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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For a particle in uniform circular motion, the velocity vector v remains constant in magnitude, but it continuously changes its direction.
Degrees and revolutions:
Arc length s, measured in radians:
And for a point at a distance R from the rotation axis:
By convention, , , are positive if they are in the counterclockwise direction.
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?
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;
If the angular acceleration is constant:
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?
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 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?
(c) If the radius of the pulley is 5.0 cm, through what distance is the mass lifted?
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.
Analogies between linear and rotational kinematics:
More Connections BetweenLinear & Rotational Quantities
This merry-go-round has both tangential and centripetal acceleration.
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?