Let x and y be measured in inches. Find the total volume of the baseball bat
generated by revolving the given curve about the x-axis. Include units on your
answer
(c) Suppose that the baseball bat has constant weight density, and that the weight
density is 0.6 ounces per cubic inch. Find the total weight of the bat whose
volume you found in (b).
(d) Because the baseball bat does not have constant cross-sectional area, we see that
the amount of weight concentrated at a location x along the bat is determined
by the volume of a slice at location x. Explain why we can think about the
function ρ(x) = 0.6π f (x) 2 (where f is the function given at the start of the
problem) as being the weight density function for how the weight of the baseball
bat is distributed from x = 0 to x = 30.
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(b) For the 30g rod, will the center of mass lie at its midpoint, to the left of the
midpoint, or to the right of the midpoint? Why?
(c) For the 30g rod, find the center of mass, and compare your prediction in (b).
(d) At what value of x should the 30g rod be cut in order to form two pieces of
equal mass?
2. Consider two thin bars of constant cross-sectional area, each of length 10 cm, with
respective mass density functions ρ(x) =
1
1+x 2 and p(x) = e −0.1x .
(a) Find the mass of each bar.
(b) Find the center of mass of each bar.
(c) Now consider a new 10 cm bar whose mass density function is f (x) = ρ(x) +
p(x).
i. Explain how you can easily find the mass of this new bar with little to no
additional work.
ii. Similarly, compute
10
0
x f (x) dx as simply as possible, in light of earlier
computations.
iii. True or false: the center of mass of this new bar is the average of the
centers of mass of the two earlier bars. Write at least one sentence to say
why your conclusion makes sense.
3. Consider the curve given by y = f (x) = 2xe −1.25x + (30 − x)e −0.25(30−x) .
(a) Plot this curve in the window x = 0 . . . 30, y = 0 . . . 3 (with constrained scaling
so the units on the x and y axis are equal), and use it to generate a solid of
revolution about the x-axis. Explain why this curve could generate a reasonable
model of a baseball bat.
(b) Let x and y be measured in inches. Find the total volume of the baseball bat
generated by revolving the given curve about the x-axis. Include units on your
answer
(c) Suppose that the baseball bat has constant weight density, and that the weight
density is 0.6 ounces per cubic inch. Find the total weight of the bat whose
volume you found in (b).
(d) Because the baseball bat does not have constant cross-sectional area, we see that
the amount of weight concentrated at a location x along the bat is determined
by the volume of a slice at location x. Explain why we can think about the
function ρ(x) = 0.6π f (x) 2 (where f is the function given at the start of the
problem) as being the weight density function for how the weight of the baseball
bat is distributed from x = 0 to x = 30.