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LECTURE 18 ROTATIONAL MOTION Instructor: Kazumi Tolich Lecture 18 2 ¨ Reading chapter 10-1 to 10-2 ¤ Angular displacement ¤ Angular velocity and speed ¤ Angular acceleration ¤ Rotational kinematics with constant angular acceleration Angular position 3 ¨ ¨ ¨ Angular position (𝜃) is the angle between a reference line on a rotating rigid body and a reference line in space. Angular position is measured counterclockwise from the reference line in space. By convention: 𝜃 > 0 counterclockwise rotation 𝜃 < 0 clockwise rotation Arc length 4 ¨ The arc length 𝑠 for an arbitrary angle 𝜃 measured in radians is given by 𝑠 = 𝑟𝜃 Quiz: 1 5 Demo 1 6 ¨ Radian Disk ¤ Demonstration of unit, radian ¤ 1 radian is approximately 57.3 degrees. Quiz: 2 7 Angular displacement 8 ¨ Angular displacement (∆𝜃) is the change in the angular position with respect to a reference line as the object rotates. ∆𝜃 = 𝜃* − 𝜃, ¨ Angular displacement is measured in radians. Angular velocity and speed 9 ¨ Angular velocity (𝜔) is defined as 𝜔./ = ∆0 ∆1 𝜔 = lim∆1→6 ¨ ¨ (average) ∆0 ∆1 (instantaneous) The magnitude of angular velocity is called angular speed. If a point on a rotating object moves in ¤ ¤ counterclockwise: 𝜔 > 0. clockwise: 𝜔 < 0. Period 10 ¨ The time to complete one revolution, 𝑇, is called period. 2𝜋 𝑇= 𝜔 Example: 1 11 ¨ One of the most studied objects in the night sky is the Crab nebula, the remains of a supernova explosion observed by the Chinese in 1054. In 1968 it was discovered that a pulser – a rapidly rotating neutron star that emits a pulse of radio waves with each revolution – lies near the center of the Crab nebula. The pulses are emitted every 33 ms. What is the angular speed of the Crab nebula pulser in rad/s, in degrees/s, and in revolutions/s? Angular acceleration 12 ¨ Angular acceleration (𝛼) is defined as ∆; ∆1 ∆; lim ∆1→6 ∆1 𝛼./ = 𝛼= ¨ (average) (instantaneous) Angular acceleration can be positive or negative, depending on the sign of change in angular velocity. Quiz: 3 & 4 13 Rotation with constant angular acceleration 14 ¨ For an object rotating with an initial angular position, 𝜃0 , initial angular velocity, 𝜔0 , and a constant angular acceleration, 𝛼, ¤ the angular velocity, 𝜔, as a function of time, 𝑡, is given by 𝜔 = 𝜔0 + 𝛼𝑡 ¤ the angular position, 𝜃, as a function of time, is given by 𝜃 = 𝜃6 + 𝜔6 𝑡 + >?𝛼𝑡 ? ¤ The angular velocity as a function of angular displacement, ∆𝜃, is given by 𝜔? = 𝜔6? + 2𝛼∆𝜃 Example: 2 15 ¨ A wheel has a constant angular acceleration α = 0.35 rad/s2. It starts from rest with an arbitrary reference line horizontal, at angular position θ0 = 0. a) b) What is the angular displacement Δθ of the reference line at t = 18 s? What is the wheel’s angular velocity at t = 18 s? Example: 3 16 ¨ The drill used by most dentists today is powered by a small air turbine that can operate at angular speeds of 350,000 rpm. These drills, along with ultrasonic dental drills, are the fastest turbines in the world – far exceeding the angular speeds of jet engines. Suppose a drill starts at rest and comes up to operating speed in 2.1 s. a) Find the angular acceleration produced by the drill, assuming it to be constant. b) How many revolutions does the drill bit make as it comes up to speed?