Plane Motion of Rigid Bodies: Forces and Accelerations. Contents
|
|
- Hannah Morton
- 7 years ago
- Views:
Transcription
1 Plane Motion of Rigid odies: Forces and Accelerations Contents Introduction Equations of Motion of a Rigid ody Angular Momentum of a Rigid ody in Plane Motion Plane Motion of a Rigid ody: d Alembert s Principle Axioms of the Mechanics of Rigid odies Problems Involving the Motion of a Rigid ody Sample Problem 6. Sample Problem 6. Sample Problem 6.3 Sample Problem 6.4 Sample Problem 6.5 Constrained Plane Motion Constrained Plane Motion: Noncentroidal Rotation Constrained Plane Motion: Rolling Motion Sample Problem 6.6 Sample Problem 6.8 Sample Problem 6.9 Sample Problem
2 Introduction In this chapter and in Chapters 7 and 8, we will be concerned with the kinetics of rigid bodies, i.e., relations between the forces acting on a rigid body, the shape and mass of the body, and the motion produced. Results of this chapter will be restricted to: - plane motion of rigid bodies, and - rigid bodies consisting of plane slabs or bodies which are symmetrical with respect to the reference plane. Our approach will be to consider rigid bodies as made of large numbers of particles and to use the results of Chapter 4 for the motion of systems of particles. Specifically, r r r &r F ma and M G H G D Alembert s principle is applied to prove that the external forces acting on a rigid body are equivalent a vector ma ṛ attached to the mass center and a couple of moment Iα 6-3 Equations of Motion for a Rigid ody Consider a rigid body acted upon by several external forces. Assume that the body is made of a large number of particles. For the motion of the mass center G of the body with respect to the Newtonian frame Oxyz, r F r ma For the motion of the body with respect to the centroidal frame Gx y z, r M G H &r G System of external forces is equipollent to the r system consisting of ma and H &r. G 6-4
3 Angular Momentum of a Rigid ody in Plane Motion Angular momentum of the slab may be computed by r n r H ( r v Δm ) G i i i n r [ ri ( ω ri ) Δmi ] i r ω ( ri Δmi ) r Iω After differentiation, r H &r G I &r ω Iα i Consider a rigid slab in plane motion. Results are also valid for plane motion of bodies which are symmetrical with respect to the reference plane. Results are not valid for asymmetrical bodies or three-dimensional motion. 6-5 Plane Motion of a Rigid ody: D Alembert s Principle Motion of a rigid body in plane motion is completely defined by the resultant and moment resultant about G of the external forces. F ma F ma M Iα x x y y G The external forces and the collective ective forces of the slab particles are equipollent (reduce to the same resultant and moment resultant) and equivalent (have the same ect on the body). d Alembert s Principle: The external forces acting on a rigid body are equivalent to the ective forces of the various particles forming the body. The most general motion of a rigid body that is symmetrical with respect to the reference plane can be replaced by the sum of a translation and a centroidal rotation
4 Axioms of the Mechanics of Rigid odies r r The forces F and F act at different points on a rigid body but but have the same magnitude, direction, and line of action. The forces produce the same moment about any point and are therefore, equipollent external forces. This proves the principle of transmissibility whereas it was previously stated as an axiom. 6-7 Problems Involving the Motion of a Rigid ody The fundamental relation between the forces acting on a rigid body in plane motion and the acceleration of its mass center and the angular acceleration of the body is illustrated in a free-body-diagram equation. The techniques for solving problems of static equilibrium may be applied to solve problems of plane motion by utilizing - d Alembert s principle, or - principle of dynamic equilibrium These techniques may also be applied to problems involving plane motion of connected rigid bodies by drawing a freebody-diagram equation for each body and solving the corresponding equations of motion simultaneously
5 Sample Problem 6. Calculate the acceleration during the skidding stop by assuming uniform acceleration. At a forward speed of 30 ft/s, the truck brakes were applied, causing the wheels to stop rotating. It was observed that the truck to skidded to a stop in 0 ft. Determine the magnitude of the normal reaction and the friction force at each wheel as the truck skidded to a stop. Draw the free-body-diagram equation expressing the equivalence of the external and ective forces. Apply the three corresponding scalar equations to solve for the unknown normal wheel forces at the front and rear and the coicient of friction between the wheels and road surface. 6-9 Sample Problem 6. ft v 30 x s 0 0ft Calculate the acceleration during the skidding stop by assuming uniform acceleration. v v 0 + a ft 0 30 s ( x x ) + a 0 ( 0ft) ft a.5 s Draw a free-body-diagram equation expressing the equivalence of the external and ective forces. Apply the corresponding scalar equations. F y ( F y ) N A + N W 0 F x ( F x ) F F μ ( N + N ) k A A ma μ W k a μk g ( W g) a
