Chapter 10: Dynamics of Rotational Motion

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Okay, so picture this.

You're at like a street fair or something and there's that classic juggler, right?

Tossing pins up effortlessly like it's no big deal.

But the thing that always gets me, especially when those pins are weighted on one end, like thicker at the bottom, they spin so smoothly, so evenly in the air.

It's like gravity just decides to take a break.

Shouldn't that spin be all messed up by gravity pulling on it differently?

Why does it stay so constant?

That's a really cool observation and actually it's that kind of question about everyday things that can lead us to some pretty deep physics.

So today we're gonna dive into what we're calling 10 .PDF, a really great resource that gives us the tools to understand the whole world of rotational motion.

Awesome, so basically we're on a mission today to break down how this chapter unpacks all the whys and hows of spin, starting with that initial push that gets things rotating, which we call torque or the twisting force.

And then diving into how that spin changes speed,

angular acceleration.

Then it's on to the energy involved in spinning things and we're gonna end with a bang exploring this key concept of angular momentum,

specifically for rigid bodies,

which is a fancy way of saying objects that don't change shape while they rotate.

And what I think is really cool about this chapter is that it doesn't just throw equations at you, it actually shows how these ideas are at work all over the place from the simplest tools to complex machines and even some surprising things in nature.

So no jargon, just clear, concise and hopefully kind of fun insights into why all this spin stuff matters.

Definitely.

Okay, to get the ball rolling, let's start with what actually gets things spinning in the first place, torque.

Yeah, so torque, think of it as the rotational brother of force.

Just a force gets something moving in a straight line.

Torque is the thing that causes something to have angular acceleration, which is a fancy way of saying it makes things start rotating faster or slower.

But, and this is super important, it's not just about how strong the force is, it's about where and how you apply it.

I'm picturing that diagram, figure 10 .1, with the wrench and the bolt.

You know instinctively that if you grab the end of the wrench handle, that's FGTB in the diagram, you get way more turning power than if you grab closer to the bolt, which is FTBA, even if you're putting in the same amount of effort.

And then there's that weird case F pulling straight along the handle.

Same strong pull as FD, same spot, but it doesn't turn the bolt at all.

Exactly, and that's all because of this thing called the lever arm.

Some folks call it the moment arm, but it's the same idea.

It's that perpendicular distance from the axis of rotation, so in this case the center of the bolt, to the line of action of the force.

So if that's part of you, you have a big lever arm, which means big torque.

For FA, it's a smaller lever arm, so less torque.

And with FTC, well the line of action of the force goes right through the center of the bolt.

No lever arm, no turning power.

Makes sense.

So we can formalize this a bit.

Torque, the symbols that Greek letter tau equals the force, F times the lever arm L, so FLO,

simple enough.

But there's this whole thing about perspective, isn't there?

It matters what point you're considering the torque around, right?

Absolutely, that's a key point the chapter really emphasizes.

Torque is always measured about a point.

So depending on where you choose that point, the torque a specific force produces can change completely.

Take a look at that F3, in fact 10 .2.

If we're thinking about the torque around point O, well that force goes right through O, zero lever arm.

So no torque.

Exactly, zero torque.

But now shift your focus to point A, now suddenly F3 has a lever arm relative to A, and it does produce a torque.

Same force, totally different torque depending on your reference point.

Okay, yeah, I see that.

And we gotta consider direction too.

Oh, absolutely.

Like clockwise versus counterclockwise.

Figure 10 .2 shows us the convention.

If a torque is trying to make something spin counterclockwise, it's positive.

Clockwise, it's negative.

It's just a consistent way to keep track of which way things are trying to turn.

Right.

And then there are units.

Torque is measured in Newton meters written newm.

Now a little heads up here.

This looks just like a joule, the unit for energy.

But be careful, torque and energy are fundamentally different things.

Okay.

So even though they happen to have the same units, stick with Newton meters when you're talking about torque.

Got it.

