Angular Kinematics Ch 9 Notes

What is a radian? 1 Θ rad = 57.3 ͦ of twist

Angular Variables Angular Variables Ɵ = S / r 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 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 Ɵ = S / r ω = v T / r α = a T / r S

Centripetal and Radial Accel 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

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? 28 rads/s 2

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? 20 m/s 2 or 20 rads/s 2

A closer look at acceleration… // to motion: tangential acceleration –Leads to more/less rpm Perp. to motion: radial acceleration or centripetal acceleration –Gives inward acceleration needed to make circular motion Acceleration can be broken down

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 f = ώ + αΔt V f 2 = V i 2 + 2ax ώ f 2 = ώ i 2 + 2αΘ x = v i t + ½at 2 Θ = ώ i 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? An 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” Measure of the unwillingness of a material to want to rotate Increases with greater overall mass, no matter what the shape of the material Shape affects moment If all mass on outside of orbit, I = mr 2 Others p. 298

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” KE rot = ½ I ω 2 Objects that are rolling tend to have both forms of KE!

Example: Total KE What is the total KE of a basketball (0.60 kg, diameter 30.0 cm) that is rolling at a speed of 2.00 m/s on the ground?

Moment of Inertia Moment of inertia generally higher the more mass is distributed further from the axis of rotation

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

Shapes & Moments of Inertia

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

Extras: Ch Angular Momentum: tendency for a system to rotate L = I ώ L = r m v units are kg m 2 /s Angular Momentum for a system is conserved Figure skaters Elliptical orbits Bike tire