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CIRCULAR MOTION AND, GRAVITATION, Angular Measure Angular Speed . and Angular Velocity, BELLWORK, 1 A tube is been placed upon the. 1 m high table and shaped into a, three quarters circle A golf ball is. pushed into the tube at one end, at high speed The ball rolls. through the tube and exits at the, opposite end Describe the path.
of the golf ball as it exits the tube , 2 If the 50g golf ball leaves the. tube with a velocity of 32 m s at, 45o a what is it s maximum. height b how long does it take to, land and c what is the impulse. force the ground exerts on the, ball to bring it to a stop in 98 s . ALGEBRA 2 REVIEW, An arc of a circle is a portion of the.
circumference of the circle , The length of an arc is simply the length of its. portion of the circumference Actually the, circumference itself can be considered an arc length . The length of an arc or arc length is traditionally. symbolized by s , The radian measure of a central angle of a circle. is defined as the ratio of the length of the, arc the angle subtends s divided by the. radius of the circle r , ALGEBRA 2 REVIEW CONT , Relationship between Degrees and.
When the arc length equals an entire, circumference we can use s r to get. 2 r r and 2 , This implies that 2 360o, So and, To change To change. from degrees to from radians to, radians degrees , multiply by multiply by. EXAMPLE PROBLEMS, 1 Convert 50 to radians , 2 Convert 6 radians to degrees . 3 How long is the arc subtended by an, angle of 7 4 radians on a circle of.
radius 20 000 cm , 1 5 18 radians, 2 30 degrees, 3 109 96 cm. ANGULAR MEASURE, Circular motion is described using. polar coordinates r q , x rcosq and y rsinq where q is. measured counterclockwise ccw , from the positive x axis . Angle is defined as q s r where s is the arc. length r is the radius and q is the angle in, radians Also expressed as s rq.
Angular distance o is measured in. degrees or radians A radian is the angle that, subtends an arc length that is equal to the. radius s r , 1 rad 57 3o or 2 rad 360o, EXAMPLE 1. When you are watching the NASCAR, Daytona 500 the 5 5 m long race car. subtends an angle of 0 31o What is, the distance from the race car to you . SOLUTION TO EXAMPLE 1, UNIFORM CIRCULAR MOTION IS THE MOTION OF AN.
OBJECT IN A CIRCLE WITH A CONSTANT OR UNIFORM SPEED . When moving in a circle an object traverses, a distance around the perimeter of the circle. The distance of one complete cycle around, the perimeter of a circle is known as the. circumference, This relationship between the circumference. of a circle the time to complete one cycle, around the circle and the speed of the object. is merely an extension of the average speed, COMBINING KINEMATICS AND CIRCULAR.
MEASUREMENT TO CALCULATE SPEED, The circumference of any circle can be. computed using from the radius, according to the equation. Circumference , 2 pi Radius, relating the speed of an object moving. in uniform circular motion to the radius, of the circle and the time to make one. cycle around the circle period T , where R radius .
ANGULAR SPEED AND VELOCITY, Instantaneous angular speed is the magnitude. of the instantaneous angular velocity, Tangential linear speed and angular speed are. related to each other through, where r is the radius . The time it takes for an object to go through one. revolution is called the period T , Then number of revolutions in one second is. called the frequency f, THE DIRECTION OF THE, VELOCITY VECTOR.
Velocity being a vector , has both a magnitude, and a direction. Since an object is, moving in a circle its, direction is continuously. direction of the velocity, vector is tangential to. the circular path, EXAMPLE 2, A bicycle wheel rotates uniformly. through 2 0 revolutions in 4 0 s , a What is the angular speed of the.
b What is the tangential speed of a, point 0 10 m from the center of. the wheel , c What is the period , d What is the frequency . ACCELERATION, an accelerating object is an object which. is changing its velocity , a change in either the magnitude or the. direction constitutes a change in the, an object moving in a circle at constant.
speed is accelerating because the, direction of the velocity vector is. CHANGE IN VELOCITY, The acceleration of the object is in the. same direction as the velocity change, Objects moving in circles at a constant. speed accelerate towards the center of, the circle . CENTRIPETAL ACCELERATION, The linear tangential velocity vector.
changes direction as the object moves, along the circle . This acceleration is called centripetal, acceleration center seeking because. it is always directed toward the center, of the circle . The magnitude of centripetal, acceleration is given by. CENTRIPETAL FORCE, From Newton s Second Law we, conclude that there MUST be a net.
