CHAPTER 8 NOTES

Linear speed: Distance divided by time when going in a straight line

Tangential speed: The speed of something moving along a circular path (review tangent line) Calculated by taking the Circumference divided by the time.

Rotational (angular speed) = revolutions per minute (rpm's)

Relationships between the two:

1.) Think about a record on a record player spinning around (or you on a merry-go-round) What happens to your rpm's as you move out from the center? (rpm's do not change they are constant)

2.) What happens to your tangential speed as you move away from the center? (your tangential speed increases) Tangential speed is proportional to the distance you are from the center of the circle (double your distance from the center, double your tangential speed)

3.) What happens to your tangential speed when you increase the rpm's? (your tangential speed increases) tangential speed is also proportional to the rpm's. (Double the rpm's, and double your speed.)

Center of gravity: The point on a body through which gravity acts. When in flight, objects will rotate about their center of gravity.

examples in class: broom, bird, meter stick on fingers, lollipops, ball with weight in it , persons center of gravity.

Rotational inertia: Just as an object in a straight line has inertia (Newton's 1st law) so do objects that rotate about an axis in a circle.

Moment of inertia (I)= How the mass is distributed about the central axis. (see page 130 for different formulas for different objects)

Torque: (T) =  F x r  r = lever arm which is the distance from the force, perpendicular to the force, and F = force so the units are newton- meter. Torques cause twisting or turning and are always involved when an object rotates.)  You have clockwise and counterclockwise torques depending on which way the force tends to make the object rotate.  If the object is in equilibrium (no rotation) then the clockwise and counterclockwise torques are balanced!.

The angular acceleration which the torque produces depends on the mass of the rotating object and upon the distribution of its mass with respect to the axis of rotation

T =  F x r is similar to F= ma except on we have angular acceleration.  The T  = F and I (moment of inertia) = m so the equation for angular acceleration becomes α = T/I  Notice that I and α are inversely related.  That is the greater the I, the smaller the α (the greater the moment of inertia, the less the angular acceleration)

examples: spinning eggs, person on the chair with bricks, rolling soup cans. tight rope walker, diving and gymanstics. hammer trick and meter sticks with clay on page 130-131, stick with mass on it. (why does a hammer balance better when the handle is on your finger but a car is more stable than a SUV?)

Rotational equilibrium and calculating Coupled forces (Not in book but in handout that I gave you. )

Record the answers and how to do them to practice problems 1 and 2 below:

Uniform circular motion:  Constant speed (remember not velocity because you are changing direction) around a circle.  Remember since your velocity is changing their must be an acceleration:  (in this case not a change in speed but a change in direction so their must be a net force causing this. Remember Newton's first law)

Centripetal acceleration: a_c = v²/r

a.) Always points towards the center of the circle
b.) Directly proportional to the square of the speed
c.) Inversely proportional to the radius

Period: (T) Time to make one complete revolution.
Frequency (f) number of revolutions per second.

The period and the frequency are inverses of each other:  T= 1/f   If the period is 2 seconds then the frequency is 1/2 (if it takes 2 seconds to make a revolution then you are making 1/2 revolution per second)

Angular momentum(L) = Iw   I = moment of inertia and w= angular velocity measured in revolutions per second. ( just like linear momentum is mass x velocity)

Remember just like linear momentum, angular momentum is conserved unless an external torque acts on it.

This explains a few things:
1.) Spinning on a chair with bricks. What happens to your velocity?

Increase the moment of inertia the angular velocity must decrease because momentum is conserved. This also applies to gymnastics and diving when they do a tumble or a dive and wish to rotate faster they tuck their body into a ball thus decreasing their moment of inertia and increasing their angular velocity.  Remember momentum is conserved their are no outside forces cause their angular velocity to increase it does because they change the distribution of their mass.
2.) Tornado tubes: Water in the top of the jar spins faster because water can only enter the mouth of the bottle by giving up angular momentum to the water above. (when water goes down the drain does it spin faster at the middle of the funnel or at the wide edge of the funnel?  Why?
3.) angular momentum for circular motion

Figure 19.14: Conservation of Angular Momentum

In summary

1.) A wheel will not start to rotate unless a torque is applied (force through lever arm)
2.) A wheel which is spinning will continue to spin at a constant angular velocity unless a torque acts on it
3.) The above are both cases of Newton's first law applied to rotary motion
4.) If we wish to change the rotation (angular velocity) we must apply a torque about the axis.

