Popular in Course
Popular in Physics 2
This 63 page Class Notes was uploaded by Miss Lyla Rolfson on Monday October 26, 2015. The Class Notes belongs to PHYS 1205 at University of Oklahoma taught by Michael Strauss in Fall. Since its upload, it has received 11 views. For similar materials see /class/229269/phys-1205-university-of-oklahoma in Physics 2 at University of Oklahoma.
Reviews for Lab
Report this Material
What is Karma?
Karma is the currency of StudySoup.
Date Created: 10/26/15
Chapter 10 Spin and Orbital Motion Some Definitions Orbital motion Motion relative to a point often periodic but not necessarily so Spin motion Motion of an object as it rotates around an axis through its center of mass Rotational motion Motion around an axis of rotation Both orbital and spin motion are examples of rotational motion Fixed axis of rotation A singlenonchanging axis around which the object rotates Rigid bodies Objects that have a de nite unchanging shape Translational motion Motion with a xed direction of net force Review of Terminolgy 9 Sr A6 d6 0 11m AHO A d Aw dw a11m AHO A d VT dSdt r dHdt r60 aT dvTdt r dwdt wag a aC aT 2 2 ac vTr or w27rv T1V Vector Cross Product CAXB C AB sin6 The direction of C is given by the right hand rule described in the text We can write the components of the cross product as Cx AyBZ A ZBy Cy AZBx AXBZ CZ Aggy A yBx This can be easily remembered by using the determinant form i j k AgtltB BgtltAAx Ay AZ Bx By 132 1sz AZBy i 1sz AZBxj AxBy AyBx k Orbital Angular Momentum y lrXp 39x Linear momentum of a single particle p Linear momentum of a system of particles P Zp Angular momentum of a single point particle l Angular momentum of a system of particles L Z Interactive Question A 6kg particle moves to the right at 4 ms as shown The magnitude of its angular momentum in kg m2 s about the point 0 is 4 ms 6kg A Zero B 24 C 144 D 288 E 249 Problem What is the angular momentum of a 40 kg object traveling with a velocity of V 23i 15 j ms when it is at 68 56 m relative to the point 30 20 Angular Momentum for Circular Motion lrxp rgtltmv The direction in this case for l is up out of the paper The magnitude is I rmv sinl9 rmv rm ar mrza And since 0 also points up out of the paper 1 mrzo For this case we de ne the moment of inertia I mr2 l In Torque for Circular Motion 2F ma Consider the tangential motion of a rotating object with a tangential force acting on the object FT maT rFT mr aT mr ra rFT ml 2 05 T mrz 0 10 TiS the torque Since the angular acceleration has a direction out of the page T on If there is more than one force with a tangential component 217 2mr2oc ZText ZTint 2mr2a Z Cext 2mr2oc 21 1 oc Where I 2mr2 is the general form of the rotational inertia or moment of inertia This is one form of Newton s second law for rotation The moment of inertia depends on the object and the aXis of rotation More on this later General De nition of Torque In the previous example we de ned T rFT and de ned the direction of the torque as up out of the page It makes sense then that the general de nition of torque is TrXF Consider now a system of particles L 2 2111 Z 2i X pi dLdt Zidridt gtlt mvi ri x m dvidt 2 Vi gtlt mvi ri gtlt mai 2 2i 0 1 i gtlt mai 2 2103 X Zij Z 212171 ZTnet ZText 2Print dLdt Z1 ext 21 dLdt This is the more general form of Newton s 2nd law for rotation ext Summary TrXF General Case Speci c Case 216 dLdt If the moment of inertia doesn t change ZText Dc 1 Z 139 X P For a point object in a circular orbit I Z 20W 1 1m Im Now let s do some problems with all of these concepts Torgue i r E IVIF cost9 E h l Axis of rotation LF sint9 TrXF x The direction of the torque is up out of the page and the magnitude is TnFn6IWim ri is often called the lever arm Problem Consider the object shown below and the forces acting on the object Calculate the torque around an aXis perpendicular to the paper through a point 0 b point C C a 30N 10N Problem A penny with a mass of 25 grams sits on a turntable a distance of 66 cm from the center The turntable starts from rest and spins up to a maximum angular speed of 33 reVmin in 22 seconds rotating counterclockwise The penny does not slide with respect to the turntable a Assuming a constant torque What is the torque on the penny with respect to the center of the turntable b What is the angular speed of the penny after 11 seconds with respect to the center of the turntable c What is the angular momentum of the penny after 11 seconds with respect to the center of the turntable d What is the magnitude of the total force of friction acting on the penny after 11 seconds Conservation of Angular Momentum XI dLdt When there is no net external torque then