1. Physics
Chapter 8 – Rotational Motion Part 1

2. Circular Motion
Tangential Speed – The linear speed of something moving along a circular path.
Symbol is the usual v and units are m/s
Rotational Speed – Number of revolutions per unit of time.
Symbol is omega, ω and units are RPM (rotations per minute)
Consider a turntable. Say the bug travels 0.8meters as he goes one time around. And say it takes 1 second to do so.
What is his tangential speed? 0.8m/s.
How many times will the record player go around in a minute? 60 So what is the bug’s rotational speed? 60 RPM

3. Circular Motion
Does tangential speed depend on where the bug sits on the record?
Does rotational speed depend on where the bug sits on the record?
Tangential speed YES – the nearer you are to the axis of rotation, the slower your tangential speed.
Rotational speed NO
Incidentally, you need to know the difference between something rotating and something revolving:
An axis is the straight line around which rotation takes place.
When an object turns about an internal axis—that is, an axis located within the body of the object—the motion is called rotation, or spin.
When an object turns about an external axis, the motion is called revolution.

4. v ~ r ω Circular Motion Is there a relationship between v and ω?
Question: Who is moving at a greater tangential speed: someone at the equator or someone in Winchester, Virginia?
Tangential speed is proportional to the radial distance times the rotational speed.
This makes sense if you think about it…. V ~ r (2x distance from center results in 2x speed)
And… v ~ w (2x RPMs means 2x speed)
DEMO: record player and pennies
Equator

5. Circular Motion

6. Check Question
At an amusement park, you and a friend sit on a large rotating disk. You sit at the edge and have a rotational speed of 4 RPM and a linear speed of 6 m/s. Your friend sits halfway to the center. What is her rotational speed? What is her linear speed?
Her rotational speed is same as yours… 4RPM
Her linear speed is half of yours… 3m/s

7. Wheels on Trains
Have you ever noticed the flanges on train wheels? What do you think they’re for? Hint: It’s NOT for holding the train on the rails!
DEMO: Single cup – since the wide part rolls faster (tang. Speed) than the narrow part, it goes in a circle.
When a car turns wheels can turn independently so one moves faster than the other (differential). But on a train, they are a rigid pair. Solidly attached to the axle.
Sooo..the taper allows for self-correcting in turns!
Which set of joined cups will self correct? DO IT! Show on meter sticks.

8. Rotational Inertia
An object rotating about an axis tends to remain rotating about the same axis unless interfered with by some external influence.
The greater the distance between an object’s mass concentration and the axis, the greater the rotational inertia.
In case I didn’t mention it already, remember everything we’ve studied up to now? You’re going to get the circular corollary in this chapter! We had speed – the corollary is rotational speed. We had Inertia - … now you get rotational inertia!
Rotational Inertia is the property of an object to resist changes in its rotational state of motion.
So the greater the rotational inertia, the harder it is to change the state of rotation.
Just like inertia for linear motion, rotational inertia depends on MASS. BUT. Unlike linear inertia… it depends on something else too: DISTRIBUTION OF THAT MASS!
DEMO: Dowel with clamps

9. Rotational Inertia
RI is why we instinctively hold our arms out to balance. Or use a pole to walk a tight rope.
And it’s changeable! If I redistribute my mass, my RI changes. If I change my axis of rotation, my RI changes.

10. Rotational Inertia
You bend your legs when you run because bent legs are much easier to swing quickly back and forth.

11. Rotational Inertia
A long baseball bat held near its thinner end has more rotational inertia than a short bat of the same mass.
Once moving, it has a greater tendency to keep moving, but it is harder to bring it up to speed.
Baseball players sometimes “choke up” on a bat to reduce its rotational inertia, which makes it easier to bring up to speed.
A bat held at its end, or a long bat, doesn’t swing as readily.

12. Rotational Inertia
A short pendulum has a shorter period, T, than a longer pendulum.

13. Rotational Inertia
For similar mass distributions, short legs have less rotational inertia than long legs.

14. I = mr2 Rotational Inertia So how can we calculate rotational inertia?
In the special case when all the mass m of an object is concentrated at the same distance r from the rotational axis:
I = mr2
When the mass is more spread out, the rotational inertia is less and the formula is different.
For this class, we’ll keep it simple. This special case applies to:
Swinging a weight on a string of negligent mass
A Pendulum
Bike wheel with negligible spoke mass
Hoop
Etc.

