Physics! #25 University Physics 4.3

4.3: Newton’s Second

Newton’s second: Force is mass time acceleration.

First, let’s get a basic understanding. I prefer to state the law this way: F = m(dv/dt). Verbally, that’s “Force equals mass times change in velocity.” Or, more succinctly “Force equals change in momentum.” The latter can be written F=(dp/dt), if you like. Physically, and mathematically, it makes no difference since the mass of whatever you’re talking about is a constant. (This gets a little wonky in relativity, but for now it’s sufficient).

I like this version because I think it tells you more clearly what’s going on. If something changes its velocity, a force must have been applied to it. The strength of that force can be determined if you know the mass and how big or little the change in velocity was.

So, as you know from real life, something with a small mass can have its velocity change very easily, while something with a large mass takes more effort. This has lots of consequences. If you roll a boulder down a hill at your ex because he cheated on you, you better be sure you’re right. Why? Not only does it take a lot of work for you to get that thing rolling, but it’d take a lot of work to get it to stop once it did start rolling. On the other hand, if you roll a ping pong ball at him, you don’t need to be sure. It doesn’t take much work to get it started, and it’s not going to kill him to attempt to stop it.

The rigorous statement of the law the book uses goes thus:

\sum \vec{F} = m\vec{a}

This is just a vectory version of what we said earlier. Here’s how I would state it in plain words:

If you see an object changing velocity and you know its mass (and you know nothing internal is causing it to move), you know the magnitude and direction of the net force acting on it.

So, if you see a nerd suddenly jerk forward, it could be the wind or a powerful magnet or that bully kicking him in the ass or some combination thereof, BUT without knowing the specifics you still know what direction the forces acting on him sum to, and how strong they are.

The book goes on to talk about how, for 2D and 3D you split your vector 2 or 3 ways, but after the last several sections I think this is second nature to you.

This rule is a very powerful one, and you’ll use it over and over. And, it’s neat too! When you move an object, it’s because your motions have told every single atom in the object to move in a certain direction. That’s crazy! When you push a rock, you’re causing every bit of matter in that rock to acceleration.

Try to think about that as you move around the world. When you walk, every atom of your body has gotten the message “move forward.” That means every single atom has some force acting on it. Otherwise, as we know from the law, it wouldn’t be moving.

We’ll get into more of the particulars soon, but I just want to note the scope of this law. Anytime something changes its velocity, there MUST be some force acting on it.

Next stop: Weight and Mass


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9 Responses to Physics! #25 University Physics 4.3

  1. NOOOOOOOOOO!!!! How could you!! F=dp/dt=d(mv)/dt, not m(dp/dt)

  2. Dave says:

    Not to be a stickler, but “force equals change in momentum” is just F=dp/dt without the “m”–since momentum is p=mv, then dp/dt=d(mv)/dt=m(dv/dt).

  3. Odysseus says:

    Nice, but you made a typo in the first paragraph. It is F=dp/dt, not F=m(dp/dt). The interesting bit is that the momentum formulation is a little more general than F=ma, since it also holds in cases of non-constant masses (e.g. a rocket ejection matter to accelerate).

  4. Hi, there’s a small mistake in the formulas. p=mv, so either you have F=dp/dt (no m since mass is already taken into account on momentum), or F=mdv/dt + vdm/dt (chain rule, without the dm/dt part astronauts wouldn’t be able to “walk” in space, since they eject mass in order to change their trajectory).

    I wrote on my blog a piece on newton’s laws. It’s in portuguese, but google translation is not so bad ->

  5. TheEngineer says:

    Just a note about F = ma or F = m * (dv/dt) …

    The distinction that it’s actually F = (dp/dt) is an EXTREMELY important detail in rocket science, where the mass is changing constantly as fuel is consumed and exhaust gases go streaming out the back nozzle.

    There’s a good chart here. Note how velocity changes over time, and the acceleration is increasing.

    • ZachWeiner says:

      Yeah, I was assuming constant mass. I think this chapter actually ends with a brief discussion of rockets where I was planning to introduce that idea.

  6. Harold Tessmann III says:

    Grammar typo: causing every bit of matter in that rock to acceleration

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