So we've got force and lever arm.

Seems pretty straightforward.

But figure 10 .3 throws three different ways to calculate torque at us.

Right, right.

Why do we need all these different formulas if it's the same basic concept?

So it's all about like having different tools for different situations.

The first one, FLL,

using the lever arm directly, that's often the easiest if the lever arm is clear.

But sometimes you might know the distance from the axis of rotation to where the force is applied and the angle between that position vector R and the force vector F.

And that's where the second formula comes in.

FFRFSIN.

If you look closely, RSIN is basically just a different way of calculating that perpendicular lever arm.

So it's the same idea, force acting at a distance to cause rotation just viewed from a slightly different geometric perspective.

Exactly.

It's about using what you know.

And then the third way involves thinking about the force in terms of its components.

So you split the force into a radial part, which is along the line connecting to the axis of rotation and a tangential part, which is perpendicular to that line.

Only the tangential component is the one actually trying to make the object rotate.

So the torque is then FtanR or Ftan equals FSIN.

And that radial component, while it's just pulling or pushing along the radius, doesn't contribute to rotation at all.

That makes sense.

Now, things get a bit more mathematically intense when we bring in vectors.

So we've got this vector definition of torque, T equals REF.

This cross product thing, what does that give us that these simpler calculations don't?

So the vector definition tells us the direction of the torque and the magnitude is still RFSIN.

The cross product means the torque vector T is always perpendicular to both the position vector R and the force vector F.

To figure out which way it points, you use the right hand rule, which is shown in FEMA 10 .4.

So point your fingers in the direction of R, curl them towards F, and your thumb will point in the direction of the torque vector T, which is also the axis of rotation the torque is trying to cause.

Example 10 .1, that plumber using a cheater bar to loosen a pipe, that really helps make this click.

He's putting 900 N of force at an angle.

And we see how using both the direct lever arm method and the RFSIN method gets you the same torque of 680 in you.

And using the right hand rule, we can confirm that the torque's direction is counterclockwise, so positive.

Exactly.

And it's a great example of how increasing the lever arm, which is what the cheater bar does, can really boost the torque, making a tough job a lot easier.

So torque is the cause with the effect on a rigid body.

That's where angular acceleration comes in, right?

You got it.

Just like a net force, which we write as INNO, causes linear acceleration, or according to Newton's second law,

in INNO, a net torque, written as, causes an angular acceleration.

And that's our rotational version of Newton's second law, INNO.

So Nest, that's the total torque acting around the Z axis, our axis of rotation,

and rises how quickly the angular velocity changes around that axis.

Yep.

And then there's INNO, moment of inertia.

It feels kind of like the rotational version of mass.

That's a good way to think about it.

Moment of inertia, or I, basically tells us how much a body resists changes in its rotational speed.

And what's interesting is that it depends not just on how much mass the object has, but also on how that mass is arranged relative to the axis of rotation.

So something with most of its mass, far from the axis, will be harder to get spinning, or to stop spinning, compared to something with the same mass, but more tightly packed around the axis.

Right.

The chapter touches on how this equation comes about.

If you think of a rigid body as,

like a zillion tiny little particles all stuck together, each little particle experiences a tiny force, which gives a tiny torque.

And when you add all those tiny torques up, that's your net torque.

And that determines how quickly the whole thing spins up or slows down.

And remember that angular acceleration has to be in radians per second squared, or rads for the units to work out correctly.

Got it.

Also, this equation, like Zayas, it really only applies to rigid bodies, where every bit of the object is spinning at the same rate.

Wouldn't work for something like, you know, water swirling in a drain, where different parts are spinning at different speeds.

Okay, so now what about gravity?

I mean, it's pulling on every atom in a rigid body.

Yeah.

How do we figure out the total torque it's creating?

The chapter gives us a handy trick for this.

When you're figuring out the torque from gravity on a rigid body, you can act as if the object's entire weight is acting at a single point.

It's center of mass.

That's super useful.

