Angular Kinematics Ch 4 Notes

What is a radian?

Angular Variables Angular Variables 1 Θ rad = 57.3 ͦ of twist 1 ω = 57.3 ͦ of twist in a second’s time = Θ rad / 1 s 1 α = change in angular speed equal to 57.3 of twist per second = Δω / 1 s The radian itself has no units S θ rad

Angular Kinematics Definitions Angular Velocity: “amount of twist per time” ώ = ΔΘ / Δt units: rad/s Angular Acceleration: “amount of change in twist per time” Angular Acceleration: “amount of change in twist per time” α = Δώ / Δt Units: rad/s 2

Linking Linear with Angular… v t = Δx/Δt = C/T = 2πr/T 1 rev = 2π radians = 6.28 radians Centripetal or radial acceleration a = a r = v 2 /r in linear units a c = a r = v 2 /r in linear units a c = a r = rώ 2 in angular units

Clockwise (CW) vs. Counterclockwise (CCW) CCW is +, CW is – throughout course

1. Finding radial acceleration- linear units What is the radial acceleration of an object that spins with a linear speed of 4.00 m/s at a 0.80 m distance from the axis of rotation?

2. Finding radial acceleration- angular units What is the radial acceleration of a motorcycle proceeding in a circular cage of diameter 14.0 m if it is moving at an angular velocity of 2.0 rad/s?

A closer look at acceleration… For objects that are moving, the acceleration can be broken down into components parallel to and perpendicular to motion. // to motion: tangential acceleration –Leads to more/less rpm, units m/s 2 Perp. to motion: radial acceleration or centripetal acceleration –Gives inward acceleration needed to make circular motion, units m/s 2

Galileo’s Formulas Work for Rotation!!!! Angular displacement: Θ(# radians spun) rad Angular velocity: ώ (radians spun per second) rad/s Angular acceleration: α (Δ in rps per sec) rad/s 2 V f = v i + aΔt ώ = ώ 0 + αΔt V f 2 = V i 2 + 2ax ώ 2 = ώ α(ΔΘ) x = v i t + ½at 2 Θ = ώ 0 t + 1/2αt 2

Rotation and Graphing Θ for x ώ for v α for a SLOPES TO GO UP AREAS UNDER CURVE TO GO DOWN

3. What is angular acceleration? A piece of tape at the edge of the disk of radius 0.19 m slows from a speed of 2.00 m/s to a speed of 1.00 m/s in a 6.00 second time frame. What is the angular acceleration of the tape? What is the angular acceleration of the inside edge of the disk (r = 0.03 m)?

4. What is the final angular velocity? An object being swung around with a radius of 0.50 m is subject to an angular acceleration of 0.25 rad/s 2 for a 3.00 second time frame. If it started with a speed of 8.00 m/s, what is its angular velocity after the three seconds have elapsed? What number of rotations will have occurred in this 3.0 sec time span?

5. What is the angular acceleration? A disk is spun for a total of 62 rotations. It begins at an angular velocity of 1.8 rad/s and due to friction this value drops to 1.2 rad/s by the 62 nd rotation. What is this disk’s angular acceleration?

Moment of Inertia “Inertia of Spin” Think of it as “Rotational Mass” Measure of the unwillingness of a material to want to rotate Units: Kg*m 2 Shape affects moment If all mass on outside of orbit, I = mr 2 Others p. 298

Moment of inertia changes for the same material according to what spin axis you choose for it.

5. Which has a greater Moment Which object requires more force to spin about its center of mass? –A hoop of diameter 15cm and 0.8kg –Or a disk of diameter 15cm and 0.8kg Justify your answer.

Torque Rotational Force τ = Torque [Nm] or [Foot-Pounds] Which F will cause the most spin?

Torque Torque depends on Force and Radius Only Forces perpendicular to Radius add torque

6. Calculate the Torque Jimmy physics slams a door that has a length (measured from the hinges) of 0.75m. He applies a 20N force at an angle of 30degrees from the door.

Newton’s 2 nd Law Rotational Force Rotational Mass F net = m*a τ net = I*α

What has more kinetic energy, a knuckleball pitched at 70 mph, or a changeup pitched at 70 mph? Close but not identical!!!!!!

Rotational Kinetic Energy “Energy of Spin” as opposed to “Energy of Motion” E k Linear = ½mv 2 E k Rotational = ½ I ω 2 Units: [Joules,J] Objects that are rolling have both forms of E k !

Example: Total KE What is the total E k of a basketball (0.60 kg, diameter 30.0 cm) that is rolling at a speed of 2.00 m/s on the ground? (I spherical shell = 2/3 mr 2 )

Which object will reach the bottom first?

The disk cylinder has a lower moment due fact that mass closer to axis of rotation is easier to rotate Τ = F r T = I α