Once an object is moving with an angular velocity is is very resistance to change because:
1.) When you attempt to tilt the axle you must apply a torque.
2.) Remember from Newtons 3rd law whey you apply a torque (which is essentially analogous to a action force) the wheel will exert an equal and opposite torque on you.
3.) An object or system of objects will maintain its angular momentum unless acted upon by an unbalanced torque (

Examples:

1.) Tilt a spinning wheel
2.) Sit on a chair on a chair and rotate to the left (which way does the chair rotate) Just as an external force is required to change the linear momentum of an object, an external torque is required to change the angular momentum of an object. When sitting on a chair any changes in the wheel must produce opposite change elsewhere so F_net = 0
3.) Falling cats

All of the above explains why:

1.) it is easier to balance a bike when it is moving
2.) A coin will not topple when rolling but will when standing still
3.) why it is easier to balance on the tips of skates why spinning rather than standing still
4.) The motion of a Top.
5.) Kicking and throwing a football
6.) Guidance systems

In depth example balancing on a bike:

Angular momentum is a vector quantity--it points in a definite direction. For example, a rolling coin has a different direction of angular momentum than a coin spinning like a top. The trouble with angular momentum is that, since it involves something that's turning, often it's not obvious what that direction is.

Fortunately, physicists have come up with a convention for the direction of angular momentum that makes angular momentum physics easy. This convention is known as the Right Hand Rule. Using your right hand, curl your fingers in the direction an object is spinning. Your thumb points in the direction of the angular momentum vector. (There are cross products and moments of inertia and other fancy physics stuff involved--if you're sufficiently fascinated, get a book or take a physics course.)

So let's look at that bike. Using your right hand, curl your fingers in the direction the bike wheels are spinning--forward and around. You should find your thumb pointing to the left. OK, now extend your left arm straight out to the left and point your left index finger. This represents the angular momentum vector of your bike wheels. Unfortunately, sticking your arm out like that also throws off your balance. Oh no, you start to tip over to the left! (Go ahead, tilt to the left--you didn't think you could do physics without waving your arms around, did you?) Notice that your wheels' angular momentum vector (i.e., your left arm) is no longer pointing directly left, but rather at some angle towards the ground. The angular momentum vector of your wheels has changed, but that's a problem because angular momentum is conserved. You need to acquire some other component of angular momentum to ensure conservation. To restore the original angular momentum of your untilted bike, you need to add a vector quantity in the "up" direction to your tilted left arm. What does that mean in terms of motion? The Right Hand Rule tells us. Using your right hand, stick your thumb in the upward direction to represent the additional angular momentum you need. Your fingers are curling around to the left. Thus, your bike will turn to the left to conserve angular momentum.

That was a whole lot of physics and gymnastics to conclude what every 7-year-old knows: when you lean your bike left left, you turn left. The magic of physics. The gyroscopic effect tends to convert a tipping-over motion into a left- or right-turning motion. You can see why gyroscopes are handy as stabilization devices in boats, where turning is preferable to tipping over.

On a moving bike, it's fairly easy to recover from a left- or right-turning motion: you can steer the bike or lean the other way. On a non-moving bike, a tipping motion is converted, thanks to gravity, into an even faster tipping motion. Your only recourse is to quickly throw your center-of-mass around to try to keep it over the bicycle's base, and since the base of a bike is very narrow, this is hard to do, especially since tipping over tends to move your center of mass rapidly off your base. http://www.straightdope.com/mailbag/mangularmo.html

A rotating bicycle wheel has angular momentum, which is a property involving the speed of rotation, the mass of the wheel, and how the mass is distributed. For example, most of a bicycle wheel's mass is concentrated along the wheel's rim, rather than at the center, and this causes a larger angular momentum at a given speed. Angular momentum is characterized by both size and direction.

The bicycle wheel, you, and the chair comprise a system that obeys the principle of conservation of angular momentum. This means that any change in angular momentum within the system must be accompanied by an equal and opposite change, so the net effect is zero.

Suppose you are now sitting on the stool with the bicycle wheel spinning. One way to change the angular momentum of the bicycle wheel is to change its direction. To do this, you must exert a twisting force, called a torque, on the wheel. The bicycle wheel will then exert an equal and opposite torque on you. (That's because for every action there is an equal and opposite reaction.) Thus, when you twist the bicycle wheel in space, the bicycle wheel will twist you the opposite way. If you are sitting on a low friction pivot, the twisting force of the bicycle wheel will cause you to turn. The change in angular momentum of the wheel is compensated for by your own change in angular momentum. The system as a whole ends up obeying the principle of conservation of angular momentum.

http://www.exploratorium.edu/snacks/bicycle_wheel_gyro/index.html

Centripetal force: According to Newton's first law of motion everything wants to continue to go in a straight line unless an outside force acts on it.  Therefore if something travels in a circle their must be a force acting on it. This is caused the centripetal force and it causes object so go in a circle.

Identify the centripetal force for the following:

1.) A bucket being swung in a circle
2.) A car going around a curve

The formula for centripetal force  Fc= mv²/r

Centrifugal force (inertial force) : The "feeling" that you are being "thrown" outward when traveling in a circle.  Remember this is not a real force. Their is nothing pushing you out of the circle. Centrifugal force is due to inertia( the tendency of you to want to travel in a straight line.) Remember if you swing a ball on a string and let go, the ball will continue off in a straight line path.