Yeah, it makes calculations a lot easier, and we'll see a more rigorous explanation for why this works a little later.

Speaking of making calculations easier, problem -solving strategy 10 .1 looks like a roadmap for these rotational motion problems.

Definitely.

It's like, you know, a checklist to make sure you approach things systematically, just like we do for linear motion problems.

So first, you figure out what physics concepts are at play.

Then you set things up visually.

Sketch the situation, draw a free -body diagram, pick a coordinate system, and decide which way you're gonna call positive rotation.

Then you actually apply the physics, Newton's second law for any linear motion, and the rotational version for angular motion.

You also have to think about any geometrical relationships.

Like, if a cable is unwinding from a spool, the linear speed of the cable is related to how fast the spool is rotating,

then you solve your equations.

And lastly, always, always check if your answer makes sense.

Does it have the right units?

Is it in the ballpark of what you'd expect?

Makes sense.

And then examples 10 .2 and 10 .3 really walk us through this with that classic unwinding cable problem.

So example 10 .2, we have a cylinder with a cable wrapped around it.

The tension in the cable creates a torque, and that makes the cylinder spin up.

Right, and what's cool there is that the weight of the cylinder itself and the normal force from whatever it's resting on, those don't create any torque around the axis of rotation because their lines of action go right through that axis, which means zero lever arm.

Right, right, no lever arm, no torque.

And then example 10 .3 adds a falling block to the end of that cable.

Now we've got the block moving in a straight line, following Christ's mom, and the cylinder rotating, falling beria einka,

and they're connected by that cable.

Exactly, it's like a chain reaction.

The block's motion directly affects the cylinder's rotation, and the key is that if the cable is unwinding without slipping,

then the linear acceleration of the block and the tangential acceleration of the point on the cylinder where the cable is leaving must be the same.

That connection lets us solve for both the acceleration and the tension in the cable.

So cool, and that tests your understanding section with a glider on a track, the hanging mass, and the pulley.

That's like a great little quiz to see if we're getting this.

Yeah.

It really shows how that net force and net torque could determine the accelerations and how the tensions in the string are all affected by the weights of the objects and how much the pulley resists spinning, which is its moment of inertia.

And it reinforces a really important relationship.

So when you have a string unwinding without slipping from a pulley,

the linear acceleration of whatever's attached to that string, let's call it a pandy, is directly related to the angular acceleration of a pulley by this simple equation, ii, where r is the radius of the pulley.

It's like the bridge between the world of straight line motion and the world of rotation.

I'm getting it.

But, okay, we've mostly been talking about rotation around a fixed point, but what if that axis of rotation is moving too?

Oh, yeah.

Like that juggler's pin or a ball rolling down a hill.

Things get a little more complicated then, right?

Right.

That's when we get into combined translation and rotation.

So the key here, as the chapter points out, is that any way a rigid body can move can be thought of as a combination of two things.

First, there's the motion of its center of mass, which is just like, you know, basic straight line motion.

And then there's rotation around an axis that passes right through that center of mass.

So like, it can be moving and spinning at the same time.

Exactly.

Figure 10 .11, with that baton being tossed in the air, that's such a good visual.

You can see the center of mass of the baton following a nice parabolic arc, just like if you chucked a ball.

But at the same time, the baton is spinning around that moving center of mass.

Right.

And because of this double motion, the object's total kinetic energy is also a sum of two parts.

Equation 10 .8 tells us total kinetic energy k is equal to the kinetic energy from the motion of the center of mass.

That's half mVcm.

Yeah.

Plus the kinetic energy from the rotation about the center of mass, which is half sm.

So it's like, the object has energy because it's moving through space, and it has additional energy because it's spinning.

Okay.

And one really common example of this combined motion is rolling without slipping.

What's special about that?

So in rolling without slipping, there's a direct relationship between the speed of the center of mass, Vcm, and how fast it's rotating, sm.

So Vcm, you want r, where r is the radius of the rolling object.

