Version 001 Rolling & Angular Momentum ramadoss (171) 1

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1 Version 001 olling & Angular omentum ramadoss (171 1 This print-out should have 76 questions ultiple-choice questions may continue on the next column or page find all choices before answering One version of this assignment with keys is on my website AP B 1998 C points A kg object moves in a circle of radius 4 m at a constant speed of 3 m/s A net force of 45 N acts on the object What is the magnitude of the angular momentum L of the object with respect to an axis perpendicular to the circle and through its center? 1 L = 18 N m kg L = 1 m s 3 L = 4 kg m correct s 4 L = 135 kg m s 5 L = 9 N m kg AP 1998 C (part 1 of 100 points A wheel with rotational inertia I is mounted on a fixed, frictionless axle The angular speed ω of the wheel is increased from zero to ω f in a time interval T What is the average net torque τ on the wheel during this time interval? 1 τ = Iω f T correct τ = ω f T 3 τ = Iω f T 003 (part of 100 points What is the average power input to the wheel during this time interval? 1 P = I ω f T P = Iω f T correct 3 P = Iω f T 4 P = I ω f T 5 P = Iω f T AP 1998 C points A wheel of mass and radius rolls on a level surface without slipping If the angular velocity of the wheel about its center is ω, what is its linear momentum relative to the surface? 1 p = ω p = ω 3 p = 0 4 p = ω 5 p = ω correct Atwood achine (part 1 of points Consider a massless, frictionless pulley attached to the ceiling A massless, inextensible string is attached to the masses b and a, where b > a ThetensionsT x, T z,t y,and the gravitational constant g are magnitudes 4 τ = Iω f T 5 τ = ω f T

2 Version 001 olling & Angular omentum ramadoss (171 T x b l T z a T y What is trueabout the tensions T x and T y? 1 T x < b g and T y < a g T x < b g and T y = a g 3 T x > b g and T y < a g 4 T x > b g and T y > a g 5 T x = b g and T y = a g 6 T x > b g and T y = a g 7 T x = b g and T y < a g 8 T x = b g and T y > a g 9 T x < b g and T y > a g correct 006 (part of points Which relationship about T z is true? 1 T z = b g + a g T z > b g + a g 3 T z < b g + a g correct 007 (part 3 of points What is the magnitude of the acceleration of the center of mass of the system? ω ( b + a 1 g b a b + a b a g 3 b + a a b g 4 b a g b + a ( b a 5 g correct b + a 6 a b b + a g Bohr odel of Hydrogen (part 1 of 100 points In the Bohr s model of the hydrogen atom, theelectronmovesinacircularorbitofradius m around the proton Assume that the orbital angular momentum of the electron is equal to h Calculate the orbital speed of the electron Correct answer: m/s 009 (part of 100 points Calculate the angular frequency of the electron s motion Correct answer: s 1 Bullet otates a od (part 1 of 100 points Awoodenblockofmass hangsfromarigid rod of length l having negligible mass The rod is pivoted at its upper end A bullet of mass m traveling horizontally and normal to the rod with speed v hits the block and gets embedded in it v m What is the angular momentum L of the block-bullet system, with respect to the pivot l

