What is Magnitude, and how to calculate it?

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Magnitude definition

In physics, objects in motion have Magnitude.

Magnitude describes the quantity or size of that motion. It is used to describe the quantity or size of objects to other objects. For example, the speed of a bicycle will be less than that of an aero plane; thus, the bicycle is said to have less magnitude than the aero plane. Moreover, magnitude relates to different applications, as seen in the topics of force, displacement, and gravity. Also, the magnitude has no units and is said to be a unit less quantity.

Magnitude can be calculated with

  • Trigonometry using the Pythagorean method
  • Geometry using a ruler
  • Algebra by adding, subtracting, multiplying, and dividing

The approach depends on whether the magnitude required is from a vector application or a scalar one. Vectors, such as displacement and force, require that magnitude and direction be known and are calculated using either the Pythagorean theorem or a ruler. In contrast, scalars, such as temperature and mass, require that the magnitude be calculated algebraically without regard for direction. The options are better explained below:

Magnitude of vectors

The Pythagorean method

This method requires that both the adjacent and opposite sides are known. The magnitude can be obtained by (1) taking the square of both the adjacent and opposite sides separately.

a^{2} + b^{2} = c^{2}

(2) adding up the result, and (3) taking the square root of the result.

c = \sqrt{a^{2} + b^{2}}

The geometric method

This method requires a ruler to measure the resultant side directly. It is best advised to draw the diagram on graph paper as each square can be scaled to the appropriate measurement.

Graph paper image by mathsbit.com

The inclusive squares can be summed, or the resultant side can be measured with a ruler. Additionally, it is advised to take several measurements, add them up, and divide them by the mean amount for an optimal approximation.

Because the value of magnitude changes with direction, it can be more or less depending on where the vector is pointing to. With a vector, the magnitude will take the convention of the XY plane.

Vectors pointing North or East will be counted as positive, whilst vectors pointing South and West will be calculated as negative. Likewise, a positive value in magnitude will be added to the total, whilst a negative value in magnitude will be subtracted from the total.

Some examples of how Magnitude relates to vectors


The Magnitude in Force

  • Is the sum of all forces on the object.
  • Depends on the mass and acceleration of each force vector
  • Has formula F=ma

The Magnitude in Velocity

  • Is the sum of all velocities that the object undergoes.
  • Depends on the displacement and time of each vector
  • Has the formula v= \frac{\Delta d}{\Delta t}

See How to calculate momentum – Easy To Calculate for more examples of how vectors can be used in similar applications.

The Magnitude in Gravity

  • Is the sum of all the gravitational forces on the object.
  • Depends on the gravitational constant, each unit of mass, and the distance between each of the masses
  • Has formula f= \frac{G m_{1} m_{2}}{r^{2}}

More information can be found here: How To Calculate Velocity – Easy To Calculate

Magnitude of scalars

The Algebraic method

This method requires that the Magnitude be algebraically calculated to its appropriate quantity (adding, subtracting, dividing, or multiplying). Hence, the Magnitude will be the quantity scaled up to a larger number or down to a smaller number.

Some examples of how Magnitude can be used with the algebraic method can be seen in the following applications:

Magnitude in Heat

  • Is the sum of an object’s total heat.
  • Depends on the mass, specific heat, and temperature change.
  • Has the formula Q= mc\Delta t

Magnitude in Speed

  • Is the sum of all speeds the object undergoes.
  • Depends on the distance and time.
  • Has the formula s=\frac{\Delta x}{\Delta t}

Examples and applications

How to calculate Magnitude in velocity using the Pythagorean or the geometric method

Pythagorean method

  1. Calculate the average velocity by dividing each distance by the time taken to complete that distance.
  2. Draw the vectors by drawing subsequent vectors from the tail of the previous vectors.

3. Take the square of both the adjacent and the opposite sides.

vector_{1}^{2},vector_{2}^{2}

4. Find the square root of the result in step 3

Resultant vector = \sqrt{vector_{1} ^{2} + vector _{2}^{2}}

Alternatively,

  1. Draw the distance vectors
  2. Calculate the resultant Magnitude
  3. Divide this answer obtained in step 2 by the time taken to get the average velocity.

Geometric method

1. Calculate the velocity for each of the adjacent and opposite vectors.

v_{adjacent} = \frac{x_{adjacent}}{t_{adjacent}}
v_{opposite} = \frac{x_{opposite}}{t_{opposite}}

2. Determine an appropriate scale for each square. (Note: if each square represents 1 m/s, a 1 m/s   arrow should cover only 1 square.)

3. Draw the first vector (adjacent) and cover the correct number of squares.

4. Draw the second vector (opposite) starting from the first vector’s arrow.

5. Use a ruler to measure the resultant vector from the first vector’s tail to the final vector’s head (see example below) or count the squares:

Alternatively,

  1. Determine an appropriate scale for each square.
  2. Draw the first vector.
  3. Draw the second vector starting from the first vector’s head.
  4. Measure the resultant distance.
  5. Convert this distance to velocity by dividing by the total time.

Example:

A man pulls a cart 10 meters northwards for 10 seconds. He then changes direction and pulls the cart more slowly for 5 meters eastwards for 15 seconds, where he gets tired and stops for a drink. Calculate the cart’s Magnitude in velocity for this period by using (a) the Pythagorean method; and (b) the geometric method.

a.).

v_{North} = \frac{10}{10} = 1 m/s
v_{East} = \frac{5}{10} = 0.33 m/s
  1.  

3.

v_{result} = \sqrt{1^{2}+ 0.33^{2}}

= 1.04 m/s

b.).

  1. Because the velocity is relatively small, and to accommodate for the 0.33, we can instead scale it by a factor of 10 such that 1 m/s takes up 10 squares instead of 1:

2. If counting the squares of the resultant magnitude proves difficult, it is best advised to use a ruler to measure this magnitude and calculate its correct square value in the following four sub-steps:

(i) Measure the size of 1 square.

(ii) Measure the magnitude of the resultant vector.

(iii) Divide the magnitude length by the length of the square to get the correct scaling factor.

(iv) Multiply the velocity for 1 square by this factor to obtain the correct resultant magnitude.

How to calculate Magnitude in speed

Now calculate the Magnitude of the average speed (a scalar) insteadby using the algebraic method (Note: the direction does not affect the outcome).

To ensure a smoother process, follow the steps below:

  1. Calculate the total distance travelled.
  2. Calculate the total time taken to travel that distance.
  3. Use the following formula to find the speed:  average speed = total distance travelled/ total time taken.

Algebraic method:

To solve the problem, we are told that to calculate the average speed; it is important to know both distance travelled, and time as follows:

average speed = total distance travelled/total time taken 

= \frac{(10+5)}{(10+15)} = \frac{15}{25} = \frac{3}{5} m/s

So, the Magnitude of the average speed is 2.67 (excluding the units).


Important points to keep in mind

● Magnitude is a relative measure; it estimates how an object relates to other objects by comparing its size or quantity.
● Vectors have both Magnitude and direction (e.g., displacement and force).
● Scalars have only Magnitude (e.g., heat and speed), and the direction is unrelated.
● If the Magnitude is calculated as positive, the number is added; if the Magnitude is calculated to be a negative number, the number gets subtracted.

What is Magnitude, and how to calculate it?

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