The Spring Pendulum
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Transcript 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 xi yj zk
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