Introduction
You might ask: How do I use the parallelogram law in real life?
My answer: all the time.
In fact, you might be using it right now! But just like inertia, it’s intimately wired into our lives that we don’t even notice when we’re using it. If you have ever rowed a boat, been to the gym, played basketball, or shot an arrow with a bow, hell if you’ve ever thrown anything at anything. Then there’s a good chance you have subconsciously referred to the parallelogram law in your mind.
Ever notice how Olympic weightlifters usually spread their arms out wide in an overhead press position? That’s not arbitrary, in addition to enhancing their stability, that pose ensures the weight of the plates on the sides is distributed equally to both arms. It is a perfect way in which the body balances off the weight of the plates between its two arms. And it is consistent with the parallelogram law.
In this article.
In this article, we will discuss the parallelogram law. My approach will be unique as we’ll dive into a short, exciting thought experiment. I assume you have some background knowledge of vectors and vector representation. If not, I have included a quick revision on vectors, just enough to make sense of the proceeding arguments involving the parallelogram law. If you don’t need vectors revision, feel free to jump to how the parallelogram works here. This article has taken me quite a while to prepare so I really hope you enjoy it.
Revision on vectors
Physical quantities
Suppose I ask you, “How many colors does a rainbow have?”
You might answer “seven”.
This simple number gives me all the information I need. It’s a perfect example of a quantity.
Suppose I ask you another question, “How long is Route 164?”
And you answer, “seven”.
This time I will look at you puzzled and ask, “Seven what?” “Meters, miles, yards or minutes’ drive?”
To which you answer, “miles”.
This time a quantity alone wasn’t enough to give me all the information I needed. The extra “miles” that was necessary to complete the information is known as a unit. Thus, I required both a quantity (seven) and a unit (miles) to get all the information I needed.
“Seven miles” is an example of a physical quantity. A physical quantity has both a number and a unit attached to it.
Suppose I ask you a third question, “How far to the nearest MacDonald’s?”
And you answer, “seven miles”.
I’d still be confused, “seven miles where?”
I will be left wondering, “North, East, West, South, or somewhere in between?” This time, even a quantity and a unit aren’t enough to give me all the information I want. This time, needed direction.
Thus some physical quantities require both magnitude and direction to be useful. Now, you might be thinking that these quantities are vectors – wrong! Unfortunately, most students assume that if a physical magnitude requires both magnitude and direction, then it must be a vector. This is inaccurate. Let’s just say, some quantities require both magnitude and direction but aren’t vectors.
Read my post on understanding vectors in physics and mathematics to understand the evolution and properties of vectors.
Vectors and scalars
For now, let us just roll with the standard definitions of vectors and scalars as follows:
Vector quantities are physical quantities that are completely specified by magnitude and direction only such as force, velocity, displacement and acceleration.
Scalar quantities are physical quantities that are completely specified by magnitude only such as mass, speed, temperature, and distance.
Because of their directional nature, vectors are commonly represented using arrow-headed line segments. The length of the line segment gives the magnitude of the vector whilst the arrow gives the direction of the vector. Thus, arrow-headed line segments completely represent vectors geometrically. More on this later.
Addition of vectors and scalars
Our basic intuition of the concept of addition is algebraic. Such as one apple and another apple equals two apples. i.e. 1 + 1 = 2, or 2 + 3 = 5, and so forth.
Usually, this arithmetic knowledge alone is sufficient to get us through much of everyday circumstances that involve numbers. Such as keeping track of your workout times or body weight, calorie intake, work hours, inventory, and financials.
But with some quantities, basic algebra just won’t cut it.
For example, consider these two puppies here pulling on a rope. Take a moment and think about the rope. It is being pulled by both puppies but it isn’t going anywhere.
Let’s get a bit quantitative with this.
Suppose each puppy is pulling on the rope at a force of 5N. This would imply that the total force on the rope is
$$5N+5N=10N$$
This may not seem like much, but 10N is a lot of force for a 20g rope.
To put this into perspective, at 10N, the rope ought to be flying off with an initial acceleration of 500m/s/s! (The same take-off speed of a soccer ball when kicked)
Since this is now what happens, it implies that there is something more than just magnitude when adding forces. Of course, we can tell that it has something to do with direction, but how that direction fits into our “5N + 5N = 10N equation” is the real question.
How the parallelogram law works
We will begin by setting it up with this thought experiment.
