Work Done on Spring

The Work Done on a Spring calculator computes the work (W) to further elongate or compress a spring based on the spring constant (k) and the initial and final positions of the spring.

INSTRUCTIONS: Choose units and enter the following:

• (k) Spring constant in Newtons per meter (N/m)
• (x₁)  Initial position of Spring
• (x₂)  Final position of Spring

Work to Elongate or Compress a Spring (W): The calculator returns the work in Newton meters (N•m).   However, this can be automatically converted to compatible units (e.g. Joules) via the pull-down menu.

The Math / Science

If you integrate the force (F) on spring over a distance, you get the following equation.

        `W = int_(x_i)^(x_f) F_x  dx =  int_(x_i)^(x_f) kx dx = 1/2 kx_2^2 - 1/2 kx_1^2`

where:

• k is the spring constant
• x_i is the initial position of the spring
• x_f is the final position of the spring

This equation is very similar in form to the equation for the potential energy of the spring and is often confused with the potential energy equation.

Work is defined to be the energy transferred by a force and mathematically work is defined in the simplest case where the force is constant to be: Work = Force * Distance.

For example: to move a mass, to just barely get it moving, might require a force of n Newtons.  If we continued to apply that force of n Newtons to move the mass some distance, d meters, then he work done would be W = n*d Joules

However, in this case of a force applied to a spring, the force is not constant.  The Force is defined to be linearly increasing with the distance, x:   `F= k*x`

Spring Equation Calculators

• Period of an Oscillating Spring: This computes the period of oscillation of a spring based on the spring constant and mass.
• Mass of a Spring: This computes the mass based on the spring constant and the period of oscillation. 
• Angular Frequency of a Spring: This computes the angular frequency based on the spring constant and the mass.
• Spring Constant: This computes a spring's constant based on the mass and period of oscillation.
• Work done on a Spring: This computes the work based on the spring constant and the two positions of a spring.
• Hooke's Law: This computes the force to change the length of a spring based on the spring constant and length of displacement.
• Force to Fully Compress a Spring: This computes the force required to fully compress a spring based on the spring's physical attributes including the Young's Modulus, wire diameter, length of spring, number of windings, Poisson ratio, and outer diameter of the spring.

References

• Khan Academy's Introduction to work and energy
• University Physics 12th Edition, Chapter 6, Equation #6.10