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Rotational Kinematics. Circular Motion. A Particle in Uniform Circular Motion. For a particle in uniform circular motion, the velocity vector v remains constant in magnitude, but it continuously changes its direction. Angular Position: θ. Angular Position q. Degrees and revolutions:.

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a particle in uniform circular motion
A Particle inUniform Circular Motion

For a particle in uniform circular motion, the velocity vector v remains constant in magnitude, but it continuously changes its direction.


Angular Position q

Degrees and revolutions:


Angular Position q

Arc length s, measured in radians:

comparison to 1 d kinematics
Comparison to 1-D Kinematics

Angular Linear

And for a point at a distance R from the rotation axis:

  • x = Rv = R a = R

By convention, , ,  are positive if they are in the counterclockwise direction.

decelerating windmill
Decelerating Windmill

As the wind dies, a windmill that had been rotating at w = 2.1 rad/s begins to slow down at a constant angular acceleration of a = -0.45 rad/s2.

How long does it take for the windmill to come to a complete stop?

angular velocity acceleration act
Angular Velocity & Acceleration ACT

The fan blade shown is slowing down. Which option describes a and w?

(c) w<0 and a>0;

(d) w<0 and a<0.

(a) w>0 and a>0;

(b) w>0 and a<0;


Rotational Kinematics

If the angular acceleration is constant:

thrown for a curve
Thrown for a Curve

To throw a curve ball, a pitchergives the ball an initial angularspeed of 157.0 rad/s. When thecatcher gloves the ball 0.795 slater, its angular speed hasdecreased (due to air resistance)to 154.7 rad/s.

(a) What is the ball’s angular acceleration, assuming it to be constant?

(b) How many revolutions does the ball make before being caught?

wheel of misfortune
Wheel of Misfortune

On a certain game show, contestants spin thewheel when it is their turn. One contestant givesthe wheel an initial angular speed of 3.40 rad/s.It then rotates through 1.25 revolutions andcomes to rest on BANKRUPT.

(a) Find the wheel’s angular acceleration,assuming it to be constant.

(b) How long does it take for the wheel tocome to rest?

a rotating crankshaft
A Rotating Crankshaft

A car’s tachometer indicates the angular velocity w of the crank shaft in rpm. A car stopped at a traffic light has its engine idling at 500 rpm. When the light turns green, the crankshaft’s angular velocity speeds up at a constant rate to 2500 rpm in a time interval of 3.0 s.

How many revolutions does the crankshaft make in this time interval?

time to rest
Time to Rest
  • A pulley rotating in the counterclockwise direction is attached to a mass suspended from a string. The mass causes the pulley’s angular velocity to decrease with a constant angular acceleration a = -2.10 rad/s2.
  • (a) If the pulley’s initial angular velocity is w0 = 5.40 rad/s, how long does it take for the pulley to come to rest?
  • Through what angle does the pulley turn during this time?

(c) If the radius of the pulley is 5.0 cm, through what distance is the mass lifted?

cd speed
CD Speed

CDs and DVDs turn with a variable w that keeps the tangential speed vt constant.

Find the angular speed w and the frequency that a CD must have in order to give it a linear speed vt= 1.25 m/s when the laser beam shines on the disk

(a) at 2.50 cm from its center, and(b) at 6.00 cm from its center.


Rotational vs. Linear Kinematics

Analogies between linear and rotational kinematics:


Speeding up

More Connections BetweenLinear & Rotational Quantities

This merry-go-round has both tangential and centripetal acceleration.

the microhematocrit
The Microhematocrit

Suppose the centrifuge is just starting up, and that it has an angular speed of 8.00 rad/s and an angular acceleration of 95.0 rad/s2.

(a) What is the magnitude of the centripetal, tangential, and total acceleration of the bottom of a tube?

(b) What angle does the total acceleration make with the direction of motion?