3 Version 001 olling & Angular omentum ramadoss (171 3 point immediately after the collision? ( m 1 L = vl +m L = vl 4 3 a m v min a ω 3 L = ( mvl 4 L = mvl correct 5 L = (m+vl 011 (part of 100 points What is the fraction K f K i (the final kinetic energy compared to the initial kinetic energy in the collision? 1 K f = K i +m K f = m K i m+ correct 3 K f K i = m m 4 K f = m K i m+ 5 K f = K i m Bullet Hits a Cube points Assume: A bullet of mass m and cube of mass undergo an inelastic collision, where m Note: The moment of inertia of this cube (with edges of length a and mass about 8 a an axis along one of its edges is 3 A solid cube is resting on a horizontal surface The cube is constrained to rotate about an axis at its bottom right edge (due to a small obstacle on the table A bullet with speed v min is shot at the left-hand face at a height of 4 a The bullet gets embedded in 3 the cube g Find the minimum value of v min required to tip the cube so that it falls its right-hand face 1 v min = m ( 5ga 1 v min = m ( ga 1 3 v min = m ( 3ga v min = m ( 3ga v min = ( 3ga 1 correct m Child on a errygoround points A child is standing on the edge of a merry-goround that is rotating with frequency f The child then walks towards the center of the merry-go-round For the system consisting of the child plus the merry-go-round, what remains constant as the child walks towards the center? (neglect friction in the bearing 1 mechanical energy and angular momentum only mechanical energy 3 neither mechanical energy nor angular momentum 4 only angular momentum correct Circular Trajectory points A particle of mass m moves in a circle of radius at a constant speed v Assume: The motion begins from the point Q, which has

4 Version 001 olling & Angular omentum ramadoss (171 4 coordinates (, 0 Determine the angular momentum of the particle about point P, which has coordinates (,0 as a function of time [ ( ] 1 L vt = mv cos +1 ˆk correct L = mv [ sin ] +1 ˆk ( vt 3 None of these [ ( ] 4 L = m v vt cos +1 ˆk [ ( 5 L vt = mv cos +π +1 ˆk [ ( ] 6 L vt = mv sin +π +1 ˆk [ ( ] 7 L = m v vt sin +π +1 ˆk [ ( ] 8 L = m v vt cos +π +1 ˆk [ ( ] 9 L = m v vt sin +1 ˆk Collision With a Cylinder points A solid cylinder of mass = 14 kg, radius = 036mand uniform density ispivotedon a frictionless axle coaxial with its symmetry axis Aparticleofmassm = 4kgandinitial velocity v 0 = 66 m/s (perpendicular to the cylinder s axis flies too close to the cylinder s edge, collides with the cylinder and sticks to it Component of Net Torque points A force F = ( Nî+ (36 Nĵ is applied to an object that is pivoted about a fixed axis aligned along the z coordinate axis The force isappliedatthepoint = (4mî+(49mĵ Find the z-component of the net torque Correct answer: 46 Nm Component of Net Torque points A force F = ( Nî+ (6 Nĵ is applied to an object that is pivoted about a fixed axis aligned along the z coordinate axis The force isappliedatthepoint r = (38mî+(48mĵ Find the z-component of the net torque Correct answer: 08 N m Conical Pendulum points A small metallic bob is suspended from the ceiling by a thread of negligible mass The ballisthen set inmotioninahorizontal circle so that the thread describes a cone 98 m/s 8 17 m Before the collision, the cylinder was not rotating What is the magnitude of its angular velocity after the collision? Correct answer: rad/s v 5 kg Calculate the magnitude of the angular momentum of the bob about a vertical axis through the supporting point The acceleration of gravity is 98 m/s Correct answer: kg m /s Cylinder rolls points

5 Version 001 olling & Angular omentum ramadoss (171 5 The coefficient of static friction between a certain cylinder and a horizontal floor is 04 If the rotational inertia of the cylinder about its symmetry axis is given by I = (1/, then the maximum acceleration the cylinder can have without slipping is: 1 08 g correct 0 g 3 04 g 4 1 g 5 01 g Decelerated Grinding Wheel 00 (part 1 of 100 points Themotordrivingagrindingwheel witharotational inertia of 05 kgm is switched off when the wheel has a rotational speed of 40 rad/s After 10 s, the wheel has slowed down to 3 rad/s What is the absolute value of the constant torque exerted by friction to slow the wheel down? Correct answer: 04 Nm 01 (part of 100 points If this torque remains constant, how long after the motor is switched off will the wheel come to rest? Correct answer: 50 s Disk and Hoop ace 0 (part 1 of 100 points A uniform solid disk and a uniform hoop are placed side by side at the top of an incline of height h If they are released from rest and roll without slipping, determine their speeds when they reach the bottom (d = disk, h = hoop 1 v d = hg, v h = hg v d = 4 3 hg, v h = hg correct 3 v d = hg, v h = hg v d = hg, v h = 3 hg 5 v d = 3 hg, v h = 3hg 03 (part of 100 points Which object reaches the bottom first? 1 the disk correct the hoop 3 both at the same time Disk and Hoop ace 0 04 (part 1 of 100 points A uniform solid disk and a uniform hoop are placed side by side at the top of an incline of height h If they are released from rest and roll without slipping, determine their speeds when they reach the bottom (d = disk, h = hoop 1 v d = 3hg, v h = hg v d = hg, v h = hg 1 3 v d = hg, 1 v h = 3 hg 4 v d = hg, v h = hg 4 5 v d = 3 hg, v h = hg correct 6 v d = 3hg, v h = hg 7 v d = 3 hg, v h = 3hg 8 v d = 1 3hg, v h = 3 hg 9 v d = 1 hg, v h = hg

