1) The document presents equations to model the motion of a particle sliding down an inclined plane, including expressions for the particle's velocity and Lagrangian. 2) Lagrange's equations are applied to derive differential equations relating the accelerations of the particle along the inclined plane. 3) Solving these differential equations provides expressions for the accelerations of the particle along the plane in terms of the mass of the particle, inclination angle of the plane, and gravitational acceleration.

1. KASUS
“Particle sliding on a smooth inclined plane”
-Septiko Aji-
x1
m
M
x2
θ
We see that the square of the speed is given by cosines;
θ
2
2 2
 
v  x1  x 2  2 x1 x 2 cos 
T 
1
2


2 2
 
m x1  x 2  2 x1 x 2 cos  
1
2

M ( x2 )
2
V   mgx 1 sin 
Then, we know;
L  T V
 L
1
2


2 2
 
m x1  x 2  2 x1 x 2 cos  
1
2

M ( x 2 )  mgx 1 sin 
2
Lagrange’s equation;
d  L  L



dt   x   x
d  L

dt   x1
 
 L

 x
1

so, we have
d  L

dt   x1
 
d  L

dt   x1
 
d
 
 L

  x and
1

d  L

dt   x 2
 
 L

 x
1




m x1  x 2 cos   mg sin 
dt
 1  2 cos   g sin 
x
x
 1  g sin   2 cos 
x
x
 L

 x
2


2. d  L

dt   x 2
 
 L

 x
2

d  L

dt   x 2
 
d
dt
 L

 x
2


m x
2




 x1 cos   M x 2  0
 2 ( m  M )  1 cos   0
x
x
Solving for 1 and 2 we find;
x
x
g sin 
1 
x
m cos 
2
1
mM
2 
x
 g sin  cos 
mM
2
 cos 
m