Vector addition: does order matter? – Play with this interactive simulation to see if the order in which you add vectors matters.
PhET Vector Addition Simulation
The PhET vector addition simulation is a great way to practice using vectors. Use the lab option to draw vectors on a 2D axis and see the components and magnitude of each vector. You can also see the sum of multiple vectors to practice vector addition.
Hello there! Welcome to lecture 2: Newton’s first law!
Newton’s first law of motion describes the property of inertia, which defines the resistance objects have to changes in their motion, whether that change is to speed up or slow down the object, or to change its direction. All objects of mass will continue to move at a constant speed, in the same direction, unless acted upon by an outside force. If an object of mass is not moving, it will stay at rest unless pushed or pulled by an outside force.
We will then introduce important topics of mass, force, and equilibrium. These concepts will form the foundation for almost all of the subsequent topics in this class. We will conclude this lecture with a tutorial on vectors.
Each of the following concepts will be discussed in this video: inertia, mass, force, mechanical equilibrium, and vectors.
Newton’s first law is sometimes referred to as the law of inertia, as it describes this property of matter. Newton’s first law says that “an object in motion tends to stay in motion, and an object at rest tends to stay at rest, unless acted upon by an outside force.”
Inertia describes how difficult it is to change the motion of an object. If an object is at rest, inertia relates to how hard it is to get the object to start moving. If an object is moving in a straight line at a constant speed, inertia relates to how hard it is to get the object to speed up, slow down, or change its direction.
Let’s consider these three objects. They appear to have similar shapes and sizes. How could we determine the inertia of each object? If inertia relates to the difficulty of getting something to change its motion, then I can figure out which, if any, of these objects has the most inertia by picking each one up and measuring the difficulty in getting each object to move.
First, I’m going to pick up one of the objects, a piece of Styrofoam, and move it back and forth as many times as I can in five seconds. This one was definitely not too hard for me to move around.
Now I’m going to repeat this experiment with the next object, which is a clay brick. I’m going to move it back and forth as many times as I can in five seconds. This one is definitely more difficult to move than the styrofoam. It had a greater resistance to changes in motion, meaning that it has more inertia.
Finally, I’m going to repeat the experiment one more time with the last object, a piece of lead. I’m going to move this one back and forth as many times as I can in five seconds. This was realllly hard for me to move. In fact, it was hard even for me to pick it up from the cart it was sitting on.
What is the difference between the three objects? What property of matter makes the Styrofoam easy to move and the lead hard to move? It can’t be volume, because the sizes are all the same. The difference is in the mass of the objects. We’ll discuss exactly what mass is in a few moments.
Unless a force acts on an object to change its motion, it will keep doing what it was already doing. Let’s take another look at this property in action.
While we have not formalized the concept of forces yet, perhaps you are aware of the force of gravity. Gravity acts on each of the objects sitting on this table: the plate, the silverware, the flower in the vase, and the glass of blue water. A support force counteracts gravity and holds each object up.
If I very quickly remove the tablecloth, I am not introducing any more forces into the mix. Gravity and the support forces are still present, and there was not much, if any, horizontal force present when I pulled the tablecloth out. Therefore, most of the objects stay put. The plate, the flower, and the glass of water pretty much stayed where they were. However, there was a little bit of force present that made the silverware move around a little bit. Inertia relates to the mass of an object. The silverware has the least amount of mass, so it makes sense that they would be the most affected by this demonstration.
Let’s recall that inertia does not only apply to objects that are at rest. It also applies to objects in motion. An object in motion will continue moving in a straight line at a constant speed unless acted upon by an outside force.
In this example, a ballistic cart contains a small yellow ping-pong ball inside. As it moves past a trigger on the track, it launches the ball into the air. Because the ball and the cart were both moving at the same speed when traveling together, they will continue to move at the same speed, even if they are separated. Unless a force can act on either object to slow them down or speed them up, then they will continue moving at the same speed. After launching into the air, the ball and cart have the same speed. That means that once the ball comes back down again, it will land right back into the cart.