6 Sample Problem 6. Apply the corresponding scalar equations. M A ( M A ) N N ( 5ft) W + ( ft) N ( 4ft) W W a 5W + 4 a g g 0.650W N A W N W ( 0. W ) Nrear N A 350 N rear 0. 75W ( 0. W ) N front N V 650 N front 0. 35W ma ( 0.690)( 0. W ) Frear μk Nrear 75 ( 0.690)( 0. W ) F rear 0. W Ffront μk N front 35 F front W 6 - Sample Problem 6. The thin plate of mass 8 kg is held in place as shown. Neglecting the mass of the links, determine immediately after the wire has been cut (a) the acceleration of the plate, and (b) the force in each link. Note that after the wire is cut, all particles of the plate move along parallel circular paths of radius 50 mm. The plate is in curvilinear translation. Draw the free-body-diagram equation expressing the equivalence of the external and ective forces. Resolve into scalar component equations parallel and perpendicular to the path of the mass center. Solve the component equations and the moment equation for the unknown acceleration and link forces. 6-6
7 Sample Problem 6. Note that after the wire is cut, all particles of the plate move along parallel circular paths of radius 50 mm. The plate is in curvilinear translation. Draw the free-body-diagram equation expressing the equivalence of the external and ective forces. Resolve the diagram equation into components parallel and perpendicular to the path of the mass center. F t ( F t ) W cos30 ma mg cos30 ( 9.8m/s ) cos a 30 a 8.50m s 60 o 6-3 Sample Problem 6. a 8.50m s 60 o Solve the component equations and the moment equation for the unknown acceleration and link forces. M ( ) G M G ( FAE sin30 )( 50mm) ( FAE cos30 )( 00mm) ( F sin 30 )( 50mm) + ( F cos30 )( 00mm) 0 F DF DF 38.4 F F F F AE +.6 F 0.85F AE DF F n ( F n ) AE AE AE + F DF 0.85F W sin 30 0 AE ( 8kg)( 9.8m s ) FDF 0.85( 47.9 N) DF W sin 30 0 F AE F DF 47.9 N 8.70 N T C 6-4 7
8 Sample Problem 6.3 Determine the direction of rotation by evaluating the net moment on the pulley due to the two blocks. Relate the acceleration of the blocks to the angular acceleration of the pulley. A pulley weighing lb and having a radius of gyration of 8 in. is connected to two blocks as shown. Assuming no axle friction, determine the angular acceleration of the pulley and the acceleration of each block. Draw the free-body-diagram equation expressing the equivalence of the external and ective forces on the complete pulley plus blocks system. Solve the corresponding moment equation for the pulley angular acceleration. 6-5 Sample Problem 6.3 Determine the direction of rotation by evaluating the net moment on the pulley due to the two blocks. note: M G W I mk k g lb 3.ft s Relate the acceleration of the blocks to the angular acceleration of the pulley. a A r A α ( 0 ft)α ( 0lb)( 6in) ( 5lb)( 0in) 0in lb rotation is counterclockwise lb ft s 8 ft a r α ( 6 ft)α 6-6 8
9 Sample Problem 6.3 Draw the free-body-diagram equation expressing the equivalence of the external and ective forces on the complete pulley and blocks system. Solve the corresponding moment equation for the pulley angular acceleration. M G ( M G ) ( )( 6 ) ( )( 0 ) ( 6 ) ( 0 0lb ft 5lb ft Iα + m ) a ft m AaA ft ( )( 6 ) ()( 0) ( ) ( 0 )( 6 )( 6 ) ( 5 )( 0)( α + α ) I lb ft s a a A ( 0 α ) ft s ( 6 α ) ft s Then, α a A r A α a r ( 0 ft )(.374rad s ) ( 6 ft )(.374rad s ) α.374rad s a A.978ft s a.87ft s 6-7 Sample Problem 6.4 A cord is wrapped around a homogeneous disk of mass 5 kg. The cord is pulled upwards with a force T 80 N. Determine: (a) the acceleration of the center of the disk, (b) the angular acceleration of the disk, and (c) the acceleration of the cord. Draw the free-body-diagram equation expressing the equivalence of the external and ective forces on the disk. Solve the three corresponding scalar equilibrium equations for the horizontal, vertical, and angular accelerations of the disk. Determine the acceleration of the cord by evaluating the tangential acceleration of the point A on the disk
10 Sample Problem 6.4 Draw the free-body-diagram equation expressing the equivalence of the external and ective forces on the disk. Solve the three scalar equilibrium equations. F x ( F x ) 0 ma x a x 0 F y ( F y ) T W ma a y y T W 80 N - m M G ( M G ) Tr Iα T α mr ( mr ) ( 5kg)( 9.8m s ) 5kg α ( 80 N) ( 5kg)( 0.5m) a y.9m s α 48.0rad s 6-9 Sample Problem 6.4 Determine the acceleration of the cord by evaluating the tangential acceleration of the point A on the disk. a r cord ( aa) t a + ( aa G ) t.9m s ( 0.5m)( 48 rad s ) + s 6.m a cord a x 0 a y.9m s α 48.0rad s 6-0 0
11 Sample Problem 6.5 A uniform sphere of mass m and radius r is projected along a rough horizontal surface with a linear velocity v 0. The coicient of kinetic friction between the sphere and the surface is μ k. Determine: (a) the time t at which the sphere will start rolling without sliding, and (b) the linear and angular velocities of the sphere at time t. Draw the free-body-diagram equation expressing the equivalence of the external and ective forces on the sphere. Solve the three corresponding scalar equilibrium equations for the normal reaction from the surface and the linear and angular accelerations of the sphere. Apply the kinematic relations for uniformly accelerated motion to determine the time at which the tangential velocity of the sphere at the surface is zero, i.e., when the sphere stops sliding. 6 - Sample Problem 6.5 Draw the free-body-diagram equation expressing the equivalence of external and ective forces on the sphere. Solve the three scalar equilibrium equations. F y ( F y ) N W 0 F x ( F x ) F ma μ mg k M G ( M G ) Fr Iα ( μ mg) r ( mr )α k 3 N W mg a μ g k 5 μ α k g r NOTE: As long as the sphere both rotates and slides, its linear and angular motions are uniformly accelerated. 6 -