force associated with centripetal, acceleration , Centripetal force is always directed. toward the center of the circle since the, net force on an object is in the same. direction as acceleration , EXAMPLE 3, A car of mass 1500 kg is negotiating a. flat circular curve of radius 50 m with a, speed of 20 m s . a What is the source of the centripetal, force on the car Explain.
b What is the magnitude of the, centripetal acceleration of the car . c What is the magnitude of the, centripetal force on the car . d What is the minimum coefficient of, static friction between the car and. Convert the following angles from, degrees to radians or from radians to. degrees to two significant figures , 4 3 , 3 4 , CONVERT THE FOLLOWING ANGLES FROM DEGREES TO.
RADIANS OR FROM RADIANS TO DEGREES TO TWO, SIGNIFICANT FIGURES . 285o 5 rad or 1 6 , 195o 3 4 rad or 1 08 , 90o 1 6 rad or 2. 4 3 rad, 270o 240o or4 3 rad, 3 4 4 7 rad or 3 2 . 165o rad, 135o or 3 4 , 2 9 rad or 0 92 , EXAMPLE 4 AFTER CLOSING A DEAL WITH A. CLIENT KENT LEANS BACK IN HIS SWIVEL CHAIR, AND SPINS AROUND WITH A FREQUENCY OF 0 5.
HZ WHAT IS KENT S PERIOD OF SPIN , Given f 0 5 Hz. Solve T 1 f, T 1 0 5 Hz, EXAMPLE 5 CURTIS FAVORITE DISCO. RECORD HAS A SCRATCH 12 CM FROM THE, CENTER THAT MAKES THE RECORD SKIP 45. TIMES EACH MINUTE WHAT IS THE LINEAR, SPEED OF THE SCRATCH AS IT TURNS . The record makes 45 revolutions every, minute 60 s so T 60 s 45 rev 1 3.
r 12 cm, v 2 r T, 2 12cm 1 3s, 58 cm s, EXAMPLE 6 MISSY S FAVORITE RIDE AT. THE TOPSFIELD FAIR IS THE ROTOR , WHICH HAS A RADIUS OF 4 0 M THE RIDE. TAKES 2 0 S TO MAKE ON E FULL, REVOLUTION , A WHAT IS MISSY S LINEAR SPEED ON. THE ROTOR , B WHAT IS MISSY S CENTRIPETAL, ACCELERATION ON THE ROTOR . Given r 4 0 m T 2 0s, v 2pir T 2pi 4 0m 2 0s 13 m s.
a v2 r 132 4 0 m 42 m s2, THE CAUSE OF CENTRIPETAL FORCE. In order to have an acceleration there, MUST be a force. What provides that force , Applied Force, Spring Force. Gravitational, WHAT HAPPENS WHEN THE FORCE VANISHES . MOTION IN A HORIZONTAL CIRCLE, The speed at which the object moves.
depends on the mass of the object and, the tension in the cord. The centripetal force is supplied by the, CIRCULAR MOTION ABOUT A CONICAL PENDULUM. The object is in equilibrium in the, vertical direction and undergoes. uniform circular motion in the, horizontal, v is independent of m. HORIZONTAL FLAT CURVE, The force of static friction.
supplies the centripetal force, The maximum speed at. which the car can negotiate, the curve is, BANKED CURVE. There is a component of, the normal force that, supplies the centripetal. FICTIONAL FORCES, From the frame of the passenger b . a force appears to push her toward the, From the frame of the Earth the car.
applies a leftward force on the, The outward force is often called a. centrifugal force, It is a fictitious force due to the. acceleration associated with the car s, change in direction. THE GREAT MISCONCEPTION, Centrifugal not to be confused with. centripetal means away from the, center or outward .
Circular motion leaves the moving, person with the sensation of being. thrown OUTWARD from the center of, the circle rather than INWARD. It s really just inertia , http www physicsclassroom com mme. dia circmot cf cfm, EFFECTS OF PRETEND FORCES, Although fictitious forces are not real. forces they can have real effects, Examples , Objects in the car do slide.