15. Rotational Inertia
Here are rotational inertias of various objects, each with mass m, around various axes. You can see how they differ.
Which have the greatest rotational inertia (resistance to change in rotation)? Pendulum, hoop normal axis
Remember, the farther away your mass is from the axis of rotation, the more RI you have. So it makes sense that a hoop or a pendulum would have greatest RI.

16. Rotational Inertia
Which will roll down an incline with greater acceleration, a hollow cylinder or a solid cylinder of the same mass and radius?
The answer is whichever has the smaller rotational inertia!
Because the cylinder with the greater rotational inertia requires more time to get rolling. So which has the smaller rotational inertia?
The solid cylinder has the smaller rotational inertia. SOLID will WIN

17. Rotational Inertia
It turns out that any solid cylinder will roll down an incline with more acceleration than any hollow cylinder, regardless of mass or radius.
A hollow cylinder has more “laziness per mass” than a solid cylinder.
Watch video of this experiment?

18. Check Question
A heavy iron cylinder and a light wooden cylinder, similar in shape, roll down an incline. Which will have more acceleration?
They have the same rotational inertia per mass so they’ll accelerate equally!
RI per mass is same for both, so both will accelerate equally.
The cylinders have different masses, but the same rotational inertia per mass, so both will accelerate equally down the incline. Their different masses make no difference, just as the acceleration of free fall is not affected by different masses. All objects of the same shape have the same “laziness per mass” ratio.

19. Rotational Inertia Rotational inertia can be used to advantage.
Industrial flywheels
Note where most of the mass is on these wheels! The crescent on the train wheel is for balance. It counters the coupling and connecting rods on the other side.

20. Torque Torque is the rotational counterpart to Force
Apply a force if you want linear motion (acceleration)
Apply a torque if you want rotation.
The counterpart to Linear Speed is Rotational Speed
The counterpart to Inertia is Rotational Inertia
Now we get the rotational counterpart to FORCE.
Anytime you turn a doorknob, turn on a faucet or use a wrench, you are exerting torque and producing rotation.
DEMO: Weight hanging from meterstick at different distances

21. Torque
Now we have to expand our definition of Mechanical Equilibrium!
∑ F = 0 …
… AND…
∑ τ = 0
That’s a lowercase tau which represents torque

22. Torque Let’s consider a door:
So why do we place a door knob where we do? (At the far side opposite the hinge?) The farther from the axis of rotation that you exert the force, the more torque you get!
And why do you push perpendicularly to the plane of the door? It gives you more rotation for less effort!

23. Torque
When the force is perpendicular, the distance from the turning axis to the point of contact is called the lever arm.
If the force is not at right angle to the lever arm, then only the perpendicular component of the force will contribute to the torque.
A brief note about units…. We multiply force (Newtons) times lever arm (meters) and so the resulting units for torque are Newton-Meters.
However, this is not JOULES because we are not talking about WORK (Fd)! We leave it as-is, Newton-Meters
Also, torque IS a vector quantity. Torque is not energy (work) because energy is scalar. With torque, we are most certainly concerned with direction as well as magnitude.

24. Torque
The same torque can be produced by a large force with a short lever arm, or a small force with a long lever arm.
The same force can produce different amounts of torque depending on how big the lever arm is.
Greater torques are produced when both the force and lever arm are large.
Describe each drawing. Be sure to show the lever arm in the first one. You have to extend the line of the force until you can draw a line (lever arm) perpendicular to the force.

25. Check Question
If you cannot exert enough torque to turn a stubborn bolt, would more torque be produced if you fastened a length of rope to the wrench handle as shown?
NO! You haven’t extended the lever arm at all, and you can’t pull with any greater force now than you did before you attached the rope. If you want more torque, add a pipe to the handle of the wrench!

26. Check Question
Extending the length of the wrench handle by placing a pipe over it will get you more torque for the same force because you increased the lever arm!