12 Sample Problem 6.5 a μk g 5 μ α k g r Apply the kinematic relations for uniformly accelerated motion to determine the time at which the tangential velocity of the sphere at the surface is zero, i.e., when the sphere stops sliding. v v + at v + ( μ g)t μk g ω ω0 + αt 0 + t r 5 μk g v0 μk gt r t r 5 μk g t r k At the instant t when the sphere stops sliding, v rω 5 μk g v r 7 μk g 0 ω 5 v r 7 r t v0 7 g μ k v 0 5 ω 7 r 0 5 v rω v v Constrained Plane Motion Most engineering applications involve rigid bodies which are moving under given constraints, e.g., cranks, connecting rods, and non-slipping wheels. Constrained plane motion: motions with definite relations between the components of acceleration of the mass center and the angular acceleration of the body. Solution of a problem involving constrained plane motion begins with a kinematic analysis. e.g., given θ, ω, and α, find P, N A, and N. - kinematic analysis yields a x and a y. - application of d Alembert s principle yields P, N A, and N. 6-4
13 Constrained Motion: Noncentroidal Rotation Noncentroidal rotation: motion of a body is constrained to rotate about a fixed axis that does not pass through its mass center. Kinematic relation between the motion of the mass center G and the motion of the body about G, a rα a rω t n The kinematic relations are used to eliminate a t and a n from equations derived from d Alembert s principle or from the method of dynamic equilibrium. 6-5 Constrained Plane Motion: Rolling Motion For a balanced disk constrained to roll without sliding, x rθ a rα Rolling, no sliding: F μs N a rα Rolling, sliding impending: F μs N a rα Rotating and sliding: F μk N a, rα independent For the geometric center of an unbalanced disk, a O rα The acceleration of the mass center, r r r ag ao + ag O r r r a + a + a O ( G O ) ( G O ) t n 6-6 3
14 Sample Problem 6.6 me 4 kg ke 85 mm mo 3 kg The portion AO of the mechanism is actuated by gear D and at the instant shown has a clockwise angular velocity of 8 rad/s and a counterclockwise angular acceleration of 40 rad/s. Determine: a) tangential force exerted by gear D, and b) components of the reaction at shaft O. Draw the free-body-equation for AO, expressing the equivalence of the external and ective forces. Evaluate the external forces due to the weights of gear E and arm O and the ective forces associated with the angular velocity and acceleration. Solve the three scalar equations derived from the free-body-equation for the tangential force at A and the horizontal and vertical components of reaction at shaft O. 6-7 Sample Problem 6.6 me 4 kg ke 85 mm mo 3 kg α 40rad s ω 8 rad/s Draw the free-body-equation for AO. Evaluate the external forces due to the weights of gear E and arm O and the ective forces. WE WO ( 4kg)( 9.8m s ) 39. N ( 3kg)( 9.8m s ) 9.4 N m k ( 4kg)( 0.085m) ( 40rad s ) IEα E Eα.56 N m m m I O O O ( a ) m ( rα ) ( 3kg)( 0.00m)( 40rad s ) O t O 4.0 N ( a ) m ( rω ) ( 3kg)( 0.00m)( 8rad s) O n O 38.4 N ( m L ) ( 3kg)( 0.400m) ( 40rad s ) α α O.600 N m 6-8 4
15 Sample Problem 6.6 Solve the three scalar equations derived from the freebody-equation for the tangential force at A and the horizontal and vertical components of reaction at O. F ( ) M O M O ( 0.0m) IEα + mo( ao ) t ( 0.00m) + IOα.56 N m + ( 4.0N)( 0.00m) +.600N m F 63.0 N W W E O 39. N 9.4 N I E α.56 N m m O( a O ) t 4.0 N ( ) 38.4 N m O a O I O α.600 N m n ( ) F x F x Rx mo O t y y ( a ) 4.0 N ( ) Fy F y R F W E W O m O ( a ) O R 63.0 N 39. N 9.4 N 38.4 N R x 4.0 N R y 4.0 N 6-9 Sample Problem 6.8 A sphere of weight W is released with no initial velocity and rolls without slipping on the incline. Determine: a) the minimum value of the coicient of friction, b) the velocity of G after the sphere has rolled 0 ft and c) the velocity of G if the sphere were to move 0 ft down a frictionless incline. Draw the free-body-equation for the sphere, expressing the equivalence of the external and ective forces. With the linear and angular accelerations related, solve the three scalar equations derived from the free-body-equation for the angular acceleration and the normal and tangential reactions at C. Calculate the friction coicient required for the indicated tangential reaction at C. Calculate the velocity after 0 ft of uniformly accelerated motion. Assuming no friction, calculate the linear acceleration down the incline and the corresponding velocity after 0 ft
16 Sample Problem 6.8 a rα Draw the free-body-equation for the sphere, expressing the equivalence of the external and ective forces. With the linear and angular accelerations related, solve the three scalar equations derived from the free-bodyequation for the angular acceleration and the normal and tangential reactions at C. M C ( M C ) ( W sinθ ) r ( ma ) r + Iα 5g sin30 a rα 7 ( mrα ) r + ( mr ) α W W rα r + r g 5 g ( s ) 5 3.ft 7 5 sin 30 α a.50ft s 5g sinθ α 7r 6-3 Sample Problem 6.8 5g sinθ α 7r a rα.50ft s Solve the three scalar equations derived from the freebody-equation for the angular acceleration and the normal and tangential reactions at C. F x ( F x ) W sinθ F ma W 5g sinθ g 7 F y ( F y ) N W cosθ 0 F W sin W 7 N W cos W Calculate the friction coicient required for the indicated tangential reaction at C. F μ N s F 0.43W μs N 0.866W μ s
