Uniform Circular Motion AP

Uniform Circular Motion AP

Uniform circular motion is motion in a circle at the same speed Speed is constant, velocity direction changes

the speed of an object moving in a circle is given by v circumference time 2r T r= radius of circle T = period, time for 1 complete circle

v 2 r T Is called the tangential velocity

Moon setting over Rio de Janerio Pictures every 6.5 minutes showing the moon is moving at a uniform speed

Centripetal Acceleration The direction of motion is changing so the object is accelerating a v v f v i t t

If you draw the vectors tail-tail, v is from v i to v f you get v v v f i

the centripetal acceleration of the object is a c 2 v r a c = centripetal acceleration (always to centre of circle)

2 a c Equations a c v 2 r 2 r T r 2 4 T r 2 2 2 r 2 4 r T 2 On data sheet too!

the velocity vector is always directed along the tangent of the circle (at right angles to the acceleration vector) centripetal means centre-seeking, the acceleration vector always points to the centre of the circle Acceleration is radially inward

Centripetal Force inertia tends to keep objects moving in a straight line, a force is needed to cause a circular motion (2nd Law of Motion) the force causing the motion is called the centripetal force and it always acts towards the centre of the circle (parallel to the acceleration vector)

Centrifuge Training

F m a & v net a c r 2 On data sheet F c 2 mv r NOT on data sheet

an object will move in a circle whenever a constant magnitude force acts on the object at right angles to its direction of motion

The force and acceleration are radially inward The velocity is tangential

The string pulls ball towards the centre of the circle 3rd Law of Motion states that a reaction force will act outward (ball pulls string) this is sometimes called centrifugal force

Important!!! centripetal force is not a fundamental force like gravity, it is the net force acting on an object moving in a curved path The force will not change the speed of the object because the force has NO component parallel to the velocity vector

Example A centrifuge for pilot training is 11 m in radius. At what speed must it rotate in order to inflict 7.0 g on a pilot?

Solution a c v r 2 v a r c v 7.09.81 m / s 11m 2

Frequency sometimes speed is given in revolutions per minute (rpm) or revolutions per second (rps) this is how many times the object goes around the circle in 1 minute or 1 second frequency is the number of cycles per second measured in hertz (Hz)

Hz 1 1 sec ond s T 1 f

v 2r 2 rf T

Example Find the period of an object that rotates at 35.0 rps. T 1 f T 0. 0286 s

Free Body Diagrams A tetherball is attached to a swivel in the ceiling by a light cord. When the ball is hit by a paddle, it swings in a horizontal circle with constant speed, and the cord makes a constant angle with the vertical direction. Write the expression for the centripetal force in terms of the other forces.

Free Body Diagrams An object is on a horizontal disc that is rotating at constant speed. Friction prevents the rock from sliding. Write the expression for the centripetal force in terms of the other forces.

Example A 3.50 kg object is swung in a 1.50 m radius horizontal circle at 40.0 rps (40.0 Hz). What magnitude force acts on the object? F net m a F c 2 4 m T r 2

Solution F net m a F c 2 4 m T r 2 or f 1 so F 4 2 rmf 2 T

Effect of radius on speed and forces all points on a rotating solid have the same period, but different speeds because the inner points have smaller distances to travel, their speeds are less the speed and force depends on the radius Rotating disc

NASA and other space agencies use this to help launch satellites at the equator, the speed is about 1667 km/h, at Edmonton, it is about 900 km/h

Angular displacement & velocity The rotational displacement of a point,, in radians 2 radians = 360 o

(in radians) Arc length Radius s r

Angular velocity, (lower case omega) radians/s Every point on the disc has the same angular velocity ave t

By convention, the angular displacement is positive if it is counterclockwise and negative if it is clockwise.

Example: Adjacent Satellites Two satellites are put into an orbit whose radius is 4.23 10 7 m. If the angular separation of the two satellites is 2.00 degrees, find the arc length that separates them.

(in radians) Arc length Radius s r 2 rad 2.00 deg 360 deg 0.0349 rad s = r = 4.23x10 7 m x 0.0349 rad s = 1.48 x 10 6 m

Angular & linear speed Linear speed, v = r Linear speed is sometimes called tangential speed

Period & angular velocity Distance (2) radians in a circle T Angular velocity 2 1 f

Direction of Angular velocity

Rolling Motion A rolling wheel has rotational and translational velocities (a) axle is at rest and the wheel is rotating at v = r

Rolling Motion A rolling wheel has rotational and translational velocities (b) the wheel is moving at v = r, every point on the wheel is moving horizontally (wheel is skidding)

Rolling Motion When the rotation and translational motions are combined

Equations of Rotational Motion 1 t t 0 0 2 2 1 x x v t a t 0 x0 x 2 2 t 0 v v a t x x0 x

Example A diver completes 1.5 rotations in 2.3 s. Determine the average angular speed of the diver

Solution =1.5 rotations x 2 rad = 9.425 radians 4.1 rad / s ave t

Example A metal cylinder of radius 0.45 m is spinning at 2000 rpm and a brake is applied slowing it to 1000 rpm in 10 seconds. What is the angular acceleration?

Solution In one revolution there are 2 radians (2000 rev/min) x (2 rad/sec) x (1 min/60 sec) = 209.4 rad/sec (1000 rev/min) x (2 rad/sec) x (1 min/60 sec) = 104.7 rad/sec

Solution f = i + at (104.7 rad/s) = (209.4 rad/s) + a(10 s) a = -10.47 rad/s 2

A bullet is fired through 2 discs rotating at 99.0 rad/s. The discs are 0.955 m apart. The angular displacement between the holes is 0.260 rad. Calculate the speed of the bullet. Example

Solution = 99.0 rad/s. d = 0.955 m = 0.260 rad. ave t t = 2.626 x 10-3 s v = 364 m/s

Small Angle Approximation For small angles, < 0.5 rad, sin tan O

P 251: #1, 2 Practice