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Circular Motion What is circular motion? Anything that rotates or revolves around a central axis is in circular motion. Rotation vs. Revolution An axis is the straight line around which rotation takes place. Rotation: an object turns about an internal axis. “Spinning” Revolution: an object turns about an external axis, “Turning” Rotation vs. Revolution The Ferris wheel turns about an axis. The Ferris wheel rotates, while the riders revolve about its axis. Rotation vs. Revolution Where is rotation? Where is revolving? Uniform Circular Motion is the motion of an object in a circle with a constant or uniform speed. We have been referring to constant speed. How would you describe the velocity? Remember velocity is a vector quantity so has magnitude and direction Velocity of Uniform Circular Motion constant magnitude changing direction-always tangential to the circle Miniature golf: where will the golf ball go? Over point A, B, or C? A C B B As an object moves in uniform circular motion (remember speed is constant) is there acceleration? Yes Although speed is constant, velocity is not!!! Acceleration is a change in velocity It is accelerating because the direction of the velocity vector is changing. Identify the three controls on an automobile which allow the car to be accelerated. In uniform circular motion, what is the direction of the acceleration? conditions for uniform circular motion The velocity vector and the acceleration vector are always perpendicular to each other. The acceleration vector changes only the direction of the velocity vector not the magnitude of it. The acceleration vector is directed inwards conditions for uniform circular motion A V In circular motion, what is the direction of the velocity vector? In circular motion, what is the direction of the acceleration vector? Question to think about: You are riding the carousel at the Woodlands Mall. How long does it take to make a complete circle? How many times does it do this in a second? Are these related? What does it mean to be periodic? Period vs Frequency T = Period Time for one cycle or revolution (seconds) f = Frequency Number of cycles or revolutions per second (hertz or sec-1) Formulas Calculating Period and Frequency sec onds T revolutions revolutions f sec ond T = period or time for one revolution (sec) f = frequency or revolutions per second (Hz or sec-1) Formulas The period is inversely proportional to the frequency 1 T f T = period or time for one revolution (sec) f = frequency or revolutions per second (Hz or sec-1) Example 1 A merry-go-round makes 6 revolutions in 10 seconds. What is its frequency? revolutions The definition of f frequency is the sec onds number of 6revolutions f revolutions per 10 sec onds second f 0.6rev / sec f 0.6 Hz Example 2 A merry-go-round makes 6 revolutions in 42 seconds. What is its period? The definition of period is time required to complete one revolution sec onds T revolutions T =revolution 42sec/6rev s f sec ond The period is 7 Example 3 A merry-go-round has a frequency .5Hz. What is its period? 1 T f T = 1rev/.5HZ Period =2sec Remember Hz = revolutions/sec If we know time of 1 revolution (T), How do we determine velocity of 1 revolution? We will consider velocity at a point tangent to the circle Velocity = d/t For distance: dependent on the circumference of the circle d = 2ᴫr (distance of 1 revolution) t = T (time of 1 revolution) Together: 2r v T Calculating speed 2r v T OR v 2rf v = speed (m/s) r = radius of circle (m) T = period or time for one revolution (sec) f = frequency Example 4: A lifeguard twirls her whistle on a string in a horizontal circle with a radius of 0.34 m. It takes 1.5 second to complete the circle. What is the average tangential speed of the whistle? Given: Example 4: A lifeguard twirls her whistle on a string in a horizontal circle with a radius of 0.34m. It takes 1.5 second to complete the circle. What is the average tangential speed of the whistle? Given: r = 0.34 m T = (1.5 sec /1rev) = 1.5 sec/rev 2r v T Example 4: A lifeguard twirls her whistle on a string in a horizontal circle with a radius of 0.34 m. It takes 1.5 second to complete the circle. What is the average tangential speed of the whistle? Given: r = 0.34 m T = (1.5 sec /1rev) = 1.5 sec/rev 2r v T 2 (0.34m) v 1.5 sec/ rev v 1.42m / sec the average speed and the radius of the circle are ________ proportional. A twofold increase in radius corresponds to a _______ increase in speed if period remains the same