Download The Spring Pendulum

The Spring Pendulum
By Nick Whitman
A Model of the Spring Pendulum
10
0
-10
z
Spring Force
-20
Air Resistance
-20
Velocity
-30
0
-40
-10
Weight
20
0
10
20
y
30
40
x
The Forces Acting on the Mass

  
Fnet  Fg  Fs  Fr
The Force Due to Gravity

ˆ
Fg   MGk
Unit Vector in the Direction of the Spring Force
ˆf 
s
1
x y z
2
2
2

ˆ
ˆ
ˆ
xi  yj  zk

The Magnitude of the Spring Force
Fs   K

x  y z L
2
2
2

The Spring Force



L
Fs  K 
 1 xiˆ  yˆj  zkˆ
 x2  y 2  x2





The Force Due to Air Resistance


ˆ
ˆ
ˆ
Fr   R xi  yj  zk

Adding the Forces, Separating into Forces on Each
Axis, and Dividing by Mass
Kx Rx
x 


2
2
2
M M
M x y z
KLx
Ky Ry
y 


2
2
2
M M
M x y z
KLy
Kz Rz
z 


G
2
2
2
M M
M x y z
KLz
Example With No Damping
10
0
z
-10
-20
-30
-40
-50
4
2
10
5
0
0
-2
y
-5
-4
-10
x
Another Without Damping
50
0
z
5
-50
-100
0
-150
-8
-6
-4
-2
0
2
4
x
6
8
-5
y
Same as Last Over a Longer Time
10
50
5
z
0
-50
0
-100
-5
-150
x
5
0
y
-5
-10
No Damping Over a Long Time From Above
10
10
5
5
0
0
-5
-5
y
x
-10
-10
With Damping
10
0
z
-10
-20
-30
-40
4
2
5
0
0
-2
y
-4
-5
x
Plots of the Motion on Each Axis vs. Time, With Damping
x
5
0
-5
0
50
100
150
t
200
250
300
0
50
100
150
t
200
250
300
0
50
100
150
t
200
250
300
y
10
0
-10
z
50
0
-50