Let the mass of the car and wheels be M. Let the mass of each wheel be m.
CLASS PROBLEMS [9.5] and [9.6] :
h
Suppose you can choose wheels of any design for a soapbox derby race car.
If the total weight of the vehicle is fixed, which type of wheels should you
The initial total energy at the top of the incline is Mgh.
choose if you want to have the best chance to win the race?
At the bottom of the incline the total energy is the sum of the translational
A: Large, massive wheels.
B:
Small, light wheels. <===
C:
Either, since it doesn’t matter.
... to stand the best chance of winning you would choose ...
kinetic energy of the car plus wheels and the rotational energy of the wheels,
1
"1
%
i.e., Mv 2 + 4$ I! 2 ' ,
#2
&
2
assuming 4 wheels. By the conservation of energy
1
"1
%
Mv 2 + 4$ I! 2 ' = Mgh .
#
&
2
2
With no slip, ! = v R , where R is the radius of the wheels.
1
v2
( Mv 2 + 2I 2 = Mgh .
2
R
A: solid wheels with the mass uniformly distributed. <===
B: wheels with most of the mass at the rim.
C: It doesn’t matter what type of wheel you use.
Multiply through by 2 and divide through by M and we get
2
v + 4I
v2
4I %
2"
= 2gh = v $ 1 +
'.
2
# MR 2 &
MR
2
Now, write the moment of inertia of each wheel as I = !mR , where
! = 1 2 for disk-like wheels and ! = 1 for hoop-like wheels. Then
"
4!m %
2 " M + 4!m %
v 2 $ 1+
' = 2gh = v $
',
#
#
M &
M &
"
2gh
M %
= 2gh$
'.
" M + 4!m %
# M + 4!m &
$
'
#
M &
i.e., v 2 =
so for maximum speed,
M
needs to be as large as possible.
M + 4!m
v2
2gh
! = 12
!=1
(
0
m (kg)
2
Plot of v 2 versus
4
6
8
M
with M = 50 kg.
M + 4!m
Three things we note from this plot. For the greatest speed:
• Make m as small as possible ([9.5]).
• Use disk-like wheels, with ! = 1 2 , not hoop-like wheels ! = 1 ([9.6]).
• The result is independent of R, the radius of the wheels.
Note that v 2 = 2gh , i.e., v = 2gh is the speed of an object sliding down the
incline with no friction.