Spring Force

Claimed by Arjun Chib

Spring Force is the non-constant, elastic force exerted by a spring upon a system.

The Main Idea

The spring force models the force in a system due to the presence of a stretched or compressed spring. This force is based upon two factors of the spring: the spring's stiffness and the distance the spring has been stretched. The spring's stiffness is a constant that represents how much force is required to stretch or compress a spring over a certain distance.

A Mathematical Model

The magnitude of the spring force is represented by the equation [math]\displaystyle{ \vert \vec{F}_{spring} \vert = k_s \vert s \vert }[/math], where [math]\displaystyle{ \vert s \vert }[/math] is the absolute value of the stretch of the spring [math]\displaystyle{ s = L - L_0 }[/math].

[math]\displaystyle{ L_0 }[/math] is the relaxed length of the spring, when the spring is neither stretched nor compressed.

[math]\displaystyle{ L }[/math] is the length that the spring after it has been stretched or compressed.

[math]\displaystyle{ k_s }[/math] is the spring stiffness, which is a constant inherent to the property of the spring.

The spring force can also be modeled as a vector by the equation [math]\displaystyle{ \vec{F}_{spring} = -k_s s \hat{L} }[/math], where [math]\displaystyle{ \hat{L} }[/math] is the direction that the spring is stretched or compressed.

A Computational Model

A GlowScript model of a spring that prints out the spring force vector: [1]

#GlowScript 1.1 VPython

L0 = 0.1 #the relaxed length of the spring
ks = 15 #spring constant

#plate holding spring end
plate= box(pos=vec(-.1,0,0), size=vec(.005,.1,.1))

# ball and spring objects
ball=sphere(pos=vec(-L0+.1,0,0), radius=0.02, color=color.red, make_trail=true)
spring=helix(pos=plate.pos, axis=ball.pos-plate.pos, radius=.02, coils=10)

ball.m = 0.1 #mass of the ball in kg
ball.p = ball.m * vec(0.5,0,0) #initial momentum

t = 0 #time
dt = 0.01 #size of the time step

#loops forever
while True: 
    rate(10) #100 calculations per second
    
    #length of the spring
    L = ball.pos - plate.pos
    
    #spring force
    Fs = -ks * (mag(L) - L0) * norm(L)
    
    #update the momentum of the ball
    ball.p = ball.p + Fs * dt
    
    #update the position of the ball
    ball.pos= ball.pos + ball.p * dt / ball.m
    
    #update the spring
    spring.axis = ball.pos - plate.pos
    
    print(Fs)

Examples

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[2]

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