the angular momentum does not change With time It is conserved ext dLdt O L nal Linitial There may be no external torque for many reasons including the case when the line of force for all forces passes through the axis of rotation 17 r gtlt F Problem Show how Kepler s second law that the area swept out by a planet moving in an ellipse is always the same for equal time intervals is a result of the conservation of angular Moment of Inertia For a single point object with mass m rotating at a distance of r around the axis of rotation I mr2 To calculate the rotational inertia for a more complicated object we simply sum up the rotational inertia of each part of the object Problem Suppose a baton is 10 m long with weights on each end weighing 03 kg Neglect the mass of the bar and consider the weights as point objects What is the moment of inertia for a baton a spinning around its center b spinning around one end of the baton The rotational inertia depends on both the aXis of rotation and how the mass is distributed Interactive Question Three thin disks with uniform mass density have the same mass and radius Rank the objects in order of increasing moment of inertia The aXis of rotation is shown as the dark dot and is perpendicular to the paper I II III A I II III B III II I C II III I D II I III E More information is needed Calculating the Moment of Inertia For an object with a continuous mass distribution the sum becomes an integral and we get IJr2dm Moments of inertia that have been calculated this way are often found in a table See Table 101 You don t have to memorize these formulas They will be given or I will ask you to derive them Problem What is the moment of inertia of a thin disk with a mass m radius R and thickness t rotating through its center of mass along an aXis perpendicular to the plane of the disk The Parallel Axis Theorem If you know the moment of inertia about any axis that passes through the center of mass of an object you can nd its moment of inertia about any other axis parallel to that axis with the parallel axis theorem which states 1 CM Mh2 where CM is the moment of inertia about the center of mass M is the mass of the object and h is the perpendicular distance between the two axes The proof of this theorem is given in the book Interactive Question Four uniform long rods with the same length but different masses are pushed with the same force as shown Rank the cases in the order of increasing angular acceleration The aXis of rotation is perpendicular to the paper and shown by the black dot 1 11 111 IV lo ID I I l M 39 M 1 1M M2 1 A 1 11 111 IV B 111 1 11 IV C 1 111 IV 11 D 1 111 11 IV E None of the above Problem A model of Uranus with the rings held on by thin rods hangs from two wires as shown What is the moment of inertia about the wire The radius of Uranus is RO25 m and its mass is Ml0 kg The ring is a thin hoop with radius of r05 m and a mass of m025 kg The rods each have negligible mass If one of the wires breaks what is the moment of inertia of the model as it rotates around the other wire Problem A cylindrical pulley with a mass of M300 kg and a radius of RO4OO In is used to lower a bucket with a mass of m200 kg into a well The bucket starts from rest and falls for 300 s a What is the linear acceleration of the falling bucket b How far does it drop c What is the angular acceleration of the cylinder Introduction to Rolling Motion Consider the motion of an object that is rolling Without slipping At point of contact v0 0 At top of Wheel vtop 2v If the rotational velocity is given by wand the radius of the rolling object is r then it is clear that v or Rotational Kinetic Energy An object spinning with angular velocity whas a rotational kinetic energy given by K 1202 exactly analogous to translations kinetic energy An object that is rolling with a center of mass velocity v and an angular velocity wabout its center of mass has a total kinetic energy given by K l2mv2 12CMa Of course when calculating gravitational potential energy the mass of an object can be considered as entirely located at the center of mass Problem A cylindrical pulley with a mass of M300 kg and a radius of RO4OO In is used to lower a bucket with a mass of m200 kg into a well The bucket starts from rest and falls for a distance of 252 In Use conservation of energy to nd the speed of the bucket after it has fallen this distance Neglect any dissipative forces Interactive Question A force F is applied to a dumbbell for a time interval At rst as in a and then as in b In which case does the dumbbell acquire the greater centerof mass speed F F a 39 b A a B b C no difference Interactive Question A force F is