This relates to the motion of any objects that are moving together. If you are riding in a train car, you could throw something up and down in the air and expect to catch it, as long as the train doesn’t speed up or slow down while the object is in the air.
However, if the train speeds up after we throw a ball into the air, the ball will continue moving at its initial speed as the train goes faster. The ball will land behind where we initially threw it. If the train slows down after we throw a ball into the air, the ball will land ahead of where we initially threw it.
Inertia also helps to explain why we wear seatbelts in cars. If we are riding or driving in a car, then we are moving at the same speed as the car. If the car is stopped abruptly by an outside force, our bodies naturally want to continue moving in the same speed in the same direction. Without a seatbelt, a collision with another object could cause us to be thrown out the windshield of a car. Even braking a car very forcefully could otherwise cause us to move forward dangerously in our seats, if we didn’t have a seatbelt on. Seatbelts are important safety devices that prevent that from happening. They apply a force to us to ensure that we slow down or stop moving as well. We’ll discuss other important safety features of cars, and how they relate to physics, in lecture six.
Mass is a property of physical objects that relates to resistance to changes in motion: inertia. Mass relates to how much “stuff” is in an object. We’ll expand upon that definition in lecture 11 when we learn about atoms. Mass has a symbol of the lowercase letter m. The units we use for mass are kilograms.
It’s important to understand that, for most physical objects, mass is a characteristic that does not change. It does not change over time, and it does not change based on where the object is located: the Earth, the moon, Jupiter, or in the vacuum of space.
I measured the mass of each of the three bricks from earlier. The Styrofoam has a mass of 0.04 kg, the clay brick has a mass of 2 kg, and the lead has a mass of 12 kg. Those masses would be exactly the same if I measured them on another planet. The inertia of these objects is independent of their location.
The situation for humans is a little different. Our mass can change, but that’s because we eat, exercise, cut our hair, and otherwise do things that may cause our mass to fluctuate. However, if we could somehow keep our mass at a constant value, that number would not change if we then traveled to the moon, or to Mars, or to any planet or celestial body with a different gravitational force than the Earth.
In other words, while mass is related to a concept called weight, it is a fundamentally different thing! Mass has to do with our resistance to changes in our motion; our inertia. Weight is a force that comes about due to gravity.
Mass is an extremely important concept in physics, in that it relates to forces, energy, and momentum, which are topics that we’ll be discussing in our next few lectures on kinematics.
Simply put, a force is a push or a pull that causes an object to change its motion. The symbol we use for force is the capital letter F. Because force is a vector, it will either appear in bold font, or with an arrow over the top. A vector is a quantity that contains both a magnitude and a direction. That means that the direction in which a force is applied is just as important as how strong the force is. We’ll discuss how we represent the direction of a vector, such as force, in the last section of this lecture video.
The units used for force are Newtons. A Newton is a unit that has other units hidden inside it. If we break it down, a Newton is equal to a kilogram times a meter divided by a second-squared. Either of these two things is interchangeable, but it’s much easier to use the unit of Newton, which is why we use it so often in physics.
Forces are used to overcome inertia and make an object change its motion. The amount that an object changes its motion due to a force is the topic of Newton’s second law, and lecture four. We’ll talk a lot more about different types of forces in that lecture.
It can be difficult to get a full appreciation for the concept of inertia, as we experience so many forces while living on the planet Earth. Gravity and friction constantly impact the motion of objects on this planet. When I push something, it eventually comes to rest. That may seem to go against the law of inertia, but it doesn’t. By accounting for all of the forces acting on an object’s motion, we can really start to appreciate Newton’s first law.
Right now, as I stand here recording this lecture, I have forces acting on me. One of those forces is gravity. Gravity is a force that pulls me toward the surface of the Earth. Another force that is acting on me is a support force from the floor, which is holding me up. These two forces are in perfect balance with each other, which is why I am able to remain at rest.
By analyzing the sum total of all forces acting on an object, we can determine whether or not that object will experience a change in its motion. When the sum of all of the forces acting on an object is equal to zero, then that object is said to be in mechanical equilibrium.
