Understanding Forces and Deformation: Hooke’s Law

Posted on Mar 17, 2025 in Physics

What are Forces?

Forces cause objects to change their position, and shifts cause deformations. Exercise by contact or at a distance.

Representation of Forces

It is not enough to state the value in Newtons; you should also specify the direction and sense. Therefore, forces are vector quantities and are represented graphically by vectors.

Characteristics of a Force

  • The strength or modulus is its value in Newtons.

  • Formula

    The direction is the straight line on which the force lies.

  • The sense is whether the force is directed towards one side or the other of the line.
  • The point of application is where the vector starts.

Forces and Deformation

Hooke’s Law

F = k * d, where:

  • F is the force applied.
  • k is the spring constant.
  • d is the deformation (elongation or compression).

The constant of proportionality, k, is specific to each spring and is expressed in Newtons per meter (N/m) in SI units. It’s called the elasticity constant, and its value indicates the force required to deform a spring by 1 meter.

Examples

A spring extends 5cm when a force of 60N is applied. Determine:

a) The spring’s elasticity constant.

We apply F = k * d. First, express the distance in meters:

5cm = 5cm * Formula m = 0.05 m

Then, substitute the data and solve for k:

60 N = 0.05 m * k => k = Formula N/m = 1200 N/m

b) The elongation when applying a force of 20N.

Reapply the same relationship, replacing the new data and the calculated value of k:

20 N = 1200 N/m * d => d = Formula m = 0.017 m = 1.7 cm

Sum or Composition of Forces

To add masses, it is enough to add their values: m1 + m2. But to combine forces, it not only depends on their value but also on their direction and sense. We proceed differently. The operation of combining forces is also called the composition of forces. The result of the sum or composition of two or more forces is called the resultant (R). Each force F1, F2, F3… that we add is a component: R = F1 + F2 + F3 + …. When we talk about concurrent forces, it means that the forces have the same point of application.

Sum of Parallel Forces

Same Direction

Add their moduli: R = F1 + F2. The direction and sense of the result are the same as their components.

Opposite Direction

R = F1 – F2 (if Formula ) The direction of the result is the same as the component with the greater modulus.

Sum of Non-Parallel Forces

Apply the parallelogram rule. First, represent the two forces. Join them so they are concurrent. Then, from the end of each force, draw a parallel line to the other force, with equal length. This forms a parallelogram. The diagonal of the parallelogram represents the resultant force.

This is a graphic method. It measures the value of the resultant, but we can only calculate it in simple special cases, such as when the forces are perpendicular.

Resultant of Perpendicular Forces

Applying the parallelogram rule to two perpendicular forces, we observe that the diagonal of the rectangle is the hypotenuse of either of the two right triangles that make up the rectangle. By the Pythagorean theorem, we know that the value of the hypotenuse, which in this case is the modulus of the resultant, is:

Formula

Examples

1. Calculate the resultant of two forces of 50N and 80N, parallel and with the same sense.

The result is obtained by adding the two forces: R = F1 + F2 = 50N + 80N = 130N.

2. Calculate the resultant of two forces of 90N and 30N, parallel and in opposite directions.

The result is obtained by subtracting the two forces: 90N – 30N = 60N.

3. Two forces of 20N and 25N form a 45-degree angle. Calculate: a) The modulus of their resultant. b) The angle that this resultant force forms with the 25N force.

Take as a scale that 2mm is equivalent to 1N.

Draw a horizontal force of 25N (the representation is simpler). Then, draw the 20N force with the required length and forming a 45-degree angle with the horizontal. Then, apply the parallelogram rule and draw the diagonal. Measure this diagonal with a graduated ruler and the angle with a protractor. The results are:

a) Length = 83mm. Since the scale is 1N = 2mm, the modulus of the result is: R = 83mm * Formula N = 41.5 N.

b) Angle = 20°