applied to a dumbbell for a time interval At rst as in a and then as in b In which case does the dumbbell acquire the greater energy F a b A a B b C no difference Interactive Question A hollow cylinder of mass M and radius R rolls down an inclined plane A block of mass M slides down an identical inclined plane If both objects are released at the same time A the block will reach the bottom rst B the cylinder will reach the bottom rst C the block will reach the bottom with greater kinetic energy D the cylinder will reach the bottom with greater kinetic energy E both the block and the cylinder will reach the bottom at the same time Interactive Question A solid sphere S a thin hoop H and a solid disk D all with the same radius are allowed to roll down an inclined plane Without slipping In Which order Will they arrive at the bottom The st one down listed rst A HDS B HSD C SDH D SHD E DHS Problem Two bicycles roll down a hill which is 20 m high Both bicycles have a total mass of 12 kg and 700 mm diameter wheels r 0350 m The rst bicycle has wheels with a mass of 060 kg each and the second bicycle has wheels with a mass of 030 kg each Neglecting air resistance which bicycle has the faster speed at the bottom of the hill Consider the wheels to be thin hoops Rotational WorkEnergy Theorem F Cons1der a force actmg ds on an object so that it V de rotates a angle d6 2 I WJFdsJFTR d6lrd6 Where is the angle between the applied force and the tangential motion and His the angle between the applied force and the radial vector from the axis of rotation to the force so that 9 900 In a similar manner P dWdt TdHdt Ta Problem a What is the kinetic energy of the earth s rotation and the kinetic energy of the earth s orbit around the sun What is the total kinetic energy of the earth b What is the angular momentum of the earth s rotation and of the earth s orbit around the sun What is the total angular momentum of the earth mE 597 x 1024 kg rE638 x106m rs 149 x lOllm More on Rolling Motion Rolling motion can be thought of as the translational motion of the center of mass plus the rotational motion around the center of mass Or it can be thought of as purely rotational motion about the point in contact with the ground Either way gives the same answer for the total kinetic energy K 1202 12mR22 msz 34mR2a2 K 12mv2 1202 12qu2 12mR22a2 12 de 12mR22a2 34 de Problem A solid sphere rolls down an incline Without slipping If the acceleration of the center of mass is 02g What angle is the incline Problem A uniform solid sphere is set rotating about a horizontal axis with an angular speed ab and is placed on the oor The coef cient of kinetic friction between the oor and the sphere is u What is the speed of the center of mass of the ball when it starts rolling Without slipping Do Problem A cylindrical rod is set rotating with an angular speed of 000 then placed on the oor If the coef cient of kinetic friction between the rod and the oor is uk What is the speed of the center of mass when the rod begins to roll Without slipping wo More on Conservation of Angular Momentum 21 dLdt ext So if there is no external torque dLdt O angular momentum is conserved When will there be no external torque We know for a point particle that l r gtlt p and that if the particle is in a circular orbit this becomes 1 0 But What is L for a rigid object rotating around a xed axis of rotation Let s show that L Zili Ziri gtlt pi 10 for any rigid body rotating around a xed axis of rotation First consider the angular momentum for a rigid body rotating around a xed axis of rotation in a plane so that the angular momentum is in a xed direction along the axis of rotation L M sianm Jr ar sin90 dm lrza dm UJFZ dm L Ia Now consider two points on an object that is symmetric about its axis of rotation The horizontal components of L1 and L2 cancel and only the vertical components contribute to the total angular momentum So just like a single plane rotating L I a Interactive Question An ice skater performs a pirouette by pulling her outstretched arms close to her body What happens to her moment of inertia about the aXis of rotation A It does not change B It increases C It decreases D It changes but it is impossible to tell which way Interactive Question An ice skater performs a pirouette by pulling her outstretched arms close to her body What happens to her angular momentum about the aXis of rotation A It does not change B It increases C It decreases D It changes but it is impossible to tell which way Problem A student is sitting on a swivel seat and is holding a 20 kg weight in each hand If he is rotating at l reVs 628 rads when the weights are held in outstretched arms 75 