In physics, we use the capital Greek letter Sigma to define “the sum of.” Therefore, Sigma F means “sum of all forces.” We can add together each force to calculate that sum.
Mechanical equilibrium defines a condition in which the sum of all forces acting on an object is zero. Sometimes we call this quantity the net force. So by saying that the net force is zero, we are saying that the sum of all forces is zero. When the net force on an object is zero, that object is in mechanical equilibrium. As a result, the object will have a constant velocity. That is, the object will have a constant speed along a straight line.
There are two types of mechanical equilibrium: static equilibrium and dynamic equilibrium. Static equilibrium defines a condition of mechanical equilibrium where an object is at rest. The velocity remains constant at zero.
In this example, a cart sits on a flat, low-friction track at rest. There are forces acting on the cart, but they are completely balanced, and because the cart is not moving, it will continue to not move unless that condition of equilibrium is disturbed somehow by a push or a pull.
The two forces acting on the cart are gravity, which pulls the cart to the Earth, and a supporting force from the track, keeping it in position on the track. These two forces are equal to each other, and point in opposite directions. Therefore, the net force is zero and the cart is in static equilibrium.
Dynamic equilibrium defines a condition of mechanical equilibrium where an object moves at a constant, but nonzero, velocity.
In this demo, the cart has been pushed until it travels at a constant speed along a low-friction track. After my hand stops pushing the cart, the only two forces acting on the cart are gravity and the support force, which remain equal and opposite, creating a net force of zero. This means that the cart will continue to move at the same speed, in the same direction.
In reality, it can be difficult to experience a condition of pure dynamic equilibrium on Earth. That’s because there are usually always other forces we have to contend with, notably air drag and friction. By minimizing those forces as much as possible, we can get close to a situation of dynamic equilibrium.
We will continue to build on this concept of equilibrium, and the concept of net force, when we discuss Newton’s second law of motion in lecture four.
A vector is a quantity that contains two pieces of information: a magnitude (or strength), and a direction. As we saw in our discussion of forces, not only can different strengths of forces be applied to an object, but they can be applied in different directions.
A quantity that does not contain direction is said to be a scalar quantity. Mass is a scalar. It has a magnitude but no direction. It wouldn’t make any sense for the brick to have a mass of 2 kg down, or up, or left, or right. It just has 2 kg of mass.
Vector quantities, in this class, will be denoted either using bold font (which is what I can use in the video editing software that I use to make these lecture videos), or with an arrow over the top. When I write out vector quantities by hand, I’ll put an arrow over the top. Either way, bold font or an arrow, those are just indications that something is a vector and includes direction.
A vector is denoted graphically with an arrow. The length of the arrow tells us about the magnitude of the vector. The positioning of the arrowhead tells us the direction of the vector. For example, I could use an arrow to demonstrate a vector with a magnitude of 10 pointing down, or a vector with a magnitude of 20 pointing down. I could use an arrow to demonstrate a vector with a magnitude of 5 pointing up, or with a magnitude of 20 pointing left, and so on.
Because a vector can be used to represent any direction in three-dimensional space, it can be kind of tricky to discuss the direction of a vector. In this class, we’ll try to keep things as straightforward as we can. Let’s limit ourselves to two-dimensions. A vector can be pointed in any of 360 degrees.
One way that we can discuss the direction of a vector is used for vectors that point completely vertical, or completely horizontal. A plus sign means that a vector points up, and a minus sign means that a vector points down. For a horizontal vector, a plus sign means that a vector points to the right, and a minus sign means that a vector points to the left. This is the same as the signs used in the Cartesian coordinate system.
Alternatively, we could use a cardinal direction such as north, south, east, or west to define the direction of a vector. This is not always relevant, but can be a nice way to talk about direction when discussing changes in location, for example
Finally, we could use angle to discuss direction. This is how direction is used for vectors in many physics classes. However, we will not be using any trigonometry in this class. Therefore, we will not be making rigorous use of angles, and I will not expect you to use sine, cosine, or tangent to do any calculations with angles to analyze vectors.