17 Sample Problem 6.8 Calculate the velocity after 0 ft of uniformly accelerated motion. a( x 0 ) ( s )( 0ft) v v0 + x ft v r 5.7ft s 5g sinθ α 7r a rα.50ft s Assuming no friction, calculate the linear acceleration and the corresponding velocity after 0 ft. ( ) MG M G 0 Iα α 0 F x ( F x ) a( x 0 ) ( s )( 0ft) v v0 + x ft W W sinθ ma a g a ( 3.ft s ) sin30 6.ft s v r 7.94ft s 6-33 Sample Problem 6.9 A cord is wrapped around the inner hub of a wheel and pulled horizontally with a force of 00 N. The wheel has a mass of 50 kg and a radius of gyration of 70 mm. Knowing μ s 0.0 and μ k 0.5, determine the acceleration of G and the angular acceleration of the wheel. Draw the free-body-equation for the wheel, expressing the equivalence of the external and ective forces. Assuming rolling without slipping and therefore, related linear and angular accelerations, solve the scalar equations for the acceleration and the normal and tangential reactions at the ground. Compare the required tangential reaction to the maximum possible friction force. If slipping occurs, calculate the kinetic friction force and then solve the scalar equations for the linear and angular accelerations
18 Sample Problem 6.9 I mk 0.45kg m ( 50kg)( 0.70m) Assume rolling without slipping, a rα ( 0.00m)α Draw the free-body-equation for the wheel,. Assuming rolling without slipping, solve the scalar equations for the acceleration and ground reactions. M C ( M C ) 00N 0.040m ma 0.00m + Iα ( )( ) ( ) 8.0 N m α rad s a ( 50kg)( 0.00m) α + ( 0.45kg m ) ( 0.00m)( 0.74rad s ).074m s F x ( F x ) F + 00 N ma F 46.3N ( ) F x F x N W 0 N mg ( 50kg)(.074m s ) ( 50kg)(.074m s ) N α 6-35 Sample Problem 6.9 Compare the required tangential reaction to the maximum possible friction force. ( N) 98.N Fmax μsn 0.0 F > F max, rolling without slipping is impossible. Without slipping, F 46.3N N 490.5N Calculate the friction force with slipping and solve the scalar equations for linear and angular accelerations. ( N) 73.6 N F F μ N 0.5 F x ( F x ) k 00 N 73.6 N k ( 50kg)a a.53m s M G ( M G ) ( 73.6 N)( 0.00m) ( 00 N)( m) α ( 0.45kg m ) 8.94rad s α α 8.94rad s
19 Sample Problem 6.0 ased on the kinematics of the constrained motion, express the accelerations of A,, and G in terms of the angular acceleration. The extremities of a 4-ft rod weighing 50 lb can move freely and with no friction along two straight tracks. The rod is released with no velocity from the position shown. Determine: a) the angular acceleration of the rod, and b) the reactions at A and. Draw the free-body-equation for the rod, expressing the equivalence of the external and ective forces. Solve the three corresponding scalar equations for the angular acceleration and the reactions at A and Sample Problem 6.0 ased on the kinematics of the constrained motion, express the accelerations of A,, and G in terms of the angular acceleration. Express the acceleration of as r r r a a + a A A With a A 4α, the corresponding vector triangle and the law of signs yields a A 5.46α a 4. 90α The acceleration of G is now obtained from r r r r a a a + a where α G A G A a G A Resolving into x and y components, a 5.46α α cos α a x y α sin 60.73α
20 Sample Problem 6.0 Draw the free-body-equation for the rod, expressing the equivalence of the external and ective forces. I.07lb ft s Iα.07α ma ma x y ml 50lb 3.ft s ( 4ft) 50 ( 4.46α ) 6.93α (.73α ).69α 3. Solve the three corresponding scalar equations for the angular acceleration and the reactions at A and. M E ( M E ) ( 50)(.73) ( 6.93α )( 4.46) + (.69α )(.73) α +.30rad s F x ( F x ) R sin 45 ( 6.93)(.30) R.5lb F y ( F y ) + R A α.30rad s R r.5lb (.5) cos45 50 (.69)(.30) R A 7.9lb +.07α 45 o
PHYSICS 111 HOMEWORK SOLUTION #10. April 8, 2013
PHYSICS HOMEWORK SOLUTION #0 April 8, 203 0. Find the net torque on the wheel in the figure below about the axle through O, taking a = 6.0 cm and b = 30.0 cm. A torque that s produced by a force can be
More informationProblem Set 1. Ans: a = 1.74 m/s 2, t = 4.80 s
Problem Set 1 1.1 A bicyclist starts from rest and after traveling along a straight path a distance of 20 m reaches a speed of 30 km/h. Determine her constant acceleration. How long does it take her to
More informationChapter 4. Forces and Newton s Laws of Motion. continued
Chapter 4 Forces and Newton s Laws of Motion continued 4.9 Static and Kinetic Frictional Forces When an object is in contact with a surface forces can act on the objects. The component of this force acting
More informationC B A T 3 T 2 T 1. 1. What is the magnitude of the force T 1? A) 37.5 N B) 75.0 N C) 113 N D) 157 N E) 192 N
Three boxes are connected by massless strings and are resting on a frictionless table. Each box has a mass of 15 kg, and the tension T 1 in the right string is accelerating the boxes to the right at a
More informationPHY231 Section 2, Form A March 22, 2012. 1. Which one of the following statements concerning kinetic energy is true?
