How to Calculate the Magnitude of Force in Physics

Have you ever wondered how you are able to jump? Understanding the basics of forces can help you to explain, because what goes up doesn't have to come down if you know how.
Have you ever wondered how you are able to jump? Understanding the basics of forces can help you to explain, because what goes up doesn't have to come down if you know how. | Source

For some reason, whether for school or out of a personal curiosity you would like to know how to calculate the magnitude or size of a force. But before getting into the Math of it all, the philosophy needs to be understood. Isaac Newton, the famous Physicist, Philosopher and discreet Alchemist, has laid the foundation for the understanding of forces and how to calculate them. His famous laws of motion and gravitation uncover the effect of forces in our Universe, but yet still there are things not understood for e.g. the inner workings of magnetic forces.

As you may already know, forces don’t just come out of thin air, and they are in a sense an expression of bodies whether living or non-living. Bodies may interact with each other through forces, and so you will not have one body acting on another without the other body reacting. This action and reaction concept is wrapped up in Newton’s 3rd Law of Motion, where he implies that:

If a body A exerts a force on a body B, then Body B will react with an equal and opposite force on Body A.

Your imagination can run wild with this principle as there are tons of examples, but the core of the matter is that forces act in pairs. There’s another principle to keep in mind, and that is the idea of the resultant or net force. At any one time, there can be two or more external forces acting on a single body. Moreover, all the external forces acting on that single body can be represented by a single resultant force having a particular magnitude and direction. Forces are vectors, and so even though you may calculate the size of a force there is also a directional path that you need to keep in mind. Since forces are vectors, they can be represented with arrows. And what is a Force without a unit – Newton (N) is the unit for a Force, of which the kilogram meter per second squared (kgms-2) is its equivalent.

Resolving Forces

The most basic calculation of forces involves vector resolution. Here’s an example:

The magnitudes of two forces A and B are 12 N and 8 N, respectively. What is the resultant force if:

(i) A and B are parallel.

Resultant Force

= A + B

= 12 N + 8 N

= 20 N (going East)

To resolve parallel forces (incl. other kinds of vectors), simply find the sum of the vectors.
To resolve parallel forces (incl. other kinds of vectors), simply find the sum of the vectors. | Source

(ii) A and B are anti-parallel.

Resultant Force

= A + B

= 12 N + (– 8 N)

= 4 N (going East)

Forces (and other vectors) that head East  and South (left and down) are positive, while forces that go West and North (right and up) are negative.
Forces (and other vectors) that head East and South (left and down) are positive, while forces that go West and North (right and up) are negative. | Source
Source

(iii) A and B are at a 90 degree angle to each other.

Resultant force

= √(A2 + B2) {Using Pythagoras Theorem}

= 14.4 N

We now need to find out the direction of this force...

Direction of Resultant Force

Trigonometrical ratios need now be applied.

Using the SINE ratio: (NB: You can use other ratios to find the angle)

sine θ = Opposite / Hypotenuse

sine θ = 12 N / 14.4 N

θ = sine-1(12 / 14.4)

θ = 56.4 degrees (from the horizontal; from force B)

Source

(iv) A and B are at 30 degrees to each other.

In this case, trigonometry again has to be applied.

STEPS:

(a) Find the horizontal and vertical component of each force.
  • Vertical Component of 8 N Force = sine 0 degrees x 8 N = 0 N
  • Vertical Component of 12 N Force = sin 30 degrees x 12 N = 6 N
  • Horizontal Component of 8 N Force = cos 0 degrees x 8 N = 8 N
  • Horizontal Component of 12 N Force = cos 30 degrees x 12 N = 10.4 N

(b) Find the sum of the horizontal components, and the sum of the vertical components (thereby simplifying the problem to that of part (iii)).
  • Sum of Vertical Components = 0 N + 6 N = 6 N
  • Sum of Horizontal Components = 8 N + 10.4 N = 18.4 N

(c) Use Pythagoras Theorem to determine the resultant force.
  • Looking back on part (iii), you can use the same process to find the resultant force as well as its angle of direction. The only difference of course is that you will have A as 6 N and B as 18.4 N.
  • Try the calculation, and you should get the Resultant Force to be 19.4 N with it's direction from the horizontal being 18 degrees.

Formulae for Forces

Forces are involved in more situations within the realm of Mechanics and without. Take for instance the following formulae:

Force = Torque (Moment of Force) / Distance (of Force from pivot/fulcrum)

Force = Mass x Acceleration

Force = Change in Momentum / Time

Force = Impulse / Time

Force = Work (Mechanical Energy) / Time

Now that you may have aquired a basic understanding of the principles of forces as well as calculating forces in parallel, anti-parallel, right-angled and other-angled situations try out these couple questions to test how well you're progressing.

Micro-Quiz: Resolving Forces

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