MyPhysicsLab – 2-Dimensional Spring

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Instructions: drag square or circle with your mouse

Does the motion look random to you? Watch the graph for a while and you'll see its actually an intricate pattern.

Physics of the 2-Dimensional Spring

An immoveable (but draggable) anchor point has a spring and bob hanging below and swinging in two dimensions. We regard the bob as a point mass. We define the following variables:

• θ = angle (0=vertical)
• S = spring stretch (displacement from rest length)
• L = length of spring
• u = position of bob
• v = u'= velocity of bob
• a = u''= acceleration of bob

and some constants:

• R = rest length of spring
• T = position of anchor point
• m = mass of bob
• k = spring constant
• b = damping constant
• g = gravitational constant

Note that for this simulation the vertical dimension increases downwards.

We'll need the standard unit vectors, i,j. We use bold and overline to indicate a vector.

• i = unit vector in horizontal direction
• j = unit vector in vertical (down) direction

There are three vector forces acting on the bob:

F_g = m g j = gravity acting straight down
F_s = -k S (sin θ i + cos θ j) = the spring pulling (or pushing) along the line from bob to anchor point.
F_b = -b (v_xi + v_yj) = damping (friction) acting opposite to the direction of motion of the bob, ie. opposite to its velocity vector.

Summing these forces and using Newton's second law we get:

m a = F_g + F_s + F_b
m (a_xi + a_yj) = m g j - k S (sin θ i + cos θ j) -b (v_xi + v_yj)

We can write the vector components of the above equation as separate equations. This gives us two simultaneous equations. We also divide each side by m.

a_x = -(k/m) S sin θ - (b/m) v_x    (eqn 1)
a_y = g - (k/m) S cos θ - (b/m) v_y

These are essentially the equations of motion. It only remains to show how S sin θ and S cos θ are functions of the position of the bob. The displacement of the spring S is the current length of the spring minus the rest length.

S = L - R

The length is easily derived using the pythagorean theorem from the position of the bob, u, and the position of the anchor point, T.

L = √((u_x - T_x)² + (u_y - T_y)²)

The sine and cosine of the angle are given by the following:

sin θ = (u_x - T_x)/L
cos θ = (u_y - T_y)/L

Numerical Solution of the 2-Dimensional Spring

To solve the equations of motion numerically, so that we can drive the simulation, we use the Runge-Kutta method for solving sets of ordinary differential equations. We only need to convert the two second order equations (1) to four first order equations.

u_x' = v_x
u_y' = v_y
v_x' = -(k/m) S sin θ - (b/m) v_x
v_y' = g - (k/m) S cos θ - (b/m) v_y

We also keep in mind that S sin θ and S cos θ are functions of the position of the bob, u_x, u_y, as given above.