1. Which one of the following statements concerning kinetic energy is true? A) Kinetic energy can be measured in watts. B) Kinetic energy is always equal to the potential energy. C) Kinetic energy is always
More informationPhysics 201 Homework 8
Physics 201 Homework 8 Feb 27, 2013 1. A ceiling fan is turned on and a net torque of 1.8 N-m is applied to the blades. 8.2 rad/s 2 The blades have a total moment of inertia of 0.22 kg-m 2. What is the
More informationPHY231 Section 1, Form B March 22, 2012
1. A car enters a horizontal, curved roadbed of radius 50 m. The coefficient of static friction between the tires and the roadbed is 0.20. What is the maximum speed with which the car can safely negotiate
More informationPhysics 2A, Sec B00: Mechanics -- Winter 2011 Instructor: B. Grinstein Final Exam
Physics 2A, Sec B00: Mechanics -- Winter 2011 Instructor: B. Grinstein Final Exam INSTRUCTIONS: Use a pencil #2 to fill your scantron. Write your code number and bubble it in under "EXAM NUMBER;" an entry
More informationPhysics 1A Lecture 10C
Physics 1A Lecture 10C "If you neglect to recharge a battery, it dies. And if you run full speed ahead without stopping for water, you lose momentum to finish the race. --Oprah Winfrey Static Equilibrium
More informationCenter of Gravity. We touched on this briefly in chapter 7! x 2
Center of Gravity We touched on this briefly in chapter 7! x 1 x 2 cm m 1 m 2 This was for what is known as discrete objects. Discrete refers to the fact that the two objects separated and individual.
More informationChapter 10 Rotational Motion. Copyright 2009 Pearson Education, Inc.
Chapter 10 Rotational Motion Angular Quantities Units of Chapter 10 Vector Nature of Angular Quantities Constant Angular Acceleration Torque Rotational Dynamics; Torque and Rotational Inertia Solving Problems
More informationPHY121 #8 Midterm I 3.06.2013
PHY11 #8 Midterm I 3.06.013 AP Physics- Newton s Laws AP Exam Multiple Choice Questions #1 #4 1. When the frictionless system shown above is accelerated by an applied force of magnitude F, the tension
More informationAngular acceleration α
Angular Acceleration Angular acceleration α measures how rapidly the angular velocity is changing: Slide 7-0 Linear and Circular Motion Compared Slide 7- Linear and Circular Kinematics Compared Slide 7-
More informationProblem 6.40 and 6.41 Kleppner and Kolenkow Notes by: Rishikesh Vaidya, Physics Group, BITS-Pilani
Problem 6.40 and 6.4 Kleppner and Kolenkow Notes by: Rishikesh Vaidya, Physics Group, BITS-Pilani 6.40 A wheel with fine teeth is attached to the end of a spring with constant k and unstretched length
More informationSerway_ISM_V1 1 Chapter 4
Serway_ISM_V1 1 Chapter 4 ANSWERS TO MULTIPLE CHOICE QUESTIONS 1. Newton s second law gives the net force acting on the crate as This gives the kinetic friction force as, so choice (a) is correct. 2. As
More informationVELOCITY, ACCELERATION, FORCE
VELOCITY, ACCELERATION, FORCE velocity Velocity v is a vector, with units of meters per second ( m s ). Velocity indicates the rate of change of the object s position ( r ); i.e., velocity tells you how
More informationPHYSICS 111 HOMEWORK SOLUTION #9. April 5, 2013
PHYSICS 111 HOMEWORK SOLUTION #9 April 5, 2013 0.1 A potter s wheel moves uniformly from rest to an angular speed of 0.16 rev/s in 33 s. Find its angular acceleration in radians per second per second.
More informationTHEORETICAL MECHANICS
PROF. DR. ING. VASILE SZOLGA THEORETICAL MECHANICS LECTURE NOTES AND SAMPLE PROBLEMS PART ONE STATICS OF THE PARTICLE, OF THE RIGID BODY AND OF THE SYSTEMS OF BODIES KINEMATICS OF THE PARTICLE 2010 0 Contents
More informationLecture L22-2D Rigid Body Dynamics: Work and Energy
J. Peraire, S. Widnall 6.07 Dynamics Fall 008 Version.0 Lecture L - D Rigid Body Dynamics: Work and Energy In this lecture, we will revisit the principle of work and energy introduced in lecture L-3 for
More informationKINEMATICS OF PARTICLES RELATIVE MOTION WITH RESPECT TO TRANSLATING AXES
KINEMTICS OF PRTICLES RELTIVE MOTION WITH RESPECT TO TRNSLTING XES In the previous articles, we have described particle motion using coordinates with respect to fixed reference axes. The displacements,
More informationSolution Derivations for Capa #11
Solution Derivations for Capa #11 1) A horizontal circular platform (M = 128.1 kg, r = 3.11 m) rotates about a frictionless vertical axle. A student (m = 68.3 kg) walks slowly from the rim of the platform
More informationLab 7: Rotational Motion
Lab 7: Rotational Motion Equipment: DataStudio, rotary motion sensor mounted on 80 cm rod and heavy duty bench clamp (PASCO ME-9472), string with loop at one end and small white bead at the other end (125
More informationwww.mathsbox.org.uk Displacement (x) Velocity (v) Acceleration (a) x = f(t) differentiate v = dx Acceleration Velocity (v) Displacement x
Mechanics 2 : Revision Notes 1. Kinematics and variable acceleration Displacement (x) Velocity (v) Acceleration (a) x = f(t) differentiate v = dx differentiate a = dv = d2 x dt dt dt 2 Acceleration Velocity
More informationPractice Exam Three Solutions
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics Physics 8.01T Fall Term 2004 Practice Exam Three Solutions Problem 1a) (5 points) Collisions and Center of Mass Reference Frame In the lab frame,
More informationPHYS 211 FINAL FALL 2004 Form A