6 Version 001 olling & Angular omentum ramadoss ( v d = 1 3 hg, v h = hg 05 (part of 100 points Whatistheratiooftheiraccelerationsasthey roll down the incline? 1 a disk a hoop = 4 3 correct a disk = a hoop 3 a disk 3 = a hoop 4 a disk 4 = a hoop 3 5 a disk a hoop = 1 6 a disk a hoop = a disk a hoop = 3 8 a disk a hoop = 3 9 a disk a hoop = 10 a disk a hoop = 3 Door Handle Torque points A door is opened in the usual way, in the direction indicated D C In which direction is the net torque vector τ due to the force applied to the door handle? The force is toward you, away from the door 1 B A B Insufficient information is given 3 C 4 D 5 A correct Figure Skater 07 (part 1 of 100 points Afigureskaterrotatingononespotwithboth arms and one leg extended has moment of inertia I i She then pulls in her arms and the extended leg, reducing her moment of inertia to 075I i What is the ratio of her final to initial kinetic energy? 1 3/8 3 8/3 4 9/16 5 3/ /3 correct 8 1/ 9 16/9 08 (part of 100 points Consider the following statements for the figure skater: I Angular momentum was conserved II echanical energy was conserved III The kinetic energy changed because of energy dissipation due to friction IV Her rotation rate changed in response toatorqueexertedbypullinginher armsand leg Which is the correct combination of statements? 1 I, II, III

7 Version 001 olling & Angular omentum ramadoss (171 7 II 3 I correct 4 I and II 5 I, II, IV Figure Skater Spin points A figure skater on ice spins on one foot She pulls in her arms and her rotational speed increases Choose the best statement below: 1 Her angular speed increases because her potential energy increases as her arms come in Her angular speed increases because by pulling inher armsshe creates a net torque in the direction of rotation 3 Her angular speed increases because her angular momentum increases 4 Her angular speed increases because her angular momentum is the same but her moment of inertia decreases correct 5 Her angular speed increases because air friction is reduced as her arms come in 6 Her angular speed increases because she is undergoing uniformly accelerated angular motion Grindstone Energy points A constant torque of 15 N m is applied to a grindstone whose moment of inertia is 014 kg m Using energy principles, and neglecting friction, find the angular speed after the grindstone has made 179 rev Correct answer: rad/s Horizontal otation 031 (part 1 of points A mass m is attached to a cord passing through a small hole in a frictionless horizontal surface The mass is initially orbiting withspeedv 0 inacircleofradiusr 0 Thecord is then slowly pulled from below, decreasing the radius of the circle to r What is the speed of the mass when the radius is r? 1 None of these v = v 0 r r 0 3 v = v 0 r 0 r 4 v = v 0 r 0 r r 0 +r 5 v = v 0 r 0 r correct 6 v = v 0 r 0 +r r 0 7 v = v 0 r 0 r 0 r 8 v = v 0 r 0 r 0 +r 9 v = v 0 r 0 r r 0 10 v = v 0 r 0 +r r 0 r 03 (part of points Find the tension in the cord as a function of r 1 T = m(rv 0 r 3 0 T = m(r 0v 0 r 3 3 None of these 4 T = m(r 0v 0 r 3 5 T = m(r 0v 0 (r+r T = m(r 0v 0 r 3 correct