Example 5: A lifeguard twirls her whistle on a string in a horizontal circle with a radius of 28 cm. It takes 4.8 second to revolve 5 times. What is the average tangential speed of the whistle? Given: r= T= Example 5: A lifeguard twirls her whistle on a string in a horizontal circle with a radius of 28 cm. It takes 4.8 second to revolve 5 times. What is the average tangential speed of the whistle? Given: r = 28 cm = 0.28 m T = (4.8 sec /5rev) = 0.96 sec/rev Example 5: A lifeguard twirls her whistle on a string in a horizontal circle with a radius of 28 cm. It takes 4.8 second to revolve 5 times. What is the average tangential speed of the whistle? Given: r = 28 cm = 0.28 m T = (4.8 sec /5rev) = 0.96 sec/rev 2r v T 2 (0.28m) v 0.96 sec/ rev v 1.83m / sec As an object moves in a uniform circle What happens to its speed? What happens to its velocity? What happens to its acceleration? Formulas Calculating Centripetal Acceleration using speed 2 v ac r ac = centripetal acceleration (m/s2) r = radius of circle (m) v = speed (m/s) What causes an object to have Centripetal Acceleration? Centripetal Force NOT centrifugal force What’s the difference? Centripetal = center seeking Centrifugal = outward seeking Without a net centripetal force, an object cannot travel in circular motion. In fact, if the forces are balanced, then an object in motion continues in motion in a straight line at constant speed. Without a centripetal force, an object in motion continues along a straight-line path. With a centripetal force, an object in motion will be accelerated and change its direction. Courtesy http://www.physicsclassroom.com/mmedia/circmot/cf.cfm frame of reference Centripetal forces are those seen by an observer in an inertial frame of reference. Centrifugal forces are those felt by an observer in an accelerating frame of reference. As a car goes around a corner, the passengers think they feel a force towards the outside of the curve, in reality this is due to inertia. Centrifugal force is a misnomer!!! What causes centripetal acceleration? To have acceleration, there must be a net force towards the center of the circle What is the force for the following: Earth circling the sun Force of Gravity Car turning a bend Force of Friction Xena warrior princess throwing a ball on a chain Force of Tension on chain Formulas Calculating centripetal force using centripetal acceleration Fc ma c OR mv Fc r 2 Fc = centripetal force (N) m = mass (kg) ac = centripetal acceleration (m/s2) Example 6 A little girl is swinging her 5 kg purse in horizontal circles using the strap that allows the purse to swing 20 cm from her hand. The girl is able to get the purse to make 10 revolutions in 8 seconds. What was the speed of the purse? What is the centripetal acceleration of the purse? How much tension is in the purse string? Example 6 Given: m= r= T= Example 6 determine velocity Given: m = 5 kg r = 20 cm = 0.2 m T = (8sec / 10rev) = 0.8 sec/rev 2r v T Example 6 determine velocity Given: m = 5 kg r = 20 cm = 0.2 m T = (8sec / 10rev) = 0.8 sec/rev 2r v T 2 (0.2m) v 0.8 sec/ rev v 1.57m / sec Example 6 determine acceleration with velocity of 1.57 m/s Given: m = 5 kg r = 20 cm = 0.2 m T = (8sec / 10rev) = 0.8 sec/rev 2 v ac r Example 6 determine acceleration with velocity of 1.57 m/s Given: m = 5 kg r = 20 cm = 0.2 m T = (8sec / 10rev) = 0.8 sec/rev 2 v ac r 2 (1.57 m / s ) ac 0 .2 m 2 ac 12.3m / s Example 6 determine FC with acceleration of 12,3 m/s2 Given: m = 5 kg r = 20 cm = 0.2 m T = (8sec / 10rev) = 0.8 sec/rev Fc ma Example 6 determine FC with acceleration of 12.3 m/s2 Given: m = 5 kg r = 20 cm = 0.2 m T = (8sec / 10rev) = 0.8 sec/rev Fc ma Fc (5kg)(12.3m / s ) 2 Fc 61.5 N Example 7: Shelby twirls her 25g whistle on a 0.450m lanyard. She twirls the cord at a uniform speed in a horizontal circle. If the speed is 4.75m/sec, what is the centripetal acceleration of the whistle? How much tension is in the lanyard? Example 7: Shelby twirls her 25g whistle on a 0.450m lanyard. She twirls the cord at a uniform speed in a horizontal circle. If the speed is 4.75m/sec, what is the centripetal acceleration of the whistle? How much tension is in the lanyard? a = 50.1 m/s2 F = 1.25 N Concept questions: An object moves in uniform horizontal circular motion. If the radius of the object triples, what happens to the speed of the object? The speed will triple How does doubling velocity affect the ac? It increases four fold How would calculating Fc change if the uniform circular motion was vertical instead of horizontal?