Any vector, regardless of its direction, can be broken down into components. Those components can be used to calculate the magnitude of the vector. We can also use the components to add vectors together. Let’s look at a couple of examples to learn more.
Our first example considers a starting position of (4,0) and an ending position of (0,0). The arrow will point from the start to the end. This is a horizontal line pointing toward the left. This vector is completely horizontal as it has no vertical component to it.
The value of the vector is negative four. The negative sign indicates that it points to the left. The magnitude of the vector is positive four, which describes the length of the vector. The magnitude of a vector will always be a positive number.
The second example considers a starting position of (-3,0) and an ending position of (-3,2). The arrow points from start to end. This is a vertical line pointing upward. This vector is completely vertical and has no horizontal component to it.
The value of the vector is positive two. The positive sign indicates that it points up. The magnitude of the vector is two, which describes the vector’s length.
Our next example has a starting location at (2,-1) and an ending location of (6,8). The arrow points from start to end. This vector has both horizontal and vertical parts. We call those components. To calculate properties of this vector, we can first determine the horizontal and vertical components.
The horizontal component defines the horizontal motion of this vector, ignoring the vertical parts. The horizontal component starts at 2 and ends at 6. Therefore, the horizontal component is a horizontal line from 2 to 6. The magnitude of the horizontal component is equal to four.
The vertical component defines the vertical motion of the vector, ignoring the horizontal parts. The vertical component starts at negative 1 and ends at 8. Therefore, the vertical component is a vertical line from -1 to 8. The magnitude of the vertical component is equal to nine.
Now that we know the value of each component, we can use the Pythagorean theorem to calculate the magnitude of the vector itself. The Pythagorean theorem states that a squared plus b squared equals c squared, where a and b are the lengths of two sides of a right triangle, and c is the hypotenuse of that triangle. By splitting a vector into horizontal and vertical components, we have found that right triangle.
In this example, the magnitude of the vector is the square root of four squared plus nine squared, which is 9.85.
Let’s consider one more example, a vector that starts at (6,3) and ends at (0,-2). This vector points down and to the left, and has both horizontal and vertical components. The horizontal component goes from 6 to zero. It points to the left and has a length of six. The vertical component goes from 3 to -2. It points downward and has a length of five.
We can use the Pythagorean theorem to calculate the magnitude of this vector. The magnitude is equal to the square root of negative six squared plus negative five squared, which is equal to 7.81. Note that when we square a negative number, we should always get a positive result. Use parenthesis on your calculator, otherwise your calculator may incorrectly give you a negative number.
What if we want to add two or more vectors together? Sometimes we will need to do this, especially when calculating the sum of all forces acting on an object, to determine whether or not it is in equilibrium. There are two methods for adding vectors together: the graphical method, and the numerical method. Either one can be used. I suggest you try both, and use the one that feels more comfortable to you.
Let’s add together the two vectors we considered earlier. The one with (4,9) components, and the one with (-6,-5) components.
We’ll start by adding them together graphically. To add vectors graphically, draw one of the vectors. Then, at the tip of that vector, draw the second vector. If there are more than two vectors, each will be connected together tail to tip, tail to tip. The order in which you draw the vectors does not matter. The sum of the vectors is the straight line from the start of the first vector to the end of the last vector. In this case, that vector has a horizontal component of negative two and a vertical component of positive four. The Pythagorean theorem can be used to calculate the magnitude of the sum of the vectors, which is 4.47.
Instead, we can add the two vectors together numerically. To add the vectors numerically, write out each vector as a set of components. The first vector is (4,9), and the second vector is (-6,-5). Add all of the x components together. 4 plus -6 is negative two. Then add all of the y components together. 9 plus -5 is four. Therefore, the sum of the vectors is (-2, 4). Note that we should, and did, get the same result using both methods. If there are more than two vectors, simply add all of the x components together to find the x component of the sum. Then add all of the y components together to find the y component of the sum.
Thanks for taking the time to learn about Newton’s first law! Until next time, stay well.