You feel pushed to the outside of a rotating, The Coriolis force is responsible for the. rotation of, weather systems and ocean currents, LOOP THE LOOP A VERTICAL. At the bottom of, the loop the, upward force, experienced by the. object is greater, than it s weight, Centripetal Force. Vector ADDS to, Normal Force, LOOP THE LOOP, At the top of the.
circle the force, exerted on the, object is less than. its weight, Centripetal Force, Vector TAKES, AWAY from the. normal force, Captain Chip the pilot of a 60500 kg jet. plane is told he must remain in a, holding pattern over the airport until it. is his turn to land If Captain Chip flies, his plane in a circle whose radius is.
50 0 km once every 30 0 min what, centripetal force must the air exert. against the wings to keep the plane, moving in a circle . EXAMPLE 7 45, Many racetracks have banked turns . which allow the cars to travel faster, around the curves than if the curves. were flat Actually cars could also, make turns on these banked curves if.
there were no friction at all , Use a free body diagram to explain how. this is possible , EXAMPLE 7 46, An indy car with a speed of 120 km hr. goes around a level circular track with, a radius of 1 00 km What is the. centripetal acceleration of the car , A 1 11 m s2, B 0 555 m s2. C 3 49 m s2, D 7 54 m s2, EXAMPLE 7 52, A car with a constant speed of 83 0 km .
hr enters a circular flat curve with a, radius of curvature of 0 400 km If the. friction between the road and the car s, tires can supply a centripetal. acceleration of 1 25 m s2 does the car, negotiate the curve safely Justify your. EXAMPLE 7 53, A student is to swing a bucket, of water in a vertical circle. without spilling any Fig , a Use a free body diagram to.
help explain how this task is, possible what provides the. centripetal force , b If the distance from his, shoulder to the center of mass. of the bucket of water is 1 0, m what is the minimum speed. required to keep the water, from coming out of the bucket. at the top of the swing , EXAMPLE 7 58, For a scene in a movie a.
stunt driver drives a, 1 50 x 103 kg SUV with a, length of 4 25 m around a circular curve. with a radius of curvature of 0 333 km, Fig 7 31 The vehicle is to be driven off. the edge of a gully 10 0 m wide and land, on the other side 2 96 m below the initial. side What is the minimum centripetal, acceleration the truck must have in going. around the circular curve to clear the gully, and land on the other side .
BELLWORK BONNIE IS ICE SKATING AT, THE OLYMPIC GAMES SHE IS MAKING A. SHARP TURN WITH A RADIUS OF 22 6 M, AND WITH A SPEED OF 16 1 M S USE. NEWTON S SECOND LAW TO DETERMINE, THE ACCELERATION AND THE ANGLE OF. LEAN OF BONNIE S 55 0 KG BODY , Given Info , m 55 0 kg. v 16 1 m s, r 22 6 m, Angle of lean , SOLUTION TO BELLWORK.
ac v2 R a 16 1, m s 2 22 6 m 11 5, Fx Fnet m a Fx. 55 0 kg 11 5 m s s , ANGULAR ACCELERATION , NON UNIFORM CIRCULAR MOTION. Angular acceleration the rate of, change of angular velocity. EXAMPLE 8 A ROTATING CD, A CD accelerates uniformly from rest to. its operational speed of 500 rpm in, A What is the angular acceleration of.
the CD during this time , B What is the angular velocity of the. CD after this time The angular, acceleration after this time . C If the CD comes uniformly to a stop, in 4 50 s what is its angular. 1 WHAT COULD THE POSITION AND VELOCITY, VECTORS FOR THE LADY BUG LOOK LIKE . 2 WHAT COULD THE ACCELERATION, AND VELOCITY VECTORS LOOK LIKE .
3 WHAT COULD THE POSITION , ACCELERATION VECTORS LOOK. THE ACCELERATION WOULD NOT BE RADIAL OR THE, PATH WOULD BE CIRCULAR THIS IS VERY DIFFICULT. TO SEE IN THE SIM , 4 IF YOU HAD TWO BUGS MOVING IN. CIRCLES LIKE THIS WHAT COULD THE, VELOCITY VECTORS AT POINT X VS. POINT Y LOOK LIKE , D Any of the above, E None of the.
circumference of the circle. The . length of an arc. is simply the length of its "portion" of the circumference. Actually, the circumference itself can be considered an arc length. The length of an arc (or arc length) is traditionally symbolized by. s. The . radian measure. of a central angle of a circle . is defined as the ratio of the length ...