Definition 8.1.0.1.
The linear momentum of an object the product of the mass of the object and its velocity vector.
Section 8.1 Definining Momentum
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Momentum is a property of objects in motion. In a sense, it measures "how much" movement there is. One way that people think about momentum is to ask the following question: "How much will this hurt if it hits me?"
It is easiest to understand by thinking about comparisons between similar situations. Suppose you have two identical objects, where one is moving slowly and the other is moving quickly. The fast-moving object has "more" movement than the slow-moving object, and so that one has more momentum. But now consider a tiny object and a massive object that are moving at the same speed. Even though the speeds are the same, there’s still "more" movement in the large object than the smaller one. And so we can see that the idea of momentum involves both the mass and the velocity.
There are two types of momentum: linear momentum and angular momentum. We will introduce both, but our focusing will be on linear momentum. Unless otherwise noted, when you see momentum in this book, it will be referring linear momentum.
Linear Momentum.
Notice from the definition that increasing either mass or velocity will increase the momentum of an object. This matches with the discussion at the start of the section. Since momentum is the product of a scalar and a vector, it is a vector quantity. This isn’t as important to think about when we are working in just one direction, but it becomes essential to understand when we are working in higher dimensions.
The momentum is calculated relative to a frame of reference, which is important because the measured velocity of an object depends on who is doing the measurement. This goes back to thinking about how the world is perceived by someone on the ground compared to someone on a moving train. An interesting aspect of this is that it’s possible for two people to calculate two different values for the momentum of an object, but somehow all of the underlying physics is unaffected by the different values.
The SI units of momentum are \(\frac{\text{kg} \cdot \text{m}}{\text{s}}\text{.}\) By itself, this doesn’t say very much. But if we compare it to the units of force, we can see that the two are very similar. The SI units of force are \(\frac{\text{kg} \cdot \text{m}}{\text{s}^2}\text{.}\) The only difference between the two is that force has an extra dimension of seconds in the denominator. In other words, force has units of momentum over time.
In fact, the connection runs much deeper. Newton’s Second Law of Motion can actually be phrased in terms of momentum:
Definition 8.1.0.2.
The net force on an object at any instant of time is equal to the rate of change of its momentum.
In other words, forces lead to changes in momentum. This idea is written mathematically in the following manner: \(F = \frac{\Delta p}{\Delta t}\text{.}\) The \(\Delta\) notation can be read as "the change of" and is a common notation in both math and physics. (You might remember seeing the formula for the slope of a line as \(m = \frac{\Delta y}{\Delta x}\text{.}\) It’s the exact same thing.) By multiplying both sides of the equation by \(\Delta t\text{,}\) we have another version of the equation: \(F \Delta t = \Delta p\text{.}\) This is the formula for impulse, which we will discuss in the next section.
Angular Momentum.
Angular momentum is the momentum of spinning objects. It’s quite a bit more complicated than linear momentum, and it’s even less intuitive when you get into the details. If you have a record spinning on a standard record player, the angular momentum is pointing up because it is spinning counter-clockwise. The exact reason it’s pointing up and the consequences of that are difficult to understand intuitively. In some sense, the reason is that this is what works out mathematically. But that’s unsatisfying.
Instead of trying to get the full flavor of angular momentum and its consequences, we are going to think about some of the basic ideas and try to build some core intuitions about it. The underlying question is about how difficult it is to get something to rotate (or to stop rotating). We will use a thought experiment. Suppose you are holding a stick with a weighted ball at the end of it. Will it be easier or harder to swing twist the stick so that the ball swings around if the stick is short or long?
Most people intuitively understand that the long stick would be more difficult, and that is the correct answer. So this tells us that angular momentum depends on how far away from the axis of rotation. It also depends on the mass, just as with linear angular momentum.
Activity 8.1. Rolling Objects.
Three objects of the same mass roll down a slope. Which one gets to the bottom first?
Instructions.
Instructions
