7.3 Solutions
299
7.3 In this system there is only one degree of freedom. The instantaneous configuration is specified by q = x. Assuming that the cord does not slip, the angular
velocity of the pulley is ˙
x/R, Fig. 7.14.
Fig. 7.14
The kinetic energy of the system is given by
T =
1
2
m 1 ˙
x
2
+
1
2
m 2 ˙
x
2
+
1
2
I
˙
x 2
R 2
(1)
The potential energy of the system is
V = −m 1 gx − m 2 g(l − x)
(2)
And the Lagrangian is
L =
1
2
m 1 + m 2 +
I
R 2
˙
x
2
+ (m 1 − m 2 )gx + m 2 gl
(3)
The equation of motion
d
dt
∂ L
∂ ˙
x
−
∂ L
∂ x
= 0
( 4 )
299
7.3 In this system there is only one degree of freedom. The instantaneous configuration is specified by q = x. Assuming that the cord does not slip, the angular
velocity of the pulley is ˙
x/R, Fig. 7.14.
Fig. 7.14
The kinetic energy of the system is given by
T =
1
2
m 1 ˙
x
2
+
1
2
m 2 ˙
x
2
+
1
2
I
˙
x 2
R 2
(1)
The potential energy of the system is
V = −m 1 gx − m 2 g(l − x)
(2)
And the Lagrangian is
L =
1
2
m 1 + m 2 +
I
R 2
˙
x
2
+ (m 1 − m 2 )gx + m 2 gl
(3)
The equation of motion
d
dt
∂ L
∂ ˙
x
−
∂ L
∂ x
= 0
( 4 )