1. Angular AccelerationObjectives:• Introduce the concept of angular acceleration• Learn how to compute and estimate angular acceleration• Understand the difference between average and instantaneous angular acceleration• Learn to use the laws of constant acceleration Angular Acceleration• The rate of change of angular velocity change in angular angular velocity acceleration = change in time• Shorthand notation: ωfinal – ωinitial ∆ω α = = tfinal – tinitial ∆t• Has units of (angular units)/time2 (e.g. radians/s2, °/s2) 1
2. Computing Angular Acceleration• Angular acceleration reflects a change in velocity of rotation ω2 – ω1 30°/s α = = = 20°/s2 t2 – t1 1.5 s ω2 = 40°/s ω1 = 10°/s ∆θ orientation at t2 = t1+1.5 s orientation at t1 axis of rotation Effects of Angular Acceleration• Velocity ω and acceleration α in same direction: magnitude of angular velocity increases• Velocity ω and acceleration α in opposite direction: magnitude of angular velocity decreases (deceleration) Velocity Acceleration Change in Velocity (+) (+) Increase in + dir. (+) (–) Decrease in + dir. (–) (–) Increase in – dir. (–) (+) Decrease in – dir. 2
3. Example Problem #1 A volleyball player spikes the ball. To bring her arm forward, she begins extending her shoulder from a flexion angle of 225°. She contacts the ball 120 ms later with her shoulder flexed to 160° and extending at 700°/s 100 ms later, at the end of follow-through, her shoulder stops extending at a flexion angle of 135° What was the average acceleration at the shoulder before and after ball impact? Instantaneous Angular Accel.• Previous formulas give the average angular acceleration between initial time (t1) and final time (t2)• Instantaneous angular acceleration is the angular acceleration at a single instant in time• Estimate instantaneous angular acceleration using the central difference method: ω (at t1 + ∆t) – ω (at t1 – ∆t) α (at t1) = 2 ∆t where ∆t is a very small change in time 3
4. Angular Acceleration as a Slope• Graph of angular velocity vs. time slope = instantaneous α at t1 ω (deg/s) slope = average α from t1 to t2 ∆ω(1→2) ∆t(1→2) ∆t t1 t2 time (s) Estimating Angular Acceleration ω (deg/s) Identify points with zero slope = points with zero acceleration 0 Portions of the curve time (s) with positive slope have positive accel. (i.e. acceleration in α (deg/s2) the + direction) Portions of the curve with negative slope 0 have negative accel. time (s) (i.e. acceleration in the – direction) 4
5. Example Problem #2 Pictured is the absolute angle of a hockey stick during a slap shot. Sketch the angular velocity and angular acceleration during the shot. 80 60 40Stick angle (deg) 20 0 -20 0 0.2 0.4 0.6 0.8 1 θ -40 -60 Time (s) Laws of Constant Angular Accel. • When angular acceleration is constant: ω2 = ω1 + α * ∆t ∆θ = ω1 * ∆t + (½) α * ( ∆ t)2 ω22 = ω 12 +2 α * (∆ θ ) where: α = angular acceleration ω 1 = angular velocity at initial (or first) time t1 ω 2 = angular velocity at final (or second) time t2 ∆ θ = angular displacement between t1 and t2 ∆ t = change in time (= t2 – t1 ) Use + values for + direction, – values for – direction 5
6. Example Problem #3 A discus thrower starts his spin while standing facing the back of the circle. He releases the discus 2 seconds later after completing 1.5 revolutions to his left. What was his angular acceleration during the throw (assuming a constant rate of acceleration)? How fast was he spinning after the first half-revolution? How fast was he spinning at the time of release? Rotation as a Vector• Rotational quantities ( θ, ω, α ) can be expressed by vectors directed along the axis of rotation.• Length of vector indicates magnitude• Direction of vector determined by “right hand rule” – Curl fingers of right hand in direction of rotational quantity – Thumb points in direction of vector z z axis of rotation ω y y x x ω 6
7. Linear & Angular Motion (part 1) Objectives: • Learn the relationship between linear and angular distance for a body in rotation • Learn the relationship between linear and angular speed for a body in rotation Linear & Angular Distance• The curvilinear distance (d) traveled by a point on a rotating body is: d d=rφ where: φ • r = radius of rotation (distance of the point r from the axis of rotation) • φ = angular distance traveled in radians! axis of rotation 7
8. Linear & Angular Speed• The linear speed (s) of a point on a rotating body is: d rφ φ s=rσ s= = =r =rσ ∆t ∆t ∆t where: d • r = radius of rotation • σ = angular speed φ in radian/s ! r axis of rotation Radius of Rotation, Distance & SpeedThe greater the radius of rotation (r):• the greater the curvilinear distance (d) traveled for a given angular distance ( φ)• the greater the linear speed (s) for a given angular speed ( σ ) d1 d=rφ d2 s=rσ if r1 > r 2 φ then d1 > d2 r2 r1 axis of rotation s1 > s2 8
9. Example Problem #1Two distance runners are racing. Runner #1 is in the inside lane. Runner #2 is on his shoulder, in the second lane.The inside radius of the track is 36.8 m. Each lane is 1.1 m wide.How much farther must Runner #2 run on each turn?In a 10,000 m race (25 laps), how much farther would Runner #2 have to run? Example Problem #2A baseball player swings a bat at a speed of 250°/s.If he makes contact with the ball 40 cm from the axis of rotation of his body, what is the speed of the bat at the point of impact?What is the speed at the point of impact if he makes contact at a distance of 80 cm instead? 9
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