A thin rod with total mass
M=4.38 kg
and length
L=2.92 m
is rotating about one end. Its linear density changes across the rod, according to:
dr
dm
​
=( L
r
​
+1)
(The rod's density is lowest at the pivot (
r=0)
, and increases moving outward.) All coefficients are given in SI units. Find the rotational inertia of this object bydirect integration. Set up the integral, and simplify it as if you were about to carry it out to the final answer, keeping track of all your steps. Then answer the following questions: (a) What is the lower limit of integration?
in 51 units (b) What is the upper limit of integration?
in 51 units (c) What is the integrand? (pull constants outside of the integral first, just give the function to be integrated) A) 1 (a constant value) B)
L
r
​
+1
C)
L
r 2
​
+r
D)
L
r 3
​
+r 2
E)
r
F)
r 2
(a) What is the lower limit of integration? in SI units (b) What is the upper limit of integration? in
5I
units (c) What is the integrand? (pull constants outside of the integral first, just give the function to be integrated) A) 1 (a constant value) B)
L
r
​
+1
C)
L
r 2
​
+r
D)
L
r 3
​
+r 2
E)
r
F)
r 2
G)
p 3
(d) What is the variable of integration? m) r I t none of these (e) Calculate the rotational inertia. (Carry out the integral, include any constants in front of the integral, plug in limits, simplify your answer etc.) in 5 ! units