AP Physics CRotational Motion

Fundamental Concepts and Applications

December 3, 2012

In mathematics and physics, a specific form of

measurement is used to describe revolution and

fractions of revolutions. In one revolution, a point on the edge travels a distance equal

to 2π times the radius ofthe object. For this reason,

the radian is defined as ½ π of a revolution. In other

words, one complete revolution is equal to 2π

radians. A radian is abbreviated “rad.”

 Angular Displacement

The Greek letter theta, Ө, is used to represent the angle of revolution. Note

that counterclockwise rotation is designated as positive, while clockwise is negative. As an object rotates, the change in the angle is called angular

displacement.

In general, for rotation through an angle, Ө, a point at a distance, r, from the center, as shown above, moves a distance

given by d = r Ө. If r is measured in meters, you might think that multiplying it by Ө rad would result in d being measured in

m•rad. However, this is not the case. Radians indicate the ratio between d and r. Thus, d is measured

in m.

Angular Velocity

How fast does a CD spin? How do you determine its speed of rotation? Recall from Chapter 2 that velocity is

displacement divided by the time taken to make the displacement. Likewise, the angular velocity of an object is angular displacement divided by the time taken to make the displacement. Thus, the angular velocity of an object is given by the following equation, where angular velocity is

represented by the Greek letter omega, ω.

It’s so easy even a six-year old can comprehend this simple concept…

If an object’s angular velocity is ω, then the linear velocity of a point a distance, r, from the axis of rotation is given by v = r ω.

The speed at which an object on Earth’s equator moves as a result of Earth’s rotation is given by v = r ω or (6.38X106 m) (7.27X 10-5 rad/s) = 464 m/s. Earth is an example of a rotating, rigid body.

Even though different points on Earth rotate different distances in each revolution, all points rotate through the same angle. All parts of a rigid body rotate at the same rate.

The Sun, on the other hand, is not a rigid body. Different parts of the Sun rotate at different rates. Most objects that we will consider in this chapter are rigid bodies.

Angular Acceleration

What if angular velocity is changing? For example, if a car were accelerated from

0.0 m/s to 25 m/s in 15 s, then the angular velocity of the wheels also would change from 0.0 rad/s to 78 rad/s in the same 15 s. The wheels would undergo

angular acceleration, which is defined as the change in angular velocity divided by the time required to make the change. Angular acceleration, a, is represented by

the following equation

         Linear                                                     Angular                                                                                            Equations

for Linear Motion and Rotational Motion are

quite similar!!

The displacement, θr, of an object in circular motion, divided by the time interval in which the displacement occurs, is the object’s average velocity during that time interval.

The direction of the change in velocity is toward the center of the circle, and so the acceleration vector

also points to the center of the circle.

As the object moves around the circle, the direction of the

acceleration vector changes, but its length remains the same. Notice that the acceleration vector of an object in uniform circular motion always points in toward the center of the

circle. For this reason, the acceleration of such an object is

called center-seeking or centripetal acceleration.

If a car makes a sharp left turn, a passenger on the right side might be thrown against the right door. Is there an outward force on the passenger?

A passenger in the car would continue to move straight ahead if it were not for the force of the door acting in the direction of the acceleration; that is, toward the center of the circle. Thus, there is no outward force on the passenger.

Another special 2D motion

• Uniform circular motion, another special case of 2D motion

• Object traveling at constant speed on a circle needs a force of fixed size, always  to motion, to keep moving in a circle

F

v

v F

v

F

v

F

How large is the acceleration in circular motion?