8 Version 001 olling & Angular omentum ramadoss ( (part 3 of points How much work W is done in moving m from r 0 to r? Note: The tension depends on r 1 W = ( r 3 mv 0 r0 W = 1 ( (r0 +r mv 0 r 1 3 W = 1 ( r mv 0 r0 1 4 W = r 0 3 0( mv r 5 W = 1 ( r 0 mv 0 r 1 correct 6 None of these 7 W = 1 r 0 3 0( mv r 1 8 W = 1 ( r 3 mv 0 1 r 0 Horizontal Circle (part 1 of 100 points A ball is rotating in a horizontal circle at the end of a string of length m at an angular velocity of 89 rad/s The string is gradually shortened to 17 m without any force being exerted in the direction of the ball s motion Find the new angular velocity of the ball Correct answer: rad/s 035 (part of 100 points Find its new linear speed Correct answer: m/s Ice Skater pulls arms in points AniceskaterwithrotationalinertiaI 0 isspinning with angular velocity ω 0 She pulls her arms in, decreasing her rotational inertia to I 0 /3 Her angular velocity becomes: 1 ω 0 /3 ω 0 3 3ω 0 correct 4 3ω 0 5 ω 0 / 3 Impulse on a Cue Ball points A cue stick strikes a cue ball and delivers a horizontal impulse in such a way that the ball rolls without slipping as it starts to move At what height above the ball s center (if the radius of the ball is 17 cm was the blow struck? Correct answer: 068 cm KE Comparison points A object of mass and radius has a rotational inertia of I C = k, where k is a constant If the object rolls without slipping, what is the ratio of its rotational kinetic energy to its linear kinetic energy? 1 K r K C = k correct K r K C = k K r 3 = k +1 K C K r 4 = k K C K r 5 = k +1 K C k K r 6 = k K C k +1 KE atio points A ball with I cm = κ, mass and radius rolls along a horizontal surface without slipping

9 Version 001 olling & Angular omentum ramadoss ( applying the force near the rim, radially outward from the axle applying the force near the axle, parallel to the tangent to the wheel What is the ratio K r K cm of rotational to center-of-mass kinetic energy? 1 κ 1; they have the same magnitude 3 1 κ 4 κ 5 κ 6 κ 7 1 κ 8 κ correct Kinetic Energy of a olling Wheel points If a steel, thin-shelled wheel of radius r and mass is moving along the road at m/s, what is its total kinetic energy? correct 5 5 aximize Torque points Aforceistobeappliedtoawheel Thetorque can be maximized by: 3 applying the force near the axle, radially outward from the axle 4 applying the force at the rim, tangent to the rim correct 5 applying the force at the rim, at 45 to the tangent Length atio points A non-uniform horizontal beam of length l andmass hasitscenter ofmass atdistance x from the left end, as shown It is supported bytworopes;theleft-endropemakesanangle of θ with the vertical and has tension T 1, and the right-hand rope makes an angle of φ with the vertical and has tension T T 1 θ x cm l φ T If sinθ = cosφ = 3 5 and cosθ = sinφ = 4 5, and therodis stationary,what isthe ratio x l? correct 7 1

