Hooke’s Law

Vinutna Veeragandham

Hooke's Law: the force needed to extend or compress a spring by some distance is proportional to that distance.

The Main Idea

Hooke's Law demonstrates the relationship between forces applied to a spring and elasticity. It states that the force needed to extend or compress a spring by some distance is proportional to that distance. This law applies to many different materials such as balloons or strings; an elastic body to which Hooke's law applies is known as linear-elastic. Hooke's Law has been the basis for the modern Theory of Elasticity, led to creation of new inventions as well as been the foundation of many different branches of science such as seismology, molecular mechanics and acoustics.

A Mathematical Model

   F = -kX

F - restoring force, force by which the free end of the spring is being pulled, SI Units: Newtons

k - spring constant, an inherent property of the string, SI Units: Meters

X - spring displacement from the spring's free end while at equilibrium position, SI Units: Newtons/Meters

A Computational Model

A computational representation of Hooke's Law can be created using VPython. The code below puts Hooke's law in action.

   # GlowScript 1.1 VPython
   
   L0=.1 #the natural length of the spring
   k=15 #spring constant
   
  #the holder is the top plate for the spring
   holder= box(pos=vec(-.1,.1,0), size=vec(.1,.005,.1))
   
  #the ball and the spring should be obvious 
   
  #note that the ball has a trail
   ball=sphere(pos=vec(-.1,-L0+.1,0), radius=0.02, color=color.red, make_trail=true)
   spring=helix(pos=holder.pos, axis=ball.pos-holder.pos, radius=.02, coils=10)
   ball.m=0.1 #mass of the ball in kg
   ball.p=ball.m*vec(0,0,0) #starting momentum mass times velocity
   dr=vec(0,-0.05,0) #this is the displacement of the spring
   ball.pos=ball.pos+dr
   g=vec(0,-9.8,0) #gravitational field
   t=0
   dt=0.01 #size of the time step
   
  #putting the loop as "while True" means it runs forever
   
  #you could change this to while t< 10: or something 
   while t<3: 
   rate(100) #this says 100 calculations per second
   
  #L is the length of the spring, from the holder to the ball
   L=ball.pos-holder.pos
   
  #remember that mag(R) gives the magnitude of vector R
   
  #remember that norm(R) gives the unit vector for R
   Fs=-k*(mag(L)-L0)*norm(L) #this is Hooke's law
   
  #this calculates the net force
   F=Fs+ball.m*g
   
  #update the momentum
   ball.p=ball.p+F*dt
   
  #update the position of the ball
   ball.pos= ball.pos +ball.p*dt/ball.m
   
  #this next line updates the spring
   spring.axis=ball.pos-holder.pos

Examples

Simple

A spring has a spring constant that is equal to 3.5. What force will make it stretch 4 cm?

 F = k * x
 F = 3.5 * 4
 F = 14 kg

Middling

A spring has a natural length of 20 cm. A 40 N force is required to stretch (and hold the spring) to a length of 30 cm. How much work is done in stretching the spring from 35 cm to 38 cm?

F = kx 40 = ((30 - 20)/100)k 40 = 0.10k k = 400

F(x) = 400x

F(x) = \int_0.15^0.18\! 400x\,dx