Rotational Kinematics Chapter 8

Rotation Motion in circles around a central axis of rotation. Analysis of rotational motion takes advantage of the fact that each point on the object rotates through the same angle of rotation in the same amount of time.

Angular displacement Angle of rotation is called the angular displacement. Units: radians

Angular velocity Angular displacement in one second. Units: radians/second rad/s 1/s s-1 Radians is a "label" and not actually units in the sense of kg, m, and s. Radians can be inserted or removed as needed. The radians label helps to distinguish between radians and revolutions of rotation.

Other common ways to describe rotation speed Revolutions per minute (rpm). Revolutions per second (rps).

Unit conversions Convert 2000 rpm to rad/s use a conversion fraction

Units: (radians/second)/second radians/second2 rad/s2 1/s2 s-2 Angular acceleration Angular velocity change in one second. Units: (radians/second)/second radians/second2 rad/s2 1/s2 s-2

Kinematic variables for constant acceleration Five linear variables: displacement, x initial velocity, vo final velocity, v acceleration, a time, t Five rotational variables: displacement, θ initial velocity, ωo final velocity, ω acceleration, α time, t

Kinematic equations: linear vs. rotational

Kinematic equations for rotational motion θ o  α t radians radians/s radians/s2 s

Strategy for successful problem solving Make a drawing. Label known and unknown quantities on your drawing. Use a curved arrow to show the positive (+) direction of rotation. Write the known values in a table. Use the correct + and - signs for values. Verify that the table has values for at least three of the kinematic variables. Select the appropriate equation and solve to unknown values. θ o  α t radians radians/s radians/s2 s

Example 5 Blending with a Blender When the “puree” button on a blender was pushed, the blades whirled with an angular velocity of 375 rad/s. Then the “blend” button was pushed, the blades accelerated at 1740 rad/s2 Find the final angular velocity of the blades and the time for the first 44 rad of accelerated rotation.

θ ωo ω α t 44 rad 375 rad/s ? 1740 rad/s2 final angular velocity

θ ωo ω α t 44 rad 375 rad/s 542 rad/s 1740 rad/s2 ?

Example Bicycle wheels were initially rotating at 20 rad/s. During braking, the wheels rotated 15 revolutions as the bicycle slowed down to a stop. Find the angular acceleration and time to stop. θ ωo ω α t 30π rad 20 rad/s ?

θ Tangential variables Particles in rotating objects move on circular arcs with • Same angular velocity • Different tangential velocities θ Tangential displacement (arc length) Tangential velocity Tangential acceleration

Example 6 A Helicopter Blade Angular velocity is 40.8 rad/s. Find the tangential velocity at point 1. Angular acceleration is 8.17 rad/s2. Find the tangential acceleration at point 1. Find the tangential velocity at point 2. Find the tangential acceleration at point 2. At any moment, all points on the blade have the same angular velocity and same angular acceleration, but points at different distances from the center have different tangential velocities and different tangential accelerations .

Tangential velocity Interconnected belts, chains, and gears have the same tangential velocity at their outer edges. Smaller radius rotates faster.

Centripetal acceleration ω tangential velocity centripetal acceleration

Centripetal acceleration example ω A 2 kg airplane takes 10 s to travel around a 5 m radius horizontal circle. Find the angular velocity. Find the centripetal acceleration. Find the centripetal force.

Total acceleration An object moving on a circular arc always has a centripetal acceleration because the direction of the tangential velocity is always changing. If the magnitude of the tangential velocity is increasing, then the object also has a forward tangential acceleration. The total acceleration is the vector sum of the centripetal and tangential accelerations. Tangential acceleration is backward if the tangential velocity is decreasing.

Example A pilot trainee experiences a total acceleration of 4.8 g when the centrifuge rotation reaches an angular velocity of 2.5 rad/s. The 5 m radius centrifuge starts from rest. Find the trainee's angular acceleration.

Rolling motion A car has a velocity of 30 m/s. The tires have a radius of 0.3 m. How fast are the tires rotating? rpm = ?

The End