Graphical Methods Continued
In a previous lesson, you learnt how to add vectors in one dimension graphically. We can expand these ideas to include vectors in two-dimensions. The following worked example shows this. Example on Finding the Magnitude of the Resultant in Two Dimensions ...
Graphical Techniques
In a previous lesson, you learnt how to add vectors in one dimension graphically. We can expand these ideas to include vectors in two-dimensions. The following worked example shows this.
Example: Finding the Magnitude of the Resultant in Two Dimensions Graphically
Question
Given two vectors, \(\vec{R}_y\) = \(\text{4}\) \(\text{N}\) in the positive \(y\)-direction and \(\vec{R}_x\) = \(\text{3}\) \(\text{N}\) in the positive \(x\)-direction, use the tail-to-head method to find the resultant of these vectors graphically.
Step 1: Choose a scale and draw axes
The vectors we have do not have very big magnitudes so we can choose simple scale, we can use \(\text{1}\) \(\text{N}\) : \(\text{1}\) \(\text{cm}\) as our scale for the drawing.
Then we draw axes that the vector diagram should fit in. The largest vector has length \(\text{4}\) \(\text{N}\) and both vectors are in the positive direction so we can draw axes from the origin to \(\text{5}\) and expect the vectors to fit.
Step 2: Draw \(\vec{R}_x\)
The magnitude of \(\vec{R}_x\) is \(\text{3}\) \(\text{N}\) so the arrow we need to draw must be \(\text{3}\) \(\text{cm}\) long. The arrow must point in the positive \(x\)-direction.
Step 3: Draw \(\vec{R}_y\)
The length of \(\vec{R}_y\) is \(\text{4}\) so the arrow we need to draw must be \(\text{4}\) \(\text{cm}\) long. The arrow must point in the positive \(y\)-direction. The important fact to note is that we are implementing the head-to-tail method so the vector must start at the end (head) of \(\vec{R}_x\).
Step 4: Draw the resultant vector, \(\vec{R}\)
The resultant vector is the vector from the tail of the first vector we drew directly to the head of the last vector we drew. This means we need to draw a vector from the tail of \(\vec{R}_{x}\) to the head of \(\vec{R}_{y}\).
Step 5: Measure the resultant, \(\vec{R}\)
We are solving the problem graphically so we now need to measure the magnitude of the vector and use the scale we chose to convert our answer from the diagram to the actual result. In the last diagram the resultant, \(\vec{R}\) is \(\text{5}\) \(\text{cm}\) long therefore the magnitude of the vector is \(\text{5}\) \(\text{N}\).
The direction of the resultant, \(\theta\), we need to measure from the diagram using a protractor. The angle that the vector makes with the \(x\)-axis is \(\text{53}\)\(\text{°}\).
Step 6: Quote the final answer
\(\vec{R}\) is \(\text{5}\) \(\text{N}\) at \(\text{53}\)\(\text{°}\) from the positive \(x\)-direction.
In the case where you have to find the resultant of more than two vectors first apply the tail-to-head method to all the vectors parallel to the one axis and then all the vectors parallel to the other axis. For example, you would first calculate \(\vec{R}_{y}\) from all the vectors parallel to the \(y\)-axis and then \(\vec{R}_{x}\) from all the vectors parallel to the \(x\)-axis. After that you apply the same procedure as in the previous worked example to the get the final resultant.
Example 2:
Question
Given the following three force vectors, determine the resultant force:
- \(\vec{F}_{1}\) = \(\text{3.4}\) \(\text{N}\) in the positive \(x\)-direction
- \(\vec{F}_{2}\) = \(\text{4}\) \(\text{N}\) in the positive \(x\)-direction
- \(\vec{F}_{3}\) = \(\text{3}\) \(\text{N}\) in the negative \(y\)-direction
Step 1: Determine \(\vec{R}_{x}\)
First we determine the resultant of all the vectors that are parallel to the \(x\)-axis. There are two vectors \(\vec{F}_{1}\) and \(\vec{F}_{2}\) that we need to add. We do this using the tail-to-head method for co-linear vectors.
The single vector, \(\vec{R}_{x}\), that would give us the same outcome is:
Step 2: Determine \(\vec{R}_{y}\)
There is only one vector in the \(y\)-direction, \(\vec{F}_{3}\), therefore \(\vec{R}_{y}\) = \(\vec{F}_{3}\).
Step 3: Choose a scale and draw axes
The vectors we have do not have very big magnitudes so we can choose simple scale, we can use \(\text{1}\) \(\text{N}\) : \(\text{1}\) \(\text{cm}\) as our scale for the drawing.
Then we draw axes that the diagram should fit on. The longest vector has length \(\text{7.4}\) \(\text{N}\). We need our axes to extend just further than the vectors aligned with each axis. Our axes need to start at the origin and go beyond \(\text{7.4}\) \(\text{N}\) in the positive \(x\)-direction and further than \(\text{3}\) \(\text{N}\) in the negative \(y\)-direction. Our scale choice of \(\text{1}\) \(\text{N}\) : \(\text{1}\) \(\text{cm}\) means that our axes actually need to extend \(\text{7.4}\) \(\text{cm}\) in the positive \(x\)-direction and further than \(\text{3}\) \(\text{cm}\) in the negative \(y\)-direction
Step 4: Draw \(\vec{R_x}\)
The magnitude of \(\vec{R}_x\) is \(\text{7.4}\) \(\text{N}\) so the arrow we need to draw must be \(\text{7.4}\) \(\text{cm}\) long. The arrow must point in the positive \(x\)-direction.
Step 5: Draw \(\vec{R_y}\)
The magnitude of \(\vec{R}_y\) is \(\text{3}\) \(\text{N}\) so the arrow we need to draw must be \(\text{3}\) \(\text{cm}\) long. The arrow must point in the negative \(y\)-direction. The important fact to note is that we are implementing the head-to-tail method so the vector must start at the end (head) of \(\vec{R}_{x}\).
Step 6: Draw the resultant vector, \(\vec{R}\)
The resultant vector is the vector from the tail of the first vector we drew directly to the head of the last vector we drew. This means we need to draw a vector from the tail of \(\vec{R}_{x}\) to the head of \(\vec{R}_{y}\).
Step 7: Measure the resultant, \(\vec{R}\)
We are solving the problem graphically so we now need to measure the magnitude of the vector and use the scale we chose to convert our answer from the diagram to the actual result. In the last diagram the resultant, \(\vec{R}\) is \(\text{8.0}\) \(\text{cm}\) long therefore the magnitude of the vector is \(\text{8.0}\) \(\text{N}\).
The direction of the resultant we need to measure from the diagram using a protractor. The angle that the vector makes with the \(x\)-axis is \(\text{22}\)\(\text{°}\).
Step 8: Quote the final answer
\(\vec{R}\) is \(\text{8.0}\) \(\text{N}\) at \(-\text{22}\)\(\text{°}\) from the positive \(x\)-direction.
This lesson is part of:
Vectors and Scalars