Angular Mechanics

Chapter 8/9

Similarities

Linear Angular

Mass Moment of Inertia

Force Torque

Momentum Angular Momentum

Center of Mass

• The center of mass of an object is the average position of mass.

• Objects tend to rotate about their center of mass.

• Examples: • Meter stick• Rotating Hammer• Rolling Double-Cone

Stability• For stability center of gravity must be over area

of support.

• Examples: • Tower of Pisa• Touching toes with back to wall• Meter stick over the edge

Otherwise we will get a rotation!

I = Rotational Inertia • An object rotating about an axis tends to

remain rotating unless interfered with by some external influence.

• This influence is called torque.

• Rotation adds stability to linear motion.– Examples:

• spinning football• bicycle tires• Frisbee

Moment of Inertia

• Defined as resistance to rotation – depends on mass– depends on

distance from axis of rotation

I = mr2

Moments For Various ObjectsObject Location of

axis Diagram Moment of

InertiaThin Hoop Center

Solid Cylinder Center

Uniform Sphere

Center

Uniform Rod Length L

Center

Uniform Rod Length L

Through End

Thin Plate Length L Width

W

Center

mr2

1

2mr2

2

5mr2

1

12ml2

1

3ml2

1

12m(l2 + w2 )

• The greater the distance between the bulk of an object's mass and its axis of rotation, the greater the rotational inertia.

• Examples: – Tightrope walker– Metronome

Ways to Measure Rotation

• Degrees: 1/360th of a revolution

• Radians: of a

revolution

1 revolution = 2 radians

1

2π

π

Angular Displacement

• Found by change in θ.

• Distance around a pivot is found by

• d = r θ

– Where the angle is measured in radians and r is the radius of the arc.

– Measured in meters

Angular Velocity

• The rate of revolution around an axis.– Measured in rads/sec

• Velocity around an axis is found by

v = rωWhere r is the radius and ω is angular velocity and is

measured in m/s.

ω =ΔθΔt

How Fast Does the Earth Spin?

• 1rev/24 hrs

• 2π radians/revolution

• Radius or earth = 6.38x106m

ω e =1rev

24hrs=

2π rad

86400s= 7.27x10−5 rad

sec

v =rω =6.38x106 m(7.27x10−5 rads

) =464ms

Angular Acceleration

• The change in angular velocity per unit of time.– Measured in rads/sec2

α =ΔωΔt

Acceleration of an object is found by

a = rα And is measured in m/s2.

Linear and Angular MeasuresQuantity Linear Angular Relationship

Displacement d (m) θ (rad) d = r θ

Velocity v (m/s) ω (rad/s) v = r ω

Acceleration a (m/s2) α (rad/s2) a = r α

Direction ofMotion

Centrifugal Force

CentripetalForce

Centripetal or Centrifugal?

No Matter What Faith Hill Says,IT’S NOT CENTRIFICAL MOTION!

Centripetal Force• …is applied by some object.

• Centripetal means "center seeking".

Centrifugal Force

• …results from a natural tendency.

• Centrifugal means "center fleeing".

• This is a fictitious force for us. Why?

Centripetal motion

ac =v2

r

Fc =mv2

r

Practice Wall and Wall

Pg 234, 6-6, Pg 243, 6-12

Examples

• water in bucket

• moon’s orbit

• car on circular path

• coin on a hanger

• jogging in a space station

Centripetal Force

• Bucket

• Earth’s gravity

• Road Friction

• Hanger

• Space Station Floor

Centrifugal Force

• Nature

• Nature

• Nature

• Nature

• Nature

Conservation of Angular Momentum

• angular momentum = rotational inertia rotational velocity

• L = I ω

• Newton's first law for rotating systems: – “A body will maintain its state of angular momentum

unless acted upon by an unbalanced external torque.”

• Examples: –1. ice skater spin–2. cat dropped on back–3. Diving–4. Collapsing Stars (neutron stars)

Torque

• Force directed on an object that has a fixed point is found by

– Where τ is torque, F is force in N, r is distance from the axis in m, and θ is measured IN DEGREES.

– (use sin θ only if force is not || to motion)

τ =Fr sinθ

Levers

• The lever arm is the distance from the axis along a θ to the direction of applied force.

• Torque here is force times the lever arm.

Lever arm (r sin θ)

rθ

τ =Fr sinθ

A Balancing Act

• Static equilibrium occurs when the sum of the torques add to equal zero.

NOW BUILD YOUR OWN!

• Must involve at least 3 different axes of rotation.

• Must hang at least 8 objects.

• No 2 objects can have the same mass.• No two hangers can have the same length.• No fulcrum can be in the middle of a hanger.