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Seventh Homework – Cancelled due to furloughs(Automatic 20 points of credit)

Campus-wide Furlough Day Monday, October 19th (in about 10 days)(Art/Phys 123 will meet on Wed., Oct. 21st)

First Term PaperWednesday, October 14th (week from today)

For full schedule, visit course website:ArtPhysics123.pbworks.com

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Tipping, Turning, and

Spinning

Understanding Rotation

Rotation was mentioned in previous lectures but now we focus on the cause of rotation.

The short answer is: torque.

The complete answer also requires you understand inertia, lever arms, etc.

Axis of RotationRotation occurs around a line called the axis of rotation.

The axel is the axis of rotation for a set of wheels.

Mass is a measure of inertia for linear motion.

Rotational inertia is similar concept for rotation.

Inertia

M mGold brick Normal brick

Difficult to move Easy to move

x xWood Bat Plastic Pee-wee Bat

Difficult to Rotate Easy to Rotate

Rotational Inertia

Rotational inertia depends on: • Total mass of the object• How the mass is distributed

The farther the object’s mass is from the axis of rotation, the larger the rotational inertia.

Demo: Inertia Sticks

Two metal pipes of the same mass

Rotate

Leadweights

Easyto

Rotate

Hardto

Rotate

Axis ofRotation

Human Rotational InertiaIn which pose does the dancer have a

larger rotational inertia?Axis of Rotation

A B

A) Pose AB) Pose BC) Same for A & B

since mass is unchanged.

Human Rotational Inertia

Axis of Rotation

A B

Pose B has larger rotational inertia since the leg is extended, putting mass further from the axis of rotation.

Demo: Long Legs

Long legs have greater rotational inertia than short legs so long legged animals have a slow walking stride.

Demo: Drop the Stick

Two meter sticks stand upright against a wall; one has a hunk of clay on the end.

Which stick will swing down and hit the floor first?

The one without the hunk of clay.

Why?Clay increases rotational

inertia, which slows the rotation.

Axis of Rotation

Demo: Drop the Stick

Axis of Rotation

X CG

X CG

Another way to understand why the weighted stick takes longer to fall is that it’s center of gravity is higher.

The higher it is, the longer it takes for an object to fall to the ground.

Longer stick tips over more slowly

Tripping and FallingIf small child trips, he

hits the ground more quickly than an adult.

Can view this two ways:*Child has small

rotational inertia.*Child’s center of gravity

is initially closer to the ground.

Axis of Rotation

X CG

X CG

Demo: Hammer Balance

BA

Axis of Rotation

In which case is the hammer easier to balance on your finger?

A)Case AB)Case BC)The same for A & B

Demo: Hammer Balance

In case B the rotation is slower and thus easier to balance.

In case B the rotational inertia is greater because most of the mass is far from the axis of rotation (at your fingertip).

BA

X CG

X CG

Tightrope Walkers

The Great Blondin Is First to Walk Across Niagara Falls (June 30, 1859)

Tightrope walkers carry a long pole to increase their rotational inertia, which slows their rotation when they’re off balance.

When a force causes a rotation, we identify this as a torque.

Torque depends on• Magnitude of Force• Direction of Force• Lever Arm

(Torque) = (Force) x (Lever Arm)

Torque

Lever ArmLever arm is the perpendicular distance from axis of rotation to the direction of the force.

Opening or Closing a DoorFor a door the hinge is the axis of rotation.

For maximum lever arm (and maximum torque), push perpendicular to the door at the edge opposite from the hinges.

DoorHinge

Pirouettes

Play

A ballet turn done on one leg, starting with one or both legs in plié and rising onto demi-pointe (usually for men) or pointe (usually for women).

A pirouette is typically done en dehors (moving leg starts at the front or the side and moves towards the back).

Torque for a Pirouette

The farther the distance between the feet, the greater the lever arm so the greater the torque for creating the rotation.

Push onFloor

ReactionForce

LeverArm

Feet apart(Easy)

Feet together(Harder)

Fouetté (Whipped) Turns

A turn executed with a quick thrust of the moving leg as it passes in front of or behind the supporting leg.

Done properly, the dancer returns to the same spot at the end of the turn.

Play

Torque for Fouetté Turns

Push onFloor

ReactionForce

LeverArm

Lower heel to the floor

Push off while swinging right leg

Lift heel and return to point

The torque first creates a rotation of the arm & leg, then whole body rotates together

Fantasia (1940)The most famous ballet scene in animation is The Dance of the Hours in Disney’s Fantasia.

Tipping and Torques

Slowing out as an object tips over is due to the torque created by the force of gravity.

X

LeverArm

GravityForce

As the tipping angle increases, the lever arm increases so the torque (and acceleration) increase as well.

Slowing In and Slowing Out

X

If brick is not moving…

…then it slows out as it tips over

X

If brick is moving upward…

…then it slows in as

it tips upward

The lever arm changes with angle so the timing of this slowing in or slowing out has a lot of texture.

Lowering the Center of GravityIf nothing prevents a stationary object from lowering its center of gravity then it will do so.Stepping off of a high tree branch, Tarzan swings downward on a vine.

The motion may continue past the lowest point, due to follow-through (inertia), but if he keeps swinging back and forth he’ll eventually come to rest with the CG as low as possible (right under the point of suspension).