1. Two boys with masses of 40 kg and 60 kg are holding onto either end of a 10 m long massless pole which is initially at rest and floating in still water. They pull themselves along the pole toward each
More informationLecture 07: Work and Kinetic Energy. Physics 2210 Fall Semester 2014
Lecture 07: Work and Kinetic Energy Physics 2210 Fall Semester 2014 Announcements Schedule next few weeks: 9/08 Unit 3 9/10 Unit 4 9/15 Unit 5 (guest lecturer) 9/17 Unit 6 (guest lecturer) 9/22 Unit 7,
More information3600 s 1 h. 24 h 1 day. 1 day
Week 7 homework IMPORTANT NOTE ABOUT WEBASSIGN: In the WebAssign versions of these problems, various details have been changed, so that the answers will come out differently. The method to find the solution
More informationCHAPTER 15 FORCE, MASS AND ACCELERATION
CHAPTER 5 FORCE, MASS AND ACCELERATION EXERCISE 83, Page 9. A car initially at rest accelerates uniformly to a speed of 55 km/h in 4 s. Determine the accelerating force required if the mass of the car
More informationRotational Inertia Demonstrator
WWW.ARBORSCI.COM Rotational Inertia Demonstrator P3-3545 BACKGROUND: The Rotational Inertia Demonstrator provides an engaging way to investigate many of the principles of angular motion and is intended
More information11. Rotation Translational Motion: Rotational Motion:
11. Rotation Translational Motion: Motion of the center of mass of an object from one position to another. All the motion discussed so far belongs to this category, except uniform circular motion. Rotational
More informationMechanical Principles
Unit 4: Mechanical Principles Unit code: F/601/1450 QCF level: 5 Credit value: 15 OUTCOME 4 POWER TRANSMISSION TUTORIAL 2 BALANCING 4. Dynamics of rotating systems Single and multi-link mechanisms: slider
More informationChapter 11 Equilibrium
11.1 The First Condition of Equilibrium The first condition of equilibrium deals with the forces that cause possible translations of a body. The simplest way to define the translational equilibrium of
More informationF f v 1 = c100(10 3 ) m h da 1h 3600 s b =
14 11. The 2-Mg car has a velocity of v 1 = 100km>h when the v 1 100 km/h driver sees an obstacle in front of the car. It takes 0.75 s for him to react and lock the brakes, causing the car to skid. If
More informationTOP VIEW. FBD s TOP VIEW. Examination No. 2 PROBLEM NO. 1. Given:
RLEM N. 1 Given: Find: vehicle having a mass of 500 kg is traveling on a banked track on a path with a constant radius of R = 1000 meters. t the instant showing, the vehicle is traveling with a speed of
More informationPhys222 Winter 2012 Quiz 4 Chapters 29-31. Name
Name If you think that no correct answer is provided, give your answer, state your reasoning briefly; append additional sheet of paper if necessary. 1. A particle (q = 5.0 nc, m = 3.0 µg) moves in a region
More informationChapter 5 Using Newton s Laws: Friction, Circular Motion, Drag Forces. Copyright 2009 Pearson Education, Inc.
Chapter 5 Using Newton s Laws: Friction, Circular Motion, Drag Forces Units of Chapter 5 Applications of Newton s Laws Involving Friction Uniform Circular Motion Kinematics Dynamics of Uniform Circular
More informationUnit 4 Practice Test: Rotational Motion
Unit 4 Practice Test: Rotational Motion Multiple Guess Identify the letter of the choice that best completes the statement or answers the question. 1. How would an angle in radians be converted to an angle
More informationHW Set VI page 1 of 9 PHYSICS 1401 (1) homework solutions
HW Set VI page 1 of 9 10-30 A 10 g bullet moving directly upward at 1000 m/s strikes and passes through the center of mass of a 5.0 kg block initially at rest (Fig. 10-33 ). The bullet emerges from the
More informationLinear Motion vs. Rotational Motion
Linear Motion vs. Rotational Motion Linear motion involves an object moving from one point to another in a straight line. Rotational motion involves an object rotating about an axis. Examples include a
More information4.2 Free Body Diagrams
CE297-FA09-Ch4 Page 1 Friday, September 18, 2009 12:11 AM Chapter 4: Equilibrium of Rigid Bodies A (rigid) body is said to in equilibrium if the vector sum of ALL forces and all their moments taken about
More informationMidterm Solutions. mvr = ω f (I wheel + I bullet ) = ω f 2 MR2 + mr 2 ) ω f = v R. 1 + M 2m
Midterm Solutions I) A bullet of mass m moving at horizontal velocity v strikes and sticks to the rim of a wheel a solid disc) of mass M, radius R, anchored at its center but free to rotate i) Which of
More informationChapter 8: Rotational Motion of Solid Objects
Chapter 8: Rotational Motion of Solid Objects 1. An isolated object is initially spinning at a constant speed. Then, although no external forces act upon it, its rotational speed increases. This must be
More informationDynamics of Rotational Motion
Chapter 10 Dynamics of Rotational Motion PowerPoint Lectures for University Physics, Twelfth Edition Hugh D. Young and Roger A. Freedman Lectures by James Pazun Modified by P. Lam 5_31_2012 Goals for Chapter
More informationAwell-known lecture demonstration1
Acceleration of a Pulled Spool Carl E. Mungan, Physics Department, U.S. Naval Academy, Annapolis, MD 40-506; mungan@usna.edu Awell-known lecture demonstration consists of pulling a spool by the free end
More informationTwo-Body System: Two Hanging Masses
Specific Outcome: i. I can apply Newton s laws of motion to solve, algebraically, linear motion problems in horizontal, vertical and inclined planes near the surface of Earth, ignoring air resistance.
