Vector Scale Diagrams

CSEC Physics Syllabus - Effective for examinations from May - June 2015

Section A - Mechanics

Vectors

Specific Objective 2.2

use scale diagrams to find the resultant of two vectors;

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Scale Diagrams

Scale diagrams are vector diagrams drawn using a suitable scale. Two or more vectors can be summed to one vector called a Resultant Vector. The resultant vector is the combined effect of all the vectors acting on a point or object. Scale Diagrams, is an accurate method of finding the Resultant Vector of two vectors. The angle of action of the resultant vector is taken from the horizontal axis.  

 How is this done?

Using a ruler, protractor and either of the following methods: 

1. Head to Tail Method
2. Parallelogram Method

Head to Tail Method

Example 1: 

Find the Resultant Vector, R and the angle of action, θ of two vectors, 30.0 m east and 20.0 m south.

Solution: 

Using a scale of 2.0 cm for 10 m, we will draw each vector head to tail:

Step 1: 30.0 m East vector

Step 2: 20.0 m South vector

Step 3: We now draw the Resultant Vector, R, from the tail of the first vector to the the head of the second vector. We identify the Resultant Vector with two arrow heads. The magnitude and direction of the Resultant Vector by using the ruler to measure its length and the protractor to measure its angle from the horizontal respectively.

Measurements & Calculation:
The Resultant Vector, R is acting at an angle, θ of 326° from the horizontal. This angle is equivalent to -34° from the horizontal (the angle of rotation will be clockwise).

The Resultant Vector was measured to be 7.2 cm. Using the scale of 2.0 cm for 10.0 m, the Resultant Vector was determined using the following conversion:

Therefore, the Resultant Vector, R was determined to be 36.0 m at an angle of 326° from the horizontal.

Example 2:

Find the Resultant Vector, R, of two vectors, 15 N east and 10 N at an angle of 45° from the horizontal. 

Solution: 

Using a scale of 2.0 cm for 5 N, we will draw each vector head to tail:

Step 1: 15 N East vector

Step 2: 10 N vector at an angle of 45° from the horizontal

Step 3: We now draw the Resultant Vector, R from the tail of the first vector to the the head of the second vector. We identify the Resultant Vector with two arrow heads. The magnitude and direction of the Resultant Vector by using the ruler to measure its length and the protractor to measure its angle from the horizontal respectively.

Measurements & Calculations:
The Resultant Vector, R is acting at an angle, θ of 18° from the horizontal.
The Resultant Vector was measured to be 9.3 cm. Using the scale of 2.0 cm for 5 N, the Resultant Vector was determined using the following conversion:

Therefore, the Resultant Vector, R was determined to be 23.25 N at an angle of 18° from the horizontal.

Parallelogram Method

Example 1:

Find the Resultant Vector of two vectors, 25 km west and 20 km 255° from the horizontal.

Solution:

Using a scale of 1.0 cm for 5 km, we will draw each vector tail to tail:

Step 1: 25 km West vector

Step 2: 20 km vector at an angle of 255° from the horizontal

Step 3: We draw dashed lines parallel to each vector, which completes a parallelogram.

Step 4: We now draw the Resultant Vector, R, which is a diagonal line from the common tail of both vectors to the opposite vertex. We identify the Resultant Vector with two arrow heads. The magnitude and direction of the Resultant Vector by using the ruler to measure its length and the protractor to measure its angle from the horizontal respectively.

Measurements & Calculation:
The Resultant Vector, R is acting at an angle, θ of 213° from the horizontal.
The Resultant Vector was measured to be 7.3 cm. Using the scale of 1.0 cm for 5 km, the Resultant Vector was determined using the following conversion:

Therefore, the Resultant Vector, R was determined to be 36.5 km at an angle of 213° from the horizontal.

Example 2:

Find the Resultant Vector of two vectors, 18 N north-west and 15 N at an angle of 25° from the horizontal.

Solution:

Using a scale of 1.0 cm for 3 N, we will draw each vector tail to tail.

Step 1: 18 N North-West vector

Step 2: 15 N vector at an angle of 25° from the horizontal

Step 3: We draw dashed lines parallel to each vector, which completes a parallelogram.

Step 4: We now draw the Resultant Vector, R, which is a diagonal line from the common tail of both vectors to the opposite vertex. We identify the Resultant Vector with two arrow heads. The magnitude and direction of the Resultant Vector by using the ruler to measure its length and the protractor to measure its angle from the horizontal respectively.

Measurements & Calculation:
The Resultant Vector, R is acting at an angle, θ of 88° from the horizontal.
The Resultant Vector was measured to be 6.4 cm. Using the scale of 1.0 cm for 3 N, the Resultant Vector was determined using the following conversion:

Therefore, the Resultant Vector, R was determined to be 19.2 N at an angle of 88° from the horizontal.