Homework 5
−
Solution
1. (a) The system is shown here again with additional variables:
k
O
m
m
r
4
r
k
T
A
θ
B
x
φ
The total kinetic energy has three parts: rotational kinetic energy of the rod, as well as
translational and rotational kinetic energies of the disk:
T
1
2
J
0
̇
2
1
2
mx
̇
2
1
2
J
̇
2
The potential energy also has three parts: deformation in the linear spring, twisting in the
torsional spring, and the rod’s gravitational potential energy:
U
1
2
k
T
2
1
2
kx
2
mg
2
r
cos
Note that the variable
x
is the displacement at the center of the disk, and the angle
is the
rotation of the disk. We can relate the motion of the disk to the rotation of the rod from
kinematics since the disk rolls without slipping. Both points
A
and
B
have the same
displacement:
4
r
2
r
2
The displacement at the disk’s center for rolling without slipping is
x
r
2
r
So the total energy can now be written as
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 Spring '08
 TSChang
 Energy, Kinetic Energy, Potential Energy, mgl cos

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