Tipping & Center of Gravity

A brick sitting on a ramp does not tip over if the rotation would raise the center of gravity.

RollingWheels roll on a slope because they

can always lower their center of gravity.

STABLE

UNSTABLE

UNSTABLE

Also notice the line of gravity and the base of support.

Class Demo: Mystery Wheel

Wheel doesn’t roll down hill. How is that possible?

CG

HiddenWeight

HoleCG

For both wheels rolling downhill would raise the center of gravity.

Wile E. Coyote & Tipping

Play

Wile E. Coyote & Tipping

X

The boulder could roll to the right if the CG was far off center (vein of uranium ore?) but removing keystone unlikely to initiate that rotation.

Center ofGravity ?

Axis ofRotationNote: This plumb

line does not mark where the boulder will actually land, even if it fell to the left.

Class Demo: Hammer & HingeHow is it that the hinged board stays up?

The center of gravity is located near the head of the hammer so the CG is lowest when the board is up.

Hinge

Mechanical Lever

Lever converts a small force into a large force by ratio of distances acting.

Push down with a small force over a large distance

Lift a large weight over a small distance

Axis of rotation goes through the fulcrum point.

Fulcrum

First Class Levers

Fulcrum

Effort ArmLoadArm

Lift a heavy load using small effort by having a long effort arm and/or short load arm.

First Class Lever: Fulcrum is located in between the input force (effort) and output force (load).

Examples of First Class Levers

Crowbar

Scissors

Trebuchet

EffortLoad

Fulcrum

Second Class Levers

Lift a heavy load using small effort by having a long effort arm and/or short load arm.

Second Class Lever: Load is located in between the effort force and the fulcrum.

Effort Arm

LoadArm

Fulcrum

Examples of Second Class Levers

Wheel Barrow

EffortLoad

Fulcrum

Hole Punch

Third Class Levers

Lift a light load using large effort when the effort arm is short and/or long load arm.

Third Class Lever: Effort force is located in between the load and the fulcrum.

Effort Arm

LoadArm

Examples of Third Class Levers

Effort

Load

Fulcrum

BroomJaw

Human Arm as a LeverBiceps muscle exerts effort force close to your elbow (fulcrum) to raise your forearm (load).This is what type of lever?

A)First classB)Second classC)Third class

Human Arm as a LeverBiceps is a third class lever so a large effort force acts over a small distance to move a small load over a large distance.

Triceps is also a third class lever, which pulls the arm in the opposite direction.Because muscles can only

contract, they’re almost always found in pairs, like biceps/triceps.

Human Foot as a Lever

EffortLoad

Fulcrum

To lift the body on the toes, the gastrocnemius (one of the strongest muscles in the body) contracts, lifting the heel upward.

This is an example of a second class lever.

Weight shift forward shortens the load arm, reducing the required effort by the muscle.

Raising heel, as in walking.

Human Neck as a Lever

Load

Fulcrum

In general, the levers in animals’ bodies sacrifice efficiency to keep a compact body form.

The neck muscles are the body’s only first class lever and even this lever is not very efficient since the effort arm is shorter than the load arm. Effort

Action/Reaction for TorquesFor every action force there is an equal and

opposite reaction force.

The action and reaction torques, caused by these forces, might not be equal since the lever arms may be different.

However, if the action torque creates a clockwise rotation, the reaction creates a counter-clockwise rotation (and vice versa).

Demo: Mid-Air Twist

Stand up and clear space around you.

When I say “Jump!”, jump.

In mid-air I’ll point left or right and I want you to try to turn so you land facing that direction.

Jump! Turn Land

How can you rotate in mid-air without pushing off of anything?

Demo: Mid-Air Twist

Jump! Turn

As you turn your legs 90 degrees, your arms and torso rotate in the opposite direction.

Sticking your arms out as you turn helps by increasing the rotational inertia of your upper body.

A large rotation of your legs is exactly cancelled by a small rotation of your outspread arms and torso.

Demo: Mid-Air Twist

Your rotation stops as soon as you stop rotating your upper body but by that time you’ve landed with your feet turned to the side.

Once on the ground you can push off on the ground to restore your arms and torso to a normal stance.

Turn Land

Front Side 180

Jump! Turn Land

The same principle is used in skateboarding tricks, such as a front side 180, in which a skater does a half turn in mid-air, turning upper and lower torso in opposite directions.

Demo: Drop the Cat Again

www.abc.net.au/science

Cat lands on its feet by clever use of action/reaction combined with changing rotational inertia by extending or pulling in legs.

Demo: Drop the Cat Again

Fouetté Turns, Analyzed

1

2

3

4

5

6

The torque from pushing off gives angular momentum to the right arm and leg, which rotate freely from #1 to #4.

Then the right arm and leg are made to rotate back in the opposite direction.

By action/reaction, the torso recoils and rotates in the original direction.

1

2

3

4

5

6

Play

Anchors Aweigh (1945)

1

The musical Anchors Aweigh has a famous scene in which Gene Kelley dances with Jerry the Mouse.

The scene climaxes with a series of fouetté turns.

@ 1:24

Next LectureSquash & Stretch

First Term Paper due onWednesday, October 14th

(In about 10 days)

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