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1、Dynamics of Uniform Circular Motion,Chapter 5,Learning Objectives-Circular motion and rotation,Uniform circular motionStudents should understand the uniform circular motion of a particle,so they can:Relate the radius of the circle and the speed or rate of revolution of the particle to the magnitude
2、of the centripetal acceleration.Describe the direction of the particles velocity and acceleration at any instant during the motion.Determine the components of the velocity and acceleration vectors at any instant,and sketch or identify graphs of these quantities.Analyze situations in which an object
3、moves with specified acceleration under the influence of one or more forces so they can determine the magnitude and direction of the net force,or of one of the forces that makes up the net force,in situations such as the following:Motion in a horizontal circle(e.g.,mass on a rotating merry-go-round,
4、or car rounding a banked curve).Motion in a vertical circle(e.g.,mass swinging on the end of a string,cart rolling down a curved track,rider on a Ferris wheel).,Table Of Contents,5.1 Uniform Circular Motion5.2 Centripetal Acceleration5.3 Centripetal Force5.4 Banked Curves5.5 Satellites in Circular O
5、rbits5.6 Apparent Weightlessness and Artificial Gravity5.7 Vertical Circular Motion,Chapter 5:Dynamics of Uniform Circular Motion,Section 1:Uniform Circular Motion,Other Effects of Forces,Up until now,weve focused on forces that speed up or slow down an object.We will now look at the third effect of
6、 a forceTurningWe need some other equations as the object will be accelerating without necessarily changing speed.,Uniform circular motion is the motion of an object traveling at a constant speed on a circular path.,DEFINITION OF UNIFORM CIRCULAR MOTION,Let T be the time it takes for the object totr
7、avel once around the circle.,Example 1:A Tire-Balancing MachineThe wheel of a car has a radius of 0.29m and it being rotatedat 830 revolutions per minute on a tire-balancing machine.Determine the speed at which the outer edge of the wheel ismoving.,Newtons Laws,1stWhen objects move along a straight
8、line the sideways/perpendicular forces must be balanced.2nd When the forces directed perpendicular to velocity become unbalanced the object will curve.3rd The force that pulls inward on the object,causing it to curve off line provides the action force that is centripetal in nature.The object will in
9、 return create a reaction force that is centrifugal in nature.,5.1.1.An airplane flying at 115 m/s due east makes a gradual turn while maintaining its speed and follows a circular path to fly south.The turn takes 15 seconds to complete.What is the radius of the circular path?a)410 mb)830 mc)1100 md)
10、1600 me)2200 m,Chapter 5:Dynamics of Uniform Circular Motion,Section 2:Centripetal Acceleration,In uniform circular motion,the speed is constant,but the direction of the velocity vector is not constant.,The direction of the centripetal acceleration is towards the center of the circle;in the same dir
11、ection as the change in velocity.,Conceptual Example 2:Which Way Will the Object Go?An object is in uniform circular motion.At point O it is released from its circular path.Does the object move along the straightpath between O and A or along the circular arc between pointsO and P?,Straight path,Exam
12、ple 3:The Effect of Radius on Centripetal AccelerationThe bobsled track contains turns with radii of 33 m and 24 m.Find the centripetal acceleration at each turn for a speed of 34 m/s.Express answers as multiples of,5.2.1.A ball is whirled on the end of a string in a horizontal circle of radius R at
13、 constant speed v.By which one of the following means can the centripetal acceleration of the ball be increased by a factor of two?a)Keep the radius fixed and increase the period by a factor of two.b)Keep the radius fixed and decrease the period by a factor of two.c)Keep the speed fixed and increase
14、 the radius by a factor of two.d)Keep the speed fixed and decrease the radius by a factor of two.e)Keep the radius fixed and increase the speed by a factor of two.,5.2.2.A steel ball is whirled on the end of a chain in a horizontal circle of radius R with a constant period T.If the radius of the cir
15、cle is then reduced to 0.75R,while the period remains T,what happens to the centripetal acceleration of the ball?a)The centripetal acceleration increases to 1.33 times its initial value.b)The centripetal acceleration increases to 1.78 times its initial value.c)The centripetal acceleration decreases
16、to 0.75 times its initial value.d)The centripetal acceleration decreases to 0.56 times its initial value.e)The centripetal acceleration does not change.,5.2.3.While we are in this classroom,the Earth is orbiting the Sun in an orbit that is nearly circular with an average radius of 1.50 1011 m.Assumi
17、ng that the Earth is in uniform circular motion,what is the centripetal acceleration of the Earth in its orbit around the Sun?a)5.9 103 m/s2b)1.9 105 m/s2c)3.2 107 m/s2d)7.0 102 m/s2e)9.8 m/s2,5.2.4.A truck is traveling with a constant speed of 15 m/s.When the truck follows a curve in the road,its c
