5.2: Using Newton’s Second Law: Dynamics of Particles
Now things are moving!
In 5.1 we constructed FBDs for systems in equilibrium. That is, systems where the net force summed to zero on every conceivable axis. Left-right, up-down, north-south, zachward-antizachward – if you’re at equilibrium, whatever axis you define with respect to the particle in question must have forces along that axis that sum to zero.
NOTE: Recall that equilibrium DOES NOT MEAN lack of motion. It means lack of change in motion. If you’re zooming through space, it may well still be the case that no forces are acting on you. Thus, you must still be at equilibrium.
Now, what happens if you’re not at equilibrium? By definition, you change notion. I like to think of it like this: At any time, a bunch of forces are acting on an object. Most of the time, there is some excess of force (aka net force) in some direction. That leftover is expressed as a change in motion (aka, acceleration) in the direction of the leftover.
So, this is essentially the same as 5.1, except in these cases there will be some leftover force, and therefore some change from equilibrium.
Fortunately, Mr. Newton made it nice and simple for us. To know what to do in 2D, you just have to know two things:
Read: The sum of forces on an object in the x direction is equal to the mass of the object times that objects acceleration.
Read: The sum of forces on an object in the y direction is equal to the mass of the object times that objects acceleration.
The book puts a helpful note of caution here. An object itself doesn’t mass-times-accelerate. On object just accelerates, and that acceleration depends on the mass. Thus, don’t write m*a on your FBD. Sum the forces, and use Newton’s laws to calculate acceleration from that. Got it?
Apparent Weight and Apparent Weightlessness
There is something obvious here that I hesitate to mention it. But, I think it’s important and physics teachers often neglect to say it. When the word “apparent” is used, it means apparent to the object in question. Physicists tend to anthropomorphize, which is sometimes useful but sometimes confusing. In this case, “apparently weightless” is like saying that under certain conditions you would feel like you were weightless.
Let’s revisit weight and mass. Mass is how much stuff is in you. Weight is the force of gravitation yanking on you. More practically, weight is how much you squish the spring of the scale in your bathroom when you stand on it.
Now, say you’re in an elevator accelerating up. Your mass hasn’t changed, but if you were standing on that scale, you’d press it down more, and it would say you weighed more. More importantly, you would feel heavier, in that you would feel your body pressing down against the ground more.
If the elevator is accelerating, the opposite would happen. The spring would depress less, you would feel like you were pushing down less hard, and the scale would return a very flattering number.
Remember, that it’s not about going up or going down. It’s about accelerating! Years ago, when I used to visit my manager, I would go to this giant intimidating pair of buildings. I think he was on the 20th floor or something (okay, not so high, but pretty high for LA). Anyway, I used to always hope the elevator was empty because by jumping at the right time, you could experience weird apparent weights.
Right when the elevator speeds up, you jump. When you do this, your jump doesn’t last very long at all because you have a higher apparent weight. This is what it’d be like if the planet had more gravitational attraction to you.
Once the elevator gets to a standard speed, you can jump, but nothing special happens. Since you’re not accelerating, it’s like a regular jump, except that it freaks out everyone else in the elevator.
Best of all, right when you get to the top, the elevator slows down. If you jump at the point of maximum deceleration, you’ll go really high. Sometimes I wish there were 10 foot high basketball hoops atop elevators for this reason. You apparently have something like the weight you would have on a smaller planet, like Mars or Mercury.
This is what is meant by “apparent weight.” In an accelerating system, the way you experience your weight varies.
(Just because: Legally, I’m not advising you jump up and down in an elevator. It’s probably stupid for 10 reasons at once).
Now, you might say “HEY! HEY! Then, isn’t your weight on the surface of the Earth also ‘apparent weight’ in this sense?” Roughly speaking, you are both wrong and ignorant.
Okay, yeah, Earth’s surface is accelerating a little, since it’s rotating. But that acceleration is so small that on the surface of the Earth, and nearby, you can ignore it. This is why rocket ship pilots worry about the spinning Earth, but ping pong players don’t.
“Apparentness” has to do with what it’s like to be inside a system (say, an elevator) that’s accelerating.
This is nicely mathematized in the book thus:
The n just refers to “normal force.” Normal is a math term, which just means orthogonal to some surface. In basic physics, it’s generally in reference to the upward force a body experiences while pushing down on some surface due to gravity. It could also be called something like “pushback force.”
Anyway, so you see that it’s nice and simple. All you do is add the acceleration of the system to the acceleration due to gravity, and voila! Apparent weight.
This is cool because it means you can figure out what acceleration would be needed, for example, to feel like you were on Mars or Venus. Pretty neat!
You may also notice that there’s a special value when acceleration up is equal gravitational acceleration. In that case, normal force is 0, and you FEEL like you are weightless. This is what is meant by “apparent weightlessness.” For example, if you were in an elevator accelerating down at 9.8 meters per second per second, you would feel no weight at all. You might feel terror. You might lose some mass when you crap yourself. But, you will not feel yourself pressing on any surface.
If you’re in orbit, you may think you don’t fall because you’re too far from the Earth for gravity to matter. WRONG. WRONG, FOOL! If that were the case, you’d just float away as Earth zoomed around the solar system.
So, let’s take stock. You know you are apparently weightless, but there is gravity on you, but you don’t fall into the Earth.
Astronomers like to say you’re “perpetually falling toward the Earth, but you keep missing.” The idea is that gravity is pulling you toward Earth, but you’re also moving parallel to the surface of Earth below. So, you get pulled down in such a way that you go down PAST the Earth. Since you’re going around in a circle, this is happening constantly. It’s the same reason when you spin a lasso, the loop never comes toward you, even though you’re constantly pulling it toward you.
Next stop: 5.3 – Frictional Forces