More informationChapter 11. h = 5m. = mgh + 1 2 mv 2 + 1 2 Iω 2. E f. = E i. v = 4 3 g(h h) = 4 3 9.8m / s2 (8m 5m) = 6.26m / s. ω = v r = 6.
Chapter 11 11.7 A solid cylinder of radius 10cm and mass 1kg starts from rest and rolls without slipping a distance of 6m down a house roof that is inclined at 30 degrees (a) What is the angular speed
More informationPhysics 11 Assignment KEY Dynamics Chapters 4 & 5
Physics Assignment KEY Dynamics Chapters 4 & 5 ote: for all dynamics problem-solving questions, draw appropriate free body diagrams and use the aforementioned problem-solving method.. Define the following
More informationStructural Axial, Shear and Bending Moments
Structural Axial, Shear and Bending Moments Positive Internal Forces Acting Recall from mechanics of materials that the internal forces P (generic axial), V (shear) and M (moment) represent resultants
More informationObjective: Equilibrium Applications of Newton s Laws of Motion I
Type: Single Date: Objective: Equilibrium Applications of Newton s Laws of Motion I Homework: Assignment (1-11) Read (4.1-4.5, 4.8, 4.11); Do PROB # s (46, 47, 52, 58) Ch. 4 AP Physics B Mr. Mirro Equilibrium,
More informationSOLID MECHANICS TUTORIAL MECHANISMS KINEMATICS - VELOCITY AND ACCELERATION DIAGRAMS
SOLID MECHANICS TUTORIAL MECHANISMS KINEMATICS - VELOCITY AND ACCELERATION DIAGRAMS This work covers elements of the syllabus for the Engineering Council exams C105 Mechanical and Structural Engineering
More informationNewton s Law of Motion
chapter 5 Newton s Law of Motion Static system 1. Hanging two identical masses Context in the textbook: Section 5.3, combination of forces, Example 4. Vertical motion without friction 2. Elevator: Decelerating
More informationSample Questions for the AP Physics 1 Exam
Sample Questions for the AP Physics 1 Exam Sample Questions for the AP Physics 1 Exam Multiple-choice Questions Note: To simplify calculations, you may use g 5 10 m/s 2 in all problems. Directions: Each
More informationChapter 3.8 & 6 Solutions
Chapter 3.8 & 6 Solutions P3.37. Prepare: We are asked to find period, speed and acceleration. Period and frequency are inverses according to Equation 3.26. To find speed we need to know the distance traveled
More informationTIME OF COMPLETION NAME SOLUTION DEPARTMENT OF NATURAL SCIENCES. PHYS 1111, Exam 3 Section 1 Version 1 December 6, 2005 Total Weight: 100 points
TIME OF COMPLETION NAME SOLUTION DEPARTMENT OF NATURAL SCIENCES PHYS 1111, Exam 3 Section 1 Version 1 December 6, 2005 Total Weight: 100 points 1. Check your examination for completeness prior to starting.
More information1. Units of a magnetic field might be: A. C m/s B. C s/m C. C/kg D. kg/c s E. N/C m ans: D
Chapter 28: MAGNETIC FIELDS 1 Units of a magnetic field might be: A C m/s B C s/m C C/kg D kg/c s E N/C m 2 In the formula F = q v B: A F must be perpendicular to v but not necessarily to B B F must be
More informationChapter 18 Static Equilibrium
Chapter 8 Static Equilibrium 8. Introduction Static Equilibrium... 8. Lever Law... Example 8. Lever Law... 4 8.3 Generalized Lever Law... 5 8.4 Worked Examples... 7 Example 8. Suspended Rod... 7 Example
More informationLecture 17. Last time we saw that the rotational analog of Newton s 2nd Law is
Lecture 17 Rotational Dynamics Rotational Kinetic Energy Stress and Strain and Springs Cutnell+Johnson: 9.4-9.6, 10.1-10.2 Rotational Dynamics (some more) Last time we saw that the rotational analog of
More informationPhysics 111: Lecture 4: Chapter 4 - Forces and Newton s Laws of Motion. Physics is about forces and how the world around us reacts to these forces.
