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]

The spring force can be computed with a loop that updates the position of an object and an attached spring. Using a small enough time step, these quantities can be calculated to a reasonable degree of accuracy. The computational approach to spring force problems can be extremely useful, as the spring force is non-constant. For every change in position, which the spring force affects, the spring force changes. Because of this nature of the spring force, iterative calculations prove rather useful in its calculation. The code below demonstrates how knowing the value of the spring stiffness and relaxed length constants and keeping track of the spring position can be used to calculate and update the the spring force.

#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
    

Connectedness

1. Springs harmonic oscillations can be entrancing, and the forces behind its motion can be just as compelling.
2. My major being CS, has many applications in modeling systems. The motion of springs is a very interesting system to model as its motion can both be harmonic and chaotic depending on the scenario.
3. Springs are often used in bungee jumping and car suspension systems.

History

The concept of spring force was first discovered in 1660 by Robert Hooke. This insight would lead Hooke to devise plans for a watch with a balance spring, that would make pocket watches much more accurate.

See also

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

Further reading

Books, Articles or other print media on this topic

External links

[2]

References

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