In from the aXis of rotation how fast is he rotating when he pulls the weights in to the aXis of rotation The rest of his body can be approximated as a cylinder with mass of 72 kg and radius of 25 In Interactive Question A ball on a string is rotating in a circle The string is shortened by pulling it through the axis of rotation What happens to the angular velocity and the tangential velocity of the ball angular velocity tangential velocitv A increases decreases B increases stays the same C increases increases D stays the same stays the same E stays the same increases Interactive Question An ice skater performs a pirouette by pulling her outstretched arms close to her body What happens to her rotational kinetic energy about the aXis of rotation A It does not change B It increases C It decreases D It changes but it is impossible to tell Which way Problem A child of mass 25 kg runs with a speed of 25 ms and jumps on a merrygoround along a path tangential to the rim The merrygoround has a moment of inertia of 500 kgm2 and a radius of 2 m What is the nal angular velocity of the merrygoround and child Interactive Question A baton with two spheres at its I 60 end is rotated around an aXis as shown As the baton spins O which of the following is true A The angular momentum of the baton doesn t change B The angular velocity points in the same direction as the angular momentum C The angular momentum changes but there is no torque applied to the baton D A net eXtemal torque causes the angular momentum to change E More than one of the above is true Interactive Question A baton with two spheres at its end is rotated around an aXis as I 60 shown Neglect the mass of the 0 rod connecting the spheres Which direction is the angular momentum pointing when the baton is in the position shown l1 A B C D E Precession of a Top TZerg along 2 L d dLL TdtL mgr dtL up d dt mgrL mgrwww Static Equilibrium The conditions for static equilibrium in two dimensions 2Fx O 2Fy 0 270 The net torque must be zero around any axis of rotation Problem A board with uniform mass density and a weight of 400 N supports two children weighing 500 N and 350 N The support is placed under the center of gravity of the board and the 500 N child is 150 In from the center a What is the force which the support exerts b Where should the 350 N child sit to balance the board Interactive Question A heavy boy and a lightweight girl are balanced on a massless seesaw If they both move forward so that they are onehalf their original distance from the pivot point what will happen to the seesaw A The side the boy is sitting on will tilt downward B The side the girl is sitting on will tilt downward C Nothing the seesaw will still be balanced D It is impossible to say without knowing the masses and the distances Problem A traf c light hangs from the end of a long pole as show The pole has a length L 75 m long and a mass of 80 kg The mass ofthe light is 110 kg Determine the tension in the horizontal cable and the vertical and horizontal components of the force on the pivot point P quot L75m 38111 ri38m M80kg m 110kg Problem A 101 kg uniform board is wedged into a corner and held by a spring attached to its end at a 5000 angle with respect to the horizontal direction The spring has a spring constant of 176 Nm By how much does the spring stretch Problem You hold your forearm out horizontally and hold a 50 N object in your hand located 035 m from the elbow joint Your bicep muscle is attached at a distance of 0030 m from the elbow joint The mass of your forearm is 13 kg with the centerofmass of the forearm located 017 m from the elbow joint What is the force of the humerous the bone between the shoulder and the forarm and the bicep on the arm PM FB 50 N Fg 13 kg98 ms2 FB 035 m FH rg017m rM 0030 m Problem The large quadriceps muscle in the upper leg terminates at its lower end in a tendon attached to the upper end of the tibia The forces on the lower leg are modeled in the gure on the next page where T is the tension in the tendon C is the weight of the lower leg F is the weight of the foot and B is the force of the femur on the tibia Assume C 300 N F 125 N and the leg is in the position shown in the gure The tendon is attached one fth of the way down the lower leg and the center of mass of the lower leg is at its geometric center a Find the tension of the tendon T b Find the x and y components of the force of the femur on the tibia B Interactive Question A 1kg rock is suspended by a massless string from one end of a 1m measuring stick What is the weight of the measuring stick if it is balanced by a support force at the 025 m mark A 025 kg B 05 kg C 1kg D 2 kg E 3kg