• We can tell just by examining the path

• Acceleration is big when v is large or the circle is small…

iv

fv

fv

iv

v

r

va

t

va

r

svvv

v

st

2

s

Uniform circular motion

• For uniform circular motion you can tell the acceleration from geometry

• The minus sign and symbol mean that the force is always pointed to the center 

• This is exactly like saying

For the case of uniform linear motion. Doesn’t tell you what the forces are or where the come from: just what the sum is….

rr

mvamF ˆ

2

r

0 amF

“Centripetal” or center-seeking force

A ball on a string travels in a horizontal circle on a table at constant speed. The string provides the centripetal force required to keep it turning in a circle. If the string is suddenly cut when the ball is in the position shown, which path does it follow?

v

r

1

2

3

4

Once the string is cut the net force is zero and the motion continues unchanged…

Sparks demo…

To keep an object going in a circle something has to supply this centripetal force…

Sometimes this is very hard… when m is big, v is big, or r is small

Realistic example: Ball swinging around a pole, like tetherball.

0cos

sin2

mgTFr

mvTF

y

x

mgr

mv

r

mvT

cossin

sin2

2

tan2

gr

v

x

yr

NOTE: this is the center of this circular motion!

Driving in a circle

Friction between tires and ground pushes car inward.

gr

v

mgFr

mvF

s

sNsf

2

2fF

Dependence on s is very unfortunate: it changes a lot when a road is wet or icy!

Driving in a circle

If you try to drive in a circle for which the combination of v2/r is larger than the sg that static friction can support, what path will the car follow?

1

2

3

4

5

There will still be some frictional force, just not enough to stay on the circle!

Stability in turns

• Large frictional force pushing to the side on cars can cause them to flip…

• Shorter, flatter cars are more stable

“Artificial gravity”: sensation of weight is really the normal force pushing up. Can be used in spaceships to provide something like gravity…

r

mvFN

2

r

mvFN

2

If you make v2/r = g, FN feels just like Earth gravity!

2001: A Space Odyssey

Technical parameters of the large TsF-18 Centrifuge:

- Radius of rotation (arm) - 18 m- Mass of rotating elements - more than 300 tons- Maximum G-load - 30 G- G-load gradient - up to 5 G/s- Quantity of parameters transmitted by telemetry from the cabin: about 100

ACG Space Technologies Corporation Sometimes v2/r is much bigger than g….

Cosmonaut training machine

Vertical circles

mg T

mg

T

T + mg = mv2/rT = mv2/r - mg

T - mg = mv2/rT = mv2/r + mg

Whatever supplies the centripetal force gets help from gravity at the top, and must work against it at the bottom….

mg T

T + mg = mv2/rT = mv2/r - mg

When a ball swings on a string in a vertical circle, the tension at the top is given by:

T = mv2/r – mgWhat happens when the required centripetal force mv2/r is less than the weight mg?

1. The tension will become negative

2. The ball while fly outside the circle

3. The ball will fall into the circle

This enhanced sensation of weight is a real problem for pilots and astronauts, who may experience large “g-forces”

mgF

mgr

mvF

r

mvmgFF

N

N

N

4

1kmr and 500mph, vIf

2

2

mg

FN

Looping coaster in 1902 Atlantic City

“Mantis” at Cedar Point in Sandusky

Real vertical circles

mg T

T + mg = mv2/rT = mv2/r - mg

Ball slows down while rising on this side

Ball speeds up while falling on this side

The fact that it travels faster at the bottom than the top makes the difference in the tension between the two larger

Speeds up

Speeds up

Slows down

Loop-the-loop demonstration

I hang a bucket of water from a rope. The water is prevented from falling by the walls of the bucket. If I now swing the bucket in a vertical circle what will happen?

1. There is no tension to pull the water into a circle, so I will get wet

2. The water will loop-the-loop and I will stay dry

The same normal force which holds the water in the bucket provides the centripetal force which keeps it on the circle.

Wine glass…

Relevance of circular motion

• Often motion can be approximated as a circle, provides useful estimates of forces

• Planets orbit in nearly circular orbits

• Rigid body rotation: rotating objects must hold themselves together…

Fin

v

Approximate circular motion

• Each part of every 2D path can be approximated by a part of a circle

• This lets you estimate the  component of the force

• Where it curves fast, r is small, and mv2/r is large; when slow, it’s small

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