18、entripetal acceleration is 4.0 m/s2.What is the radius of the curve?a)3.8 mb)14 mc)56 md)120 me)210 m,5.2.5.Consider the following situations:(i)A minivan is following a hairpin turn on a mountain road at a constant speed of twenty miles per hour.(ii)A parachutist is descending at a constant speed 1
19、0 m/s.(iii)A heavy crate has been given a quick shove and is now sliding across the floor.(iv)Jenny is swinging back and forth on a swing at the park.(v)A football that was kicked is flying through the goal posts.(vi)A plucked guitar string vibrates at a constant frequency.In which one of these situ
20、ations does the object or person experience zero acceleration?a)i onlyb)ii onlyc)iii and iv onlyd)iv,v,and vi onlye)all of the situations,Chapter 5:Dynamics of Uniform Circular Motion,Section 3:Centripetal Force,Recall Newtons Second Law,When a net external force acts on an objectof mass m,the accel
21、eration that results is directly proportional to the net force and hasa magnitude that is inversely proportional tothe mass.The direction of the acceleration isthe same as the direction of the net force.,Recall Newtons Second Law,Thus,in uniform circular motion there must be a net force to produce t
22、he centripetal acceleration.The centripetal force is the name given to the net force required to keep an object moving on a circular path.The direction of the centripetal force always points toward the center of the circle and continually changes direction as the object moves.,Problem Solving Strate
23、gy Horizontal Circles,Draw a free-body diagram of the curving object(s).Choose a coordinate system with the following two axes.a)One axis will point inward along the radius(inward is positive direction).b)One axis will point perpendicular to the circular path(up is positive direction).Sum the forces
24、 along each axis to get two equations for two unknowns.a)FRADIUS:+FIN FOUT=m(v2)/r b)F:FUP FDOWN=0Do the math of two equations with two unknowns.,R,Just in case,The third dimension in these problems would be a direction tangent to the circle and in the plane of the circle.We choose to ignore this di
25、rection for objects moving at constant speed.If an object moves along the circle with changing speed then the forces tangent to the circle have become unbalanced.You can sum the tangential forces to find the rate at which speed changes with time,aTAN.The linear kinematics equations can then be used
26、to describe motion along or tangent to the circle.FTAN:FFORWARD FBACKWARD=m aTAN,R,tan,Example 5:The Effect of Speed on Centripetal ForceThe model airplane has a mass of 0.90 kg and moves at constant speed on a circle that is parallel to the ground.The path of the airplane and the guideline lie in t
27、he samehorizontal plane because the weight of the plane is balancedby the lift generated by its wings.Find the tension in the 17 mguideline for a speed of 19 m/s.,5.3.1.A boy is whirling a stone at the end of a string around his head.The string makes one complete revolution every second,and the tens
28、ion in the string is FT.The boy increases the speed of the stone,keeping the radius of the circle unchanged,so that the string makes two complete revolutions per second.What happens to the tension in the sting?a)The tension increases to four times its original value.b)The tension increases to twice
29、its original value.c)The tension is unchanged.d)The tension is reduced to one half of its original value.e)The tension is reduced to one fourth of its original value.,5.3.2.An aluminum rod is designed to break when it is under a tension of 600 N.One end of the rod is connected to a motor and a 12-kg
30、 spherical object is attached to the other end.When the motor is turned on,the object moves in a horizontal circle with a radius of 6.0 m.If the speed of the motor is continuously increased,at what speed will the rod break?Ignore the mass of the rod for this calculation.a)11 m/sb)17 m/sc)34 m/sd)88
31、m/se)3.0 102 m/s,5.3.3.A ball is attached to a string and whirled in a horizontal circle.The ball is moving in uniform circular motion when the string separates from the ball(the knot wasnt very tight).Which one of the following statements best describes the subsequent motion of the ball?a)The ball
32、immediately flies in the direction radially outward from the center of the circular path the ball had been following.b)The ball continues to follow the circular path for a short time,but then it gradually falls away.c)The ball gradually curves away from the circular path it had been following.d)The
33、ball immediately follows a linear path away from,but not tangent to the circular path it had been following.e)The ball immediately follows a line that is tangent to the circular path the ball had been following,5.3.4.A rancher puts a hay bail into the back of her SUV.Later,she drives around an unban
34、ked curve with a radius of 48 m at a speed of 16 m/s.What is the minimum coefficient of static friction for the hay bail on the floor of the SUV so that the hay bail does not slide while on the curve?a)This cannot be determined without knowing the mass of the hay bail.b)0.17c)0.33d)0.42e)0.54,5.3.5.