27. Balanced Torques
A pair of torques can balance each other. Balance is achieved if the torque that tends to produce clockwise rotation by the boy equals the torque that tends to produce counterclockwise rotation by the girl.
Notice the magnitudes of the torques produced by the boy and girl:
Girl = 200N x 3m = 600Nm Clockwise
Boy = 400N x 1.5m = 600NM Counterclockwise
So the vector sum of the torques is zero and thus we have mechanical equilibrium.
Would anything change if the girl hung way beneath the see-saw? No. (Like fig 8.19 in book)

28. Check Question
This meter stick is suspended in mechanical equilibrium. What is the weight of the block hanging at the 10 cm mark?
(?) (40m) = (20N) (30m)
? = 15N

29. Balanced Torques
Scale balances that work with sliding weights are based on balanced torques, not balanced masses.

30. Center of Mass
A baseball thrown into the air follows a smooth parabolic path.
DEMO: Toss a ball around – path is always parabolic, always smooth

31. Center of Mass
A bat thrown in the air, however, wobbles about a special point. This point stays on a parabolic path, even though the rest of the bat does not.
The motion of the bat is the sum of two motions:
a spin around this point, and
a movement through the air as if all the mass were concentrated at this point.
This point, called the center of mass, is where all the mass of an object can be considered to be concentrated.

32. Center of Mass
Center of Mass is the average position of all mass that makes up the object
Center of Gravity is the average position of all the particles of weight that make up an object
This is a powerful statement and extremely useful. It means that we don’t have to take into account the non-uniform mass distribution of objects when we are trying to calculate where the force of gravity is acting, etc.
We can consider all the force of gravity to be acting at the center of mass regardless of what shape or mass distribution the object has. This is very convenient!
Sometimes you’ll hear people say, “center of gravity”. CG is just the average position of weight distribution. Since weight and mass are proportional, we can use either term to express this concept.
Center of mass is often called center of gravity, the average position of all the particles of weight that make up an object.
For almost all objects on and near Earth, these terms are interchangeable.
There can be a small difference between center of gravity and center of mass when an object is large enough for gravity to vary from one part to another.
The center of gravity of the Sears Tower in Chicago is about 1 mm below its center of mass because the lower stories are pulled a little more strongly by Earth’s gravity than the upper stories.

33. Center of Mass Location of the Center of Mass
For a symmetrical object, such as a baseball, the center of mass is at the geometric center of the object.
For an irregularly shaped object, such as a baseball bat, the center of mass is toward the heavier end.

34. Center of Mass
Objects not made of the same material throughout may have the center of mass quite far from the geometric center.
Consider a hollow ball half filled with lead. The center of mass would be located somewhere within the lead part.
The ball will always roll to a stop with its center of mass as low as possible.
Pass around and show CG bird

35. Center of Mass
The center of mass can even be located at a position where there IS no mass!
Or consider a donut - a torus

36. Center of Mass Motion About the Center of Mass
As an object slides across a surface, its center of mass follows a straight-line path.
The center of mass of the rotating wrench follows a straight-line path as it slides across a smooth surface.
Show CM Video
The motion of the wrench is a combination of straight-line motion of its center of mass and rotation around its center of mass.
If the wrench were tossed into the air, its center of mass would follow a smooth parabola.

37. Spin A force must be applied to the edge of an object for it to spin.
If the football is kicked in line with its center of mass, it will move without rotating.
If it is kicked above or below its center of mass, it will rotate.
When you throw a ball and apply spin to it, or when you launch a plastic flying disk, a force must be applied to the edge of the object.
This produces a torque that adds rotation to the projectile.
A skilled pool player strikes the cue ball below its center to put backspin on the ball.

38. Center of Gravity
So how can we precisely locate the center of gravity (CG)?
The CG of a uniform object is at the midpoint, its geometric center.
The CG is the balance point.
Supporting that single point supports the whole object.
The weight of the entire stick behaves as if it were concentrated at its center. The small vectors represent the force of gravity along the meter stick, which combine into a resultant force that acts at the CG.

39. Center of Gravity
If you suspend any object at a single point, the CG of the object will hang directly below (or at) the point of suspension.
To locate an object’s CG:
Construct a vertical line beneath the point of suspension.
The CG lies somewhere along that line.
Suspend the object from some other point and construct a second vertical line.
The CG is where the two lines intersect.
You can use a plumb bob to find CG for an irregularly shaped object.

40. Center of Gravity
The CG of an object may be located where no actual material exists.
Check Question: Can an object have more than one CG?
No. A rigid object has one CG. If it is nonrigid, such as a piece of clay or putty, and is distorted into different shapes, then its CG may change as its shape is changed. Even then, it has one CG for any given shape.
The CG of a ring lies at the geometric center where no matter exists.
The same holds true for a hollow sphere such as a basketball.

