7.3 Solutions
303
system is fixed on the smooth table, Fig. 7.16. The x- and y-components of the
velocity of the block are given by
Fig. 7.16
v x = ˙
x
+ ˙
x cos θ
(1)
v y = − ˙
x sin θ
(2)
∴ v
2
= v
2
x + v
2
y = ˙
x
2
+ 2 ˙
x ˙
x
cos θ + ˙
x
2
(3)
Hence, the kinetic energy of the system will be
T =
1
2
M ˙
x
2
+
1
2
m( ˙
x
2
+ 2 ˙
x ˙
x
cos θ + ˙
x
2
)
(4)
while the potential energy takes the form
V = −mgx sin θ
(5)
and the Lagrangian is given by
L =
1
2
M ˙
x
2
+
1
2
m( ˙
x
2
+ 2 ˙
x ˙
x
cos θ + ˙
x
2
) + mgx sin θ
(6)
The equations of motion
d
dt
∂ L
∂ ˙
x
−
∂ L
∂ x
= 0
( 7 )
d
dt
∂ L
∂ ˙
x
−
∂ L
∂ x = 0
( 8 )
yield
m( ¨
x
cos θ + ¨
x) − mg sin θ = 0
( 9 )
M ¨
x
+ m( ¨
x
+ ¨
x cos θ) = 0
(10)