The period T is the time for one

The period T is the time for one complete revolution. So the linear speed can be found by dividing the period into the circumference: Units: m/s Another useful parameter in engineering problems is the rotational speed, expressed in revolutions per minute (rpm), revolutions per second (rev/s) or Hz (s-1). This quantity is called the frequency f of rotation and is given by the reciprocal of the period. Units: s-1 = Hz

• Acceleration involves a change in speed and/or direction; it is caused by an unbalanced force. • In UCM, the object moves at constant speed but is accelerating because its direction is constantly changing. • An object moving in a circle of radius r with constant speed v has an acceleration whose direction is toward the center of the circle.

v and a are perpendicular at every point.

An object in uniform circular motion has centripetal acceleration (ac): *the direction of the acceleration is toward the center of the circle *Remember, you can feel acceleration: a Which way do they feel pulled when the car speeds up? Which way do they feel pulled when the car slows down? a a What about when the car goes around a corner? ?

A ball is whirled at the end of a string in a horizontal circle 60 cm in radius at the rate of 1 revolution every 2 s. Find the ball's centripetal acceleration. UCM r = 0. 6 m T=2 s = 5. 92 m/s 2

Equation Theory Similar Triangles v 2 Dt r 2 Dq Dr r 1 Centripetal Acceleration v 1 -v 1 Dq Dv Divide by time v 2

CENTRIPETAL ACCELERATION An object experiencing uniform circular motion is continually accelerating. The position and velocity of a particle moving in a circular path of radius r are shown at two instants in the figure. When the particle is at point A, its velocity is represented by vector v 1. After a time interval t, its velocity is represented by the vector v 2.

The acceleration is given by: as Δt becomes smaller and smaller, the chord length becomes equal to the arc length s = v Δt

CENTRIPETAL FORCE The inward force necessary to maintain uniform circular motion is defined as centripetal force. From Newton's Second Law, the centripetal force is given by:

Centripetal force keeps an object in circular motion. Centripetal forces can be exerted in a variety of ways. • The “string” that holds the moon on its almost circular path, for example, is gravity. • Electrical forces provide the centripetal force acting between an orbiting electron and the atomic nucleus in an atom. • Anything that moves in a circular path is acted on by a centripetal force. • The centripetal force is not a 'special' kind of force, but is provided by the force that keeps the object in a circle.

a. A car makes a turn, what force is required to keep it in circular motion? As a car makes a turn, the force of friction acting upon the tires of the car provide the centripetal force required for circular motion.

b. A bucket of water is tied to a string and spun in a circle, what force is required to keep it in circular motion? As a bucket of water is tied to a string and spun in a circle, the force of tension acting upon the bucket provides the centripetal force required for circular motion. c. The moon orbits the Earth, what force is required to keep it in circular motion? As the moon orbits the Earth, the force of gravity acting upon the moon provides the centripetal force required for circular motion.

The Forbidden F-Word When the subject of circular motion is discussed, it is not uncommon to hear mention of the word "centrifugal. " Centrifugal, not to be confused with centripetal, means away from the center or outward. The use of this word combined with the common sensation of an outward force when experiencing circular motion, often creates or reinforces a deadly student misconception.

The Forbidden F-Word The deadly misconception, is the notion that objects in circular motion are experiencing an outward force. "After all, " a well-meaning student may think, "I can recall vividly the sensation of being thrown outward away from the center of the circle on that roller coaster ride. Therefore, circular motion must be characterized by an outward force. "

Centrifugal Force is often used to describe why mud gets spun off a spinning tire, or water gets pushed out of the clothes during the spin dry cycle of your washer. It is also used to describe why we tend to slide to the outer side of a car going around a curve. Let’s imagine that you are riding in Granny’s car going around a curve. Sitting on your dashboard is a cassette tape. As you go around the curve, the tape moves to outside edge of the car. Because you don't want to blame it on ghosts, you say “centrifugal force pushed the tape across the dashboard. ” Wwrroonngg!!

The animation shows both views at the same time. The top window shows you the passenger's view of the car and the tape, while the shows you the bird's eye view.

The tape on the slippery dashboard does not have enough friction to act as a centripetal force, so in the absence of a centripetal force the tape follows straight line motion. If the car you are riding in has the windows rolled down, then the tape will leave the car (or does the car leave the tape? ) as it follows its straight line path. If the windows are rolled up, then the window will deliver a centripetal force to the tape and keep it in a circular path.

Without a centripetal With a centripetal force, an object in motion will continues along a be accelerated and straight-line path. change its direction.

We can see that the force must be inward by thinking about a ball on a string:

There is no centrifugal force pointing outward; what happens is that the natural tendency of the object to move in a straight line must be overcome. If the centripetal force vanishes, the object flies off tangent to the circle.

Spin Cycle on a Washer How is the water removed from clothes during the spin cycle of a washer? Think carefully before answering. . . Does the centripetal force throw water off the clothes? NO. Actually, it is the LACK of a force that allows the water to leave the clothes through holes in the circular wall of the rotating washer.

The clothes in a washing machine are forced into a circular path, but the water is not, and it flies off tangentially.

Ex: Ball on a string If the ball’s mass is 2. 0 kg, its speed is 3. 0 m/s, and the radius of the circle is 1. 0 m, find a) the ball’s acceleration b) the tension in the string

Problems: • Convert 10 rpm (rev/min) into m/s for a horizontal circle of 50 cm radius. • A moving ball is spun in a circle with a diameter of 1 m at a speed of 3 m/s. What is its centripetal acceleration?

• A 250 g mass is spun in a horizontal circle. It is held at the end of a 1 m length of string. If it is spun at 15 m/s, what force is applied? • A ball weighing 5 N is attached to a 1 m string and swung in a horizontal circle above one's head at a frequency of 5 Hz. What centripetal force is required?