41. Check Question
A uniform meter stick supported at the 25-cm mark balances when a 1 kg rock is suspended at the 0-cm end. What is the mass of the meterstick?
Remember, all the mass of the meterstick can be thought of as acting at the CG. Where is the CG of a meterstick? 50-cm mark.
Draw in the force vectors. The meterstick is 1 kg.

42. Torque & Center of Gravity
If the center of gravity of an object is above the area of support, the object will remain upright.
If the CG extends outside the area of support, an unbalanced torque exists, and the object will topple.
If the CG extends outside the area of support, an unbalanced torque exists, and the object will topple.

43. Torque & Center of Gravity
This “Londoner” double-decker bus is undergoing a tilt test.
So much of the weight of the vehicle is in the lower part that the bus can be tilted beyond 28° without toppling.

44. Torque & Center of Gravity
The Leaning Tower of Pisa does not topple because its CG does not extend beyond its base.
A vertical line below the CG falls inside the base, and so the Leaning Tower has stood for centuries.
If the tower leaned far enough that the CG extended beyond the base, an unbalanced torque would topple the tower.
DEMO: L-shaped wood on incline. Use a plumb bob to show?

45. Torque & CG
Note that the support base does not have to be solid. Take this chair for instance. It’s support base is shown by the shaded area under the chair. As long as the CG is somewhere over that shaded area, the chair will not topple.

46. Balancing
Question: Which way do you suppose it will be easier to balance a baseball bat on your finger?
DEMO: Balancing BBall bat. Two concepts… keeping your finger beneath the CG. And RI – When more of the mass is farther from the axis of rotation, more RI. This means slower to start rotating giving me more time to move my finger under CG.
Find CG of bat by balancing horizontally.

47. Question Where is the CG of a person?
Standing normally, it is usually 2 to 3 cm below your navel, and midway between your front and back.
But it can change!
Your CG may be located outside your body altogether.
The CG of a human is not fixed! It depends on body position.
Standing straight and tall, it’s usually slightly below the belly button, though it tends to be lower for women and higher for men.
Raise your arms vertically overhead. Your CG rises 5 to 8 cm.
DEMO: Everybody stand with your feet against the wall. Now without moving your feet, touch your toes. You can’t do it! Your CG moves forward as you bend forward and at some point acts beyond your base of support (your feet).
Why can we do it when not backed against a wall? We compensate for the CG movement by shifting weight back.

48. Center of Gravity

49. Center of Gravity
A high jumper executes a “Fosbury flop” to clear the bar while his CG nearly passes beneath the bar.

50. Center of Gravity
When you stand, your CG is somewhere above your support base, the area bounded by your feet.
In unstable situations, as in standing in the aisle of a bumpy-riding bus, you place your feet farther apart to increase this area.
Standing on one foot greatly decreases this area.
In learning to walk, a baby must learn to coordinate and position the CG above a supporting foot.

51. Stability
When an object is toppled, the center of gravity of that object is raised, lowered, or unchanged.
Equilibrium is unstable when the CG is lowered with displacement.
Equilibrium is stable when work must be done to raise the CG.
Equilibrium is neutral when displacement neither raises nor lowers the CG.

52. Stability Write in: a) Unstable – CG lowers b) Stable – CG raises
c) Neutral – CG remains at same level

53. Stability
For the pen to topple when it is on its flat end, it must rotate over one edge. During the rotation, the CG rises slightly and then falls.
Toppling the upright book requires only a slight raising of its CG. Toppling the flat book requires a relatively large raising of its CG.
An object with a low CG is usually more stable than an object with a relatively high CG.

54. Stability
A pencil balanced on the edge of a hand is in unstable equilibrium.
The CG of the pencil is lowered when it tilts.
When the ends of the pencil are stuck into long potatoes that hang below, it is stable because its CG rises when it is tipped.
DEMO: Pencil with clothes pins on each side?

55. Stability
The CG of a building is lowered if much of the structure is below ground level.
This is important for tall, narrow structures.
The Seattle Space Needle is so “deeply rooted” that its center of mass is actually below ground level.
It cannot fall over intact because falling would not lower its CG at all. If the structure were to tilt intact onto the ground, its CG would be raised!

56. Rolling Spools
If time, demo Spool of ribbon.