Physics 111: Lecture 4: Chapter 4 - Forces and Newton s Laws of Motion Physics is about forces and how the world around us reacts to these forces. Whats a force? Contact and non-contact forces. Whats a
More informationAP Physics Circular Motion Practice Test B,B,B,A,D,D,C,B,D,B,E,E,E, 14. 6.6m/s, 0.4 N, 1.5 m, 6.3m/s, 15. 12.9 m/s, 22.9 m/s
AP Physics Circular Motion Practice Test B,B,B,A,D,D,C,B,D,B,E,E,E, 14. 6.6m/s, 0.4 N, 1.5 m, 6.3m/s, 15. 12.9 m/s, 22.9 m/s Answer the multiple choice questions (2 Points Each) on this sheet with capital
More informationv v ax v a x a v a v = = = Since F = ma, it follows that a = F/m. The mass of the arrow is unchanged, and ( )
Week 3 homework IMPORTANT NOTE ABOUT WEBASSIGN: In the WebAssign versions of these problems, various details have been changed, so that the answers will come out differently. The method to find the solution
More informationE X P E R I M E N T 8
E X P E R I M E N T 8 Torque, Equilibrium & Center of Gravity Produced by the Physics Staff at Collin College Copyright Collin College Physics Department. All Rights Reserved. University Physics, Exp 8:
More informationD Alembert s principle and applications
Chapter 1 D Alembert s principle and applications 1.1 D Alembert s principle The principle of virtual work states that the sum of the incremental virtual works done by all external forces F i acting in
More informationFric-3. force F k and the equation (4.2) may be used. The sense of F k is opposite
4. FRICTION 4.1 Laws of friction. We know from experience that when two bodies tend to slide on each other a resisting force appears at their surface of contact which opposes their relative motion. The
More informationEXPERIMENT: MOMENT OF INERTIA
OBJECTIVES EXPERIMENT: MOMENT OF INERTIA to familiarize yourself with the concept of moment of inertia, I, which plays the same role in the description of the rotation of a rigid body as mass plays in
More informationTorque and Rotary Motion
Torque and Rotary Motion Name Partner Introduction Motion in a circle is a straight-forward extension of linear motion. According to the textbook, all you have to do is replace displacement, velocity,
More informationLecture L2 - Degrees of Freedom and Constraints, Rectilinear Motion
S. Widnall 6.07 Dynamics Fall 009 Version.0 Lecture L - Degrees of Freedom and Constraints, Rectilinear Motion Degrees of Freedom Degrees of freedom refers to the number of independent spatial coordinates
More informationChapter 21 Rigid Body Dynamics: Rotation and Translation about a Fixed Axis
Chapter 21 Rigid Body Dynamics: Rotation and Translation about a Fixed Axis 21.1 Introduction... 1 21.2 Translational Equation of Motion... 1 21.3 Translational and Rotational Equations of Motion... 1
More information8.012 Physics I: Classical Mechanics Fall 2008
MIT OpenCourseWare http://ocw.mit.edu 8.012 Physics I: Classical Mechanics Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. MASSACHUSETTS INSTITUTE
More informationAP Physics: Rotational Dynamics 2
Name: Assignment Due Date: March 30, 2012 AP Physics: Rotational Dynamics 2 Problem A solid cylinder with mass M, radius R, and rotational inertia 1 2 MR2 rolls without slipping down the inclined plane
More informationTEACHER ANSWER KEY November 12, 2003. Phys - Vectors 11-13-2003
Phys - Vectors 11-13-2003 TEACHER ANSWER KEY November 12, 2003 5 1. A 1.5-kilogram lab cart is accelerated uniformly from rest to a speed of 2.0 meters per second in 0.50 second. What is the magnitude
More informationMECHANICAL PRINCIPLES OUTCOME 4 MECHANICAL POWER TRANSMISSION TUTORIAL 1 SIMPLE MACHINES
MECHANICAL PRINCIPLES OUTCOME 4 MECHANICAL POWER TRANSMISSION TUTORIAL 1 SIMPLE MACHINES Simple machines: lifting devices e.g. lever systems, inclined plane, screw jack, pulley blocks, Weston differential
More informationB Answer: neither of these. Mass A is accelerating, so the net force on A must be non-zero Likewise for mass B.
CTA-1. An Atwood's machine is a pulley with two masses connected by a string as shown. The mass of object A, m A, is twice the mass of object B, m B. The tension T in the string on the left, above mass
More informationCopyright 2011 Casa Software Ltd. www.casaxps.com. Centre of Mass
Centre of Mass A central theme in mathematical modelling is that of reducing complex problems to simpler, and hopefully, equivalent problems for which mathematical analysis is possible. The concept of
More informationKyu-Jung Kim Mechanical Engineering Department, California State Polytechnic University, Pomona, U.S.A.
MECHANICS: STATICS AND DYNAMICS Kyu-Jung Kim Mechanical Engineering Department, California State Polytechnic University, Pomona, U.S.A. Keywords: mechanics, statics, dynamics, equilibrium, kinematics,
More informationRotation: Moment of Inertia and Torque
Rotation: Moment of Inertia and Torque Every time we push a door open or tighten a bolt using a wrench, we apply a force that results in a rotational motion about a fixed axis. Through experience we learn
More informationLecture L5 - Other Coordinate Systems
S. Widnall, J. Peraire 16.07 Dynamics Fall 008 Version.0 Lecture L5 - Other Coordinate Systems In this lecture, we will look at some other common systems of coordinates. We will present polar coordinates
More informationPhysics 1120: Simple Harmonic Motion Solutions
Questions: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Physics 1120: Simple Harmonic Motion Solutions 1. A 1.75 kg particle moves as function of time as follows: x = 4cos(1.33t+π/5) where distance is measured
More informationPhysics: Principles and Applications, 6e Giancoli Chapter 4 Dynamics: Newton's Laws of Motion
Physics: Principles and Applications, 6e Giancoli Chapter 4 Dynamics: Newton's Laws of Motion Conceptual Questions 1) Which of Newton's laws best explains why motorists should buckle-up? A) the first law
More informationF N A) 330 N 0.31 B) 310 N 0.33 C) 250 N 0.27 D) 290 N 0.30 E) 370 N 0.26
Physics 23 Exam 2 Spring 2010 Dr. Alward Page 1 1. A 250-N force is directed horizontally as shown to push a 29-kg box up an inclined plane at a constant speed. Determine the magnitude of the normal force,
More informationAcceleration due to Gravity
Acceleration due to Gravity 1 Object To determine the acceleration due to gravity by different methods. 2 Apparatus Balance, ball bearing, clamps, electric timers, meter stick, paper strips, precision
More informationRecitation Week 4 Chapter 5
Recitation Week 4 Chