338
7 Lagrangian and Hamiltonian Mechanics
∴ T (system) =
7
10
m( ˙
x
2
+ ˙
y
2
+ 2 ˙
x ˙
y cos α) +
1
2
(M + m) ˙
x
2
(6)
V (system) = V (ball) = −mgy sin α
(7)
L =
7
10
m( ˙
x
2
+ ˙
y
2
+2 ˙
x ˙
y cos α)+
1
2
(M + m) ˙
x
2
+ mgy sin α (8)
Lagrange’s equations
d
dt
∂ L
∂ ˙
x
−
∂ L
∂ x
= 0,
d
dt
∂ L
∂ ˙
y
−
∂ L
∂ y
= 0
( 9 )
become
7m
5
¨
x +
7
5
m ¨
y cos α + (M + m) ¨
x = 0
(10)
7m
5
¨
y +
7
5
m ¨
x cos α − mg sin α = 0
(11)
Solving (10) and (11) and simplifying
¨
x = −
5mg sin α cos α
5M + (5 + 7 sin
2
α)m
(for the wedge)
¨
y =
5(5M + 12m)g sin α
7(5M + (5 + 7 sin
2
α)m)
(for the ball)
For M = m and α = π/4
¨
x =
5g
27
¨
y =
85
√
2
189