10 Version 001 olling & Angular omentum ramadoss ( omentum of a Particle points A particle whose mass is kg moves in xyplane with a constant speed of 9 m/s in the positive x-direction along y = 4 m Find the magnitude of its angular momentum relative to the point (x 0,y 0, where x 0 = 1 m and y 0 = 1 m of mass = m and radius That is, the distance between the trajectory of the particle and the center of the disk is If the disk is initially at rest and is pivoted about a frictionless axle through its center O perpendicular to the plane of the paper, the angular velocity of the disk plus particle system after the collision in terms of v 0 and? O m Correct answer: 144 kgm /s omentum of Airplane points An airplane of mass 998 kg flies level to the groundataconstantspeedof175m/srelative to the Earth An observer on the ground along the path of the plane sees the plane a distance 1455 m away at an angle above the horizontal of 605 What is the magnitude of the airplane s angular momentum relative to a ground observer directly below the airplane? Correct answer: kgm /s omentum of oon points There is a moon orbiting an Earth-like planet The mass of the moon is kg, the center-to-center separation of the planet and the moon is km, the orbital period ofthemoonis57days,andtheradiusofthe moon is 1930 km What is the angular momentum of the moon about the planet? Correct answer: kgm /s Projectile Hits a Disk 046 (part 1 of 100 points A particle of mass m and speed v 0 collides with and sticks to the edge of a uniform disk 1 3v v 0 8 v v 0 5 v 0 correct 047 (part of 100 points What is the loss of kinetic energy due to the collision process? mv 0 correct 1 1 mv mv mv mv 0 Projectile Sticks to a od

11 Version 001 olling & Angular omentum ramadoss ( (part 1 of 100 points A projectile of mass m = 134 kg moves to the right with speed v 0 = 53 m/s The projectile strikes and sticks to the end of a stationary rod of mass = 44 kg and length d = 95mthatispivotedaboutafrictionless axle through its center v 0 ω m O Find the angular speed of the system right after the collision Correct answer: rad/s 049 (part of 100 points Determine the ratio of the kinetic energy lost to the initial kinetic energy Correct answer: Qualitative String on ope points A particle, held by a string whose other end is attached to a fixed point C, moves counterclockwise in a circle on a horizontal frictionless surface If the string is cut, the angular momentum of the particle about the point C: 1 changes direction but not magnitude decreases 3 increases 4 does not change correct 5 none of these ace Down a Plane 051 (part 1 of points A thin cylindrical shell and a solid cylinder have the same mass and radius The two are released side by side and roll down, without d O slipping, from the top of an inclined plane that is 14 m above the ground Find the final linear velocity of the thin cylindrical shellthe acceleration of gravity is 98 m/s Correct answer: m/s 05 (part of points Find the final linear velocity of the solid cylinder Correct answer: m/s 053 (part 3 of points When the first object reaches the bottom, what is the height above the ground of the other object? Correct answer: 035 m igid Body Torque points A force F = F 0 (î+ĵ+ˆk acts on a rigid body at a point r = r 0 (î ĵ away from the axis of rotation What is the resulting torque on the body? 1 r 0 F 0 (ĵ î ˆk r 0 F 0 (î+ĵ+ˆk 3 r 0 F 0 (ˆk î ĵ correct 4 r 0 F 0 (î ˆk ĵ 5 Zero 6 r 0 F 0 (ˆk î ĵ 7 r 0 F 0 (ˆk î ĵ 8 r 0 F 0 (ˆk î ĵ 9 r 0 F 0 (ˆk î ĵ ing and Disk ace points