This is equation 10 .11.

And this happens because the point where the object touches the surface is at that very moment at rest.

So it's like that point becomes a temporary pivot point.

It's wild how that works, that it's not actually moving at that instant.

Figure 10 .13 really shows this well.

Yeah, it's a great visual.

The top of the wheel is actually moving forward twice as fast as the center while the contact point is momentarily still.

So cool.

Now example 10 .4 is really interesting.

It's that primitive yo -yo falling and unwinding.

They use energy conservation to figure out how fast it's falling and spinning.

As the yo -yo drops, its potential energy gets turned into both that straight down kinetic energy and the rotational kinetic energy from spinning.

And they show that the total kinetic energy is three -quarter MVC -cobin, so both types of motion are contributing.

Right.

And then we get into the forces and torques at play in this combined motion.

Newton's second law, fixed FSAWM, that still applies to the overall motion of the center of mass.

That's equation 10 .12.

And the rotation around that moving center of mass, that's governed by its own version of Newton's second law, Tubbs and Wiese -Campus.

That's equation 10 .13.

But there are a couple of important things to keep in mind for this to work right.

So first, the axis of rotation needs to be an axis of symmetry, meaning the object has to be balanced around that axis.

And second, the axis can't change direction in space.

Got it.

So example 10 .6 goes back to that primitive yo -yo, but this time they use forces and torques to figure out the acceleration and the tension in the string.

It's really cool that you can get to the same answer using both energy conservation and Newton's laws.

Right, two paths leading to the same truth.

Exactly.

Example 10 .7 is even trickier.

It's a solid sphere rolling down an incline.

You gotta think about gravity, but also static friction acting at the point of contact.

And that friction is what keeps it from just sliding down without rolling.

Right, right.

It's that static friction that causes the rotation.

And by applying both the linear and rotational versions of Newton's second law, we can figure out how fast the sphere's center of mass accelerates.

And I love that little bio -application about the Russian thistle.

Oh yeah.

It actually uses rolling to spread its seeds.

That's a cool example of how physics is at work in nature, even though in this case, rolling friction eventually brings it to a stop.

Yeah, that's nature for you, always finding clever ways to use physics.

And that test your understanding with a yo -yo again highlights how the distribution of mass matters.

That hollow cylinder, because its mass is further out from the center, has a bigger moment of inertia, which means it accelerates slower and has greater tension in the string compared to a solid cylinder that has the same mass and radius.

Right, right.

Okay, we're getting a really good picture of how objects move when they're rotating and translating at the same time.

Yeah.

Let's talk about the energy involved in all this spinning work and power.

Okay, so just like a force doing work can change an object's kinetic energy when it's moving in a straight line, a torque doing work can change the rotational kinetic energy.

Makes sense.

Equation 10 .20 gives us the definition of work done by a torque.

It's an integral, W equals its pasta, where W is a tiny bit of angular displacement.

Oh, an integral.

But if the torque's constant, it simplifies, right?

Yeah, if the torque's constant becomes Wt, that's equation 10 .21, where toss is the total angular displacement measured in radians.

And the work done by the net torque equals the change in rotational kinetic energy according to the rotational work energy theorem.

What toss I across, half I lo.

That's equation 10 .22.

So we have this nice connection between the work done and how the rotational motion changes.

What about power?

Power, which is how quickly work is done, also has a rotational counterpart.

So it's P equation 10 .23.

Or toss is the angular velocity.

Basically, it tells us how quickly energy is being transferred to or from the rotating object.

Figure 10 .21 shows us a good visual.

It's a tangential force acting on a rotating body.

Only the component of the force that's tangential to the circular path contributes to the torque and does work in rotating the object.

Forces acting radially, they don't do any work in rotation.

Okay,

example 10 .8, the one with a grinding wheel being spun up by a motor.

That's a nice practical application.

They show us how to calculate things like the angular displacement, the work done by the torque, the final kinetic energy of the wheel, and the average power of the motor.

And then there's that test your understanding where they compare two cylinders with different moments of inertia, and they apply the same torque to each for one full rotation.

And what we see is that because the torque and angular displacement are the same, the work done and therefore the gain in rotational kinetic energy is also the same for both cylinders.

It doesn't matter how their mass is distributed.

If the torque and the amount of rotation are the same, they gain the same amount of rotational kinetic energy.

So cool.

All right, we're on to the final stretch.

Angular momentum.

It sounds like it's the rotational equivalent of linear momentum.

You got it.

For a single particle, its angular momentum L relative to a point O is defined as the vector product of its position vector R, which points from O to the particle and its linear momentum P, which is just mass times velocity or MV.

So the equation is LREP, which is also equal to R8MV.

That's equation 10 .24.

And remember, angular momentum is a vector, so it has both magnitude and direction.

You find the direction using the right -hand rule, just like with torque.

Figure 10 .23 helps visualize this.

We've got a particle moving in the side plane.

The magnitude of its angular momentum turns out to be LMVL, which is also equal to Murrow, where L is the perpendicular distance from the line of motion to the origin, which is like the lever arm of the momentum.

Okay, so now what about a rigid body, something that's like a solid object rotating?

So when a rigid body is rotating about an axis of symmetry, meaning its mass is balanced around that axis, its total angular momentum, L, is directly proportional to its angular velocity O.

The equation is L equals IO.

That's equation 10 .28.

And I is the moment of inertia about that axis of symmetry.

Got it.

Figure 10 .26 shows that the angular momentum vector, L, and the angular velocity vector point in the same direction along that axis of symmetry.

And you might wonder, why is that?

Yeah, why?

Well, even though each little bit of the rotating object might have a component of angular momentum that isn't perfectly along the axis, if it's a symmetrical object, all those off -axis components cancel each other out.

Figure 10 .25 shows this nicely.

It's all thanks to the balanced distribution of mass.

I see.

Now, one of the most important things about angular momentum is that just like a net force changes an object's linear momentum, a net external torque changes its angular momentum.

Equation 10 .29 gives us this relationship.

X equals DLD.

Right.

So that means the net external torque is equal to the rate at which the total angular momentum is changing.

And this applies to any system of particles, whether it's a rigid body or a bunch of individual particles flying around.

And it's the rotational version of Newton's second law in terms of momentum, which is Thex equals DPDT.

And just like internal forces within a system don't affect the total linear momentum,

internal torques within the system also cancel each other out.

Only external torques can change the total angular momentum.

And for a rigid body rotating around a fixed axis of symmetry, this general relationship simplifies back to our good old dusty.

But this more fundamental version, the one with angular momentum, is really important when we're dealing with systems where the moment of inertia might be changing over time.

That makes sense.

Example 10 .9, the turbine fan that's slowing down shows how you can calculate both the angular momentum and the net torque acting on the fan as functions of time if we know how its angular velocity is changing.

Yeah.

And then there's that test your understanding

with the ball on a string being swung in a circle.

That one's a great illustration.

Because even though the ball's linear momentum is constantly changing, since its direction is always changing, its angular momentum around the center of the circle is constant.

Because there's no torque.

Exactly.

The force from the string acts towards the center, so there's no torque about that point.

Okay, so all this leads to this really fundamental idea, the conservation of angular momentum.

One of the biggies in physics.

Equation 10 .31 tells us,

if the net external torque acting on a system is zero, then the total angular momentum of the system doesn't change.

L 'initial sedext equals L final.

It's a conservation law, just like conservation of energy and conservation of linear momentum.

Right, and it all comes from that equation, text DLDT.

If there's no net external torque, then the rate of change of angular momentum has to be zero, meaning the total angular momentum stays constant.

Exactly.

And this is ultimately tied to Newton's third law and the fact that internal torques within a system always cancel each other out.

Example 10 .10, the one where a professor is spinning on a platform holding dumbbells, that's a classic demonstration.

When they pull the dumbbells in closer to their body, the moment of inertia of the system decreases.

But to keep the total angular momentum constant, their angular speed has to increase and they start spinning much faster.

It's the same principle that figure skaters use to spin really fast.

So cool.

Example 10 .19 shows this conservation law even in a collision.

It's two rotating discs that stick together after they collide.

Even though some kinetic energy is lost because of friction, the total angular momentum is still conserved as long as there are no external torques acting on the two -disc system.

Right.

And example 10 .12, a bullet hitting a swinging door and getting embedded, that's another good one.

The angular momentum of the bullet before it hits is equal to the angular momentum of the door plus bullet after the collision.

So you can use that to figure out how fast the door starts swinging.

So the big takeaway here is that the total angular momentum of any system, whether it's rigid bodies, particles, or a mix, you just add up the angular momentum of all the pieces to get the total.

Exactly.

And then there's that test your understanding about the melting polar ice caps.

That one makes you think.

So if the ice caps melt and the water spreads out, that actually increases the Earth's moment of inertia.

And to conserve the Earth's total angular momentum, the Earth's rotation would have to slow down ever so slightly, making our days a tiny bit longer.

It's amazing how these concepts apply on such a huge scale.

Right.

Okay, so last but not least, those mind -boggling gyroscopes.

How do they seem to defy gravity and just spin without falling over?

It's all about the vector nature of angular momentum and torque.

So first, imagine a flywheel that's not spinning, like in figure 10 .33.

If you let go, gravity creates a torque pointing downwards, which causes the angular momentum to change in the same direction, so the wheel just falls down.

But now imagine a spinning gyroscope, like in figure 10 .34.

The initial angular momentum vector, Li, is pointing along the axis of the spin.

Gravity still creates a torque, but now that torque vector is perpendicular to the angular momentum.

And since the net external torque equals the rate of change of angular momentum, that means the change in angular momentum, DL, is also perpendicular to the original Li.

So when you add a little DL to Li, the overall magnitude of the angular momentum doesn't change much, but the direction does.

And this continuous change in the direction of the angular momentum vector is what causes precession.

Precession.

Yeah, so the gyroscope's axis of rotation slowly sweeps out a cone shape around the vertical.

And that's why the gyroscope seems to circle around instead of just toppling over.

It's like that spin makes it resistant to gravity.

Exactly.

Equation 10 .33 gives us an approximate formula for how fast this precession happens.

LAMGR, where M is the mass, G is acceleration due to gravity, R is the distance from the pivot to the center of mass, I is the moment of inertia, and I is how fast it's spinning.

So the faster it spins and the larger its moment of inertia, the slower the precession.

Example 10 .33 shows us how to use the right -hand rule to figure out which way the precession goes and how to calculate the angular speed of the spinning wheel if you know the precession right.

Awesome.

Okay, we have really gone deep into the world of rotational motion.

Yeah, we have.

We covered torque, how to calculate it, and how it relates to angular acceleration through the moment of inertia.

We explored how objects move and spin at the same time.

We talked about work and power and rotation.

Right.

And we delved into angular momentum,

its conservation law, and even those bizarre gyroscopes.

And going back to your question at the beginning about the juggler's pin, we can answer now.

Oh yeah.

Because the pin's weight acts at its center of mass, there's no net external torque acting around that point.

So its angular momentum about its center of mass stays constant while it's in the air, which means its spin rate also stays constant, assuming we ignore air resistance.

That's so cool.

So for you, the listener, think about all the things you use every day that rely on these ideas.

Car wheels, hard drives, fans, motors, the list goes on.

And then look around and see what other seemingly simple things might actually be governed by these same laws of physics.

It's a whole new way of looking at the world.

And if you're curious to learn more, remember this is just the start.

There's so much more to discover in rotational mechanics.

Definitely.

Thanks for joining us on this deep dive.

Until next time, keep those brains spinning.