Suppose, after an ordinary day at work/school you are on a bus heading home. And sitting there, you notice a bug scuttling across the floor of the moving bus.
“Cute”, you think. “The lucky bug didn’t have to pay a dime for the ride”.
Absentmindedly, you begin to wonder, how exactly this free ride means for the bug. How much ground is the bug covering while moving in a moving bus?
You pull out your pen and notebook and trace the bug’s sprint across the bus. You end up with a sketch looking like the figure below.
After scrutinizing your figure for a minute or so, several things become apparent.
- Since the bug is moving in a moving bus, the bug is moving much faster relative to the ground than to the bus.
- Relative to the ground, the bug is in a combination of velocities. One from its own and the other from the bus. These two velocities are independent of each other.
- If the bus weren’t moving, the bug would cover the same distance on the bus as on the ground in a given interval of time.
- If you wish to calculate the true “advantage” of the bug’s velocity over the ground, you are going to need numerical values.
After deliberating with yourself for a minute or so, you come up with the approximate speeds of the bus and the bug. You modify your sketch as follows.
Here, you have assumed the bug to be scuttling across the bus at 2 feet/second, and the bus to be travelling at a mere 10 feet/second (about 7mph). You wish to know the speed and direction that the bug travels relative to the ground.
Using the parallelogram law
The procedure for using the parallelogram law here includes representing the vector quantities appropriately in magnitude and direction using arrow-headed line segments starting at a common point and then completing the parallelogram. The resulting diagonal represents the resultant in magnitude and direction of the vector quantity.
In the above figure, the velocities are represented with a scale of 1:1. The units could be anything, centimetres, or inches. The resultant here (diagonal) is 11 units, which translates to a velocity of 11 feet/second. The direction is as shown by the arrow, about 9° from the horizontal.
Therefore, the bug is moving at a velocity of 11 feet/second, traversing diagonally at an angle of 9° to the horizontal.
You may now skip to the conclusion of this article and avoid the step-by-step process that I describe in the next section.
The systematic process may be useful to students who need to know the bolts and nuts of how the parallelogram law works. However, it is not all that important for the general understanding of the parallelogram law, which is the objective here. Nevertheless, here it is.
Step-by-step application of the parallelogram law.
- Note the magnitude and directions of the quantities that you seek to combine. In our case, the magnitudes are 2 feet/second and 10 feet/second. The direction is as indicated in the figure above.
- Select an appropriate scale to represent the quantities. For our case, we will select a 1:1 scale i.e. 1 unit on paper will represent 1 foot/second of the quantities.
- Select an appropriate point on the paper and use it as your starting point. Proceed to draw each arrow-headed line segment as defined by the scale in the given direction of the quantity.
4. Complete the parallelogram by drawing parallel lines appropriately.
5. Draw the diagonal OR.
6. We then obtain by measurement the length of the arrow-headed line segment OR and the direction. And use the scale to convert it back to the physical quantity it represents.
Then, when taken together the two vectors represented by OP and OQ are equivalent to a single vector represented by the arrow-headed line segment OR. This vector is called the resultant of the vectors OQ and OP.
Statement of the parallelogram law
Steps 1 to 6 may be summed up together to form the statement of the parallelogram law of vector addition.
If two vector quantities a and b are acting simultaneously on a particle. They can be represented in both magnitude and direction by the adjacent sides of a parallelogram drawn from a point. Their resultant (a + b) is also represented in both magnitude and direction by the diagonal of that parallelogram drawn from that point.
If two vectors a and b combine to form a resultant vector r, we usually write;
$$a ⃗+ b ⃗= r ⃗$$
or
a+b=r
Conclusion
You might argue, “but, I’m a bank clerk, why the heck would I want to calculate the velocity of a bug in a bus, isn’t that for nerds?”
To which I answer, “true, but you occasionally play basketball with friends on weekends right?” (For the purpose of this argument, let us suppose that you do)
You: “Yeah, so?”
Me: “When you are running down the court during a break, how do you anticipate where your buddy Mark is going to be so you can throw the ball at him?”
You: *shrugging* “Instinct, I dunno”
Me: *smiling* “It’s parallelogram law”
Me (continues): See, your brain is taking into account the speed and direction at which both you and Mark are running down the court. And it correctly predicts where and with how much force should you throw the ball so that it would intercept Mark’s path. These calculations are essentially the parallelogram law! All done in your head! Pretty cool huh?
You: *to yourself* Nerds!
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