12 Version 001 olling & Angular omentum ramadoss (171 1 A solid disk (thin cylindrical object with uniform density and a ring (all of its mass located at the radius of the ring roll down an incline without slipping Let m ring, m disk be the inertial masses and r ring and r disk the radii The ring is slower than the disk only if 1 m ring > m disk m ring = m disk and r ring = r disk 3 r ring > r disk 4Theringisalwaysslower regardlessofthe relative values of m and r correct olling Basketball points A 180 g basketball has a 30 cm diameter and may be approximated as a thin spherical shell A 69 kg mass is on a spring of length 70 cm and stiffness constant 44 N/cm It is spun at a speed of 033 1/s in the absence of gravity How far will the spring stretch? Correct answer: cm System Angular omentum points Suppose a physics instructor seated himself at rest on a low-friction piano stool and heldaspinningbikewheelwithitsangularvelocity vector initially horizontal Suppose the instructor then turned the bike wheel so that its angular velocity vector pointed vertically downward When this turn was completed, the instructor would be spinning rapidly on the stool What would be the final, total angular momentum of the system of instructor plus bike wheel, projected along the vertical axis, the only axis along which the system can spin? 1 Twice its initial value Equalbutoppositetowhatthewheelhad initially 8 m 180 g µ = The same, nonzero value it had initially 4 Zero, as it was initially correct Starting from rest, how long will it take a basketball to roll without slipping 8 m down an incline that makes an angle of 46 with the horizontal? The moment of inertia of a thin spherical shell of radius and mass m is I = 3 m, the acceleration due to gravity is 98 m/s, and the coefficient of friction is 04 Correct answer: s Supernova Explosion points A star of radius km rotates about its axiswithaperiodof34 days Thestar undergoes a supernova explosion, whereby its core collapses into a neutron star of radius 4 km Estimate the period of the neutron star (assume the mass remains constant Correct answer: s Yo Yo on table with friction points Ayo-yo,arrangedasshown,restsonasurface evolving ass on a Spring points

13 Version 001 olling & Angular omentum ramadoss ( h 8 m ω When a force F is applied to the string as shown, the yo-yo: 1 moves to the right and rotates clockwise correct moves to the right and does not rotate 3 moves to the right and rotates counterclockwise 4 moves to the left and rotates counterclockwise 5 moves to the left and rotates clockwise Yo Yo N (part 1 of points Assume: Drag (friction is negligible Given: g = 981 m/s The density of this large Yo-Yo like solid is uniform throughout The Yo-Yo like solid has a mass of 6 kg 4 m 6 kg Cross sectional Side View Calculate the moment of inertia about the center of mass (axis of rotation Correct answer: 707 kgm 06 (part of points What is the vertical acceleration of the center of mass of the Yo-Yo? Correct answer: m/s 063 (part 3 of points What is magnitude of the torque the cord exerts on the center of mass of the Yo-Yo? Correct answer: N m 064 (part 4 of points The Yo-Yo is released from rest at height h Find the velocity v of the center of mass of the disk at the height h = 9 m Correct answer: m/s l l l Front View A cord is wrapped around the stem of the Yo-Yo like solid and attached to the ceiling The radius of the stem is 4 m and the radius of the disk is 8 m Yo Yo C (part 1 of 100 points You are playing with a Yo-Yo which you can describe as a uniform disk with mass m and radiusr Thestring ofthe Yo-Yohas a length L The Yo-Yo rolls down vertically h Determine the tension T of the string at a height h below the original position h as indicated in the sketch m r ω

14 Version 001 olling & Angular omentum ramadoss ( T = 5 mg T = mg 3 T = 3mg 4 T = 1 4 mg 5 T = L h r 6 T = 1 mg 7 T = 5 mg 8 T = 7 4 mg 9 T = T = r L mg mg mg correct 066 (part of 100 points After the Yo-Yo has been elastically reflected at length L, it moves upward Determine the speed of itscenter when it is at a height h below the original position h r 1 v = L g h v = g h gl 3 v = 3g h 5 4 v = 3 g h 3 5 v = g h 1 6 v = 8 g h 7 v = g h 4 8 v = g h correct 3 9 v = g h 10 v = 1 g h Yo Yo C (part 1 of points A yo-yo consists of two identical, uniform disks whose total mass is and radius connected between the disks by a shaft of radius r and negligible mass See the figure below r h The combined mass of the three disks (the yo-yo is The yo-yo rolls down an inextensible, thin string of length h + l, where l Neglect the mass and moment of inertia of the shaft The string is initially wound around the shaft and unwinds as the yo-yo rollsdown the string What is the moment of inertia, I, of the yo-yo about the axis through the centers of mass of the two disks? 1 I = 1 4 (r+ I = 1 (r+ 3 I = 1 correct 4 I = ( +r 5 I = 1 r 6 I = 1 4 ( r 7 I = (r + 8 I = 9 I = r ω

