Angular Mechanics Chapter 8/9

Similarities LinearAngular MassMoment of Inertia ForceTorque MomentumAngular 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 = mr 2

Moments For Various Objects ObjectLocation of axis DiagramMoment of Inertia Thin HoopCenter Solid CylinderCenter Uniform Sphere Center Uniform Rod Length L Center Uniform Rod Length L Through End Thin Plate Length L Width W Center

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

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 = rω Where r is the radius and ω is angular velocity and is measured in m/s.

How Fast Does the Earth Spin? 1rev/24 hrs 2π radians/revolution Radius or earth = 6.38x10 6 m

Angular Acceleration The change in angular velocity per unit of time. –Measured in rads/sec 2 Acceleration of an object is found by a = rα And is measured in m/s 2.

Linear and Angular Measures QuantityLinearAngularRelationship Displacement d (m)θ (rad)d = r θ Velocity v (m/s)ω (rad/s)v = r ω Acceleration a (m/s 2 )α (rad/s 2 )a = r α

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 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

Conservation of Angular Momentum angular momentum = rotational inertia  rotational velocity L = I  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)

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) r θ

A Balancing Act Static equilibrium occurs when the sum of the torques add to equal zero.