35、Imagine you are swinging a bucket by the handle around in a circle that is nearly level with the ground(a horizontal circle).What is the force,the physical force,holding the bucket in a circular path?a)the centripetal forceb)the centrifugal forcec)your hand on the handled)gravitational forcee)None o
36、f the above are correct.,5.3.6.Imagine you are swinging a bucket by the handle around in a circle that is nearly level with the ground(a horizontal circle).Now imagine theres a ball in the bucket.What keeps the ball moving in a circular path?a)contact force of the bucket on the ballb)contact force o
37、f the ball on the bucketc)gravitational force on the balld)the centripetal forcee)the centrifugal force,5.3.7.The moon,which is approximately 4 109 m from Earth,has a mass of 7.4 1022 kg and a period of 27.3 days.What must is the magnitude of the gravitational force between the Earth and the moon?a)
38、1.8 1018 Nb)2.1 1022 Nc)1.7 1013 Nd)5.0 1022 Ne)4.2 1020 N,5.3.8.The Rapid Rotor amusement ride is spinning fast enough that the floor beneath the rider drops away and the rider remains in place.If the Rotor speeds up until it is going twice as fast as it was previously,what is the effect on the fri
39、ctional force on the rider?a)The frictional force is reduced to one-fourth of its previous value.b)The frictional force is the same as its previous value.c)The frictional force is reduced to one-half of its previous value.d)The frictional force is increased to twice its previous value.e)The friction
40、al force is increased to four times its previous value.,Chapter 5:Dynamics of Uniform Circular Motion,Section 4:Banked Curves,On an unbanked curve,the static frictional forceprovides the centripetal force.,Unbanked curve,On a frictionless banked curve,the centripetal force is the horizontal componen
41、t of the normal force.The vertical component of the normal force balances the cars weight.,Banked Curve,Example 8:The Daytona 500The turns at the Daytona International Speedway have a maximum radius of 316 m and are steely banked at 31degrees.Suppose these turns were frictionless.As what speed would
42、 the cars have to travel around them?,5.4.1.Complete the following statement:The maximum speed at which a car can safely negotiate an unbanked curve depends on all of the following factors excepta)the coefficient of kinetic friction between the road and the tires.b)the coefficient of static friction
43、 between the road and the tires.c)the acceleration due to gravity.d)the diameter of the curve.e)the ratio of the static frictional force between the road and the tires and the normal force exerted on the car.,5.4.2.A 1000-kg car travels along a straight portion of highway at a constant velocity of 1
44、0 m/s,due east.The car then encounters an unbanked curve of radius 50 m.The car follows the curve traveling at a constant speed of 10 m/s while the direction of the car changes from east to south.What is the magnitude of the acceleration of the car as it travels the unbanked curve?a)zero m/s2b)2 m/s
45、2c)5 m/s2d)10 m/s2e)20 m/s2,5.4.3.A 1000-kg car travels along a straight portion of highway at a constant velocity of 10 m/s,due east.The car then encounters an unbanked curve of radius 50 m.The car follows the curve traveling at a constant speed of 10 m/s while the direction of the car changes from
46、 east to south.What is the magnitude of the frictional force between the tires and the road as the car negotiates the unbanked curve?a)500 Nb)1000 Nc)2000 Nd)5000 Ne)10 000 N,5.4.4.You are riding in the forward passenger seat of a car as it travels along a straight portion of highway.The car continu
47、es traveling at a constant speed as it follows a sharp,unbanked curve to the left.You feel the door pushing on the right side of your body.Which of the following forces in the horizontal direction are acting on you?a)a static frictional force between you and the seat b)a normal force of the door c)a
48、 force pushing you toward the doord)answers a and be)answers a and c,Chapter 5:Dynamics of Uniform Circular Motion,Section 5:Satellites in Circular Orbits,There is only one speed that a satellite can have if the satellite is to remain in an orbit with a fixed radius.,Dont worry,its only rocket scien
49、ce,Example 9:Orbital Speed of the Hubble Space TelescopeDetermine the speed of the Hubble Space Telescope orbitingat a height of 598 km above the earths surface.,Period to orbit the Earth,Geosynchronous Orbit,5.5.1.A satellite is in a circular orbit around the Earth.If it is at an altitude equal to
50、twice the radius of the Earth,2RE,how does its speed v relate to the Earths radius RE,and the magnitude g of the acceleration due to gravity on the Earths surface?a)b)c)d)e),5.5.2.It is the year 2094;and people are designing a new space station that will be placed in a circular orbit around the Sun.