15 Version 001 olling & Angular omentum ramadoss ( I = 1 ( +r 068 (part of points What is the ratio of the magnitude of the gravitational acceleration g to the magnitude of the acceleration a of the center of mass of the yo-yo? 1 g = a r g = 1+ 1 ( correct a r 3 g ( = a r 4 g = 1+ 1 a r 5 g = 1+ 1 a r 6 g ( = 1+ a r 7 g = 1+ a r 8 g = 1 a r 9 g = 1+ a r 10 g = 1 a r 069 (part 3 of points Assume: The yo-yo started from rest What is the ratio of the final kinetic energy K f totherotationalkineticenergyk rot about the axis through the center of mass when the yo-yo has fallen the entire length of the string? K f r 1 = 1+ K rot 3 4 K f = 1+ r K rot K f r = K rot K f K rot = 1+ r K f K rot = 1+( r K f K rot = 1+ r K f K rot = ( r K f r 8 = 1+( K rot K f 9 = r K rot correct 070 (part 4 of points If E is the total mechanical energy, p is the linear momentum of the center of mass and L is the angular momentum relative to the rotational axis through the center of mass, mark the correct statement 1 E is conserved but not p and L correct E and L are conserved, but not p 3 p and L are conserved, but not E 4 E and p are conserved but not L 5 E, p, and L are all conserved 6 Neither E, p, or L is conserved 7 L is conserved, but not E and p 8 p is conserved, but not E and L 071 (part 5 of points What is the SI unit of torque τ? 1 [τ] = Nm/s [τ] = kgm/s 3 [τ] = kgm/s 4 [τ] = kgm /s correct 5 [τ] = kgs /m 6 [τ] = Nkg

16 Version 001 olling & Angular omentum ramadoss ( [τ] = N/m 8 [τ] = kg s/m Yo Yo C (part 1 of 100 points Given:, g, and A string is wound around a uniform disk of radius and mass The disk is released fromrest ataheight hwiththestringvertical and its tip end tied to a fixed support as in figure h Derive an expression for the tension, T, on the string at height h + h, as the disk descends 1 T = g ( h T = g h 3 T = g ω parameters? 1 a = ( h h a = g 3 a = g h 4 a = 1 4 g 5 a = 4g 6 a = ( h 7 a = 3 4 g g 8 a = 3 g correct 9 a = 4 3 g g Yo Yo C (part 1 of points A string is wrapped around the stem of a Yo- Yo which has a mass and a moment of inertia I about its center axis The radius of the stem is r 1 and the radius of the disk is r Denote thetension ofthe string T andthe descending acceleration a 4 T = g h 5 T = 3 g h r ω 6 T = g h 7 T = 1 3 g correct 8 T = g ( h h 9 T = 1 g+ g h 10 T = g 073 (part of 100 points What is the equation for the vertical accelerationofthecenterofmassintermsofthegiven r 1 The correct torque equation is given by 1 T r = ai r T r r 1 = ai r 1 +r 3 T r 1 = ai r 1 correct 4 T r 1 = ai r 1 +r

17 5 T r = ai r 1 +r 6 T r 1 +r 7 T r 1 +r 8 T r 1 +r 9 T r 1 +r Version 001 olling & Angular omentum ramadoss ( = ai r 1 = ai r = ai r 1 +r = ai r r (part of points The correct vertical force equation is given by 1 g +T = a g T = a correct 3 T g = a 076 (part 3 of points elease the Yo-Yo from rest It reaches a center of mass speed v when it falls for a height h Using the same notation as in the previous questions, the correct work-energy equation is given by 1 1 v = 1 I ( v r 1 v = 1 Iv 3 1 v = 1 I ( v r 1 4 g h = 1 v 5 g h = 1 I ( v r 1 6 g h = 1 I ( v r 7 g h = 1 v + 1 I ( v r 1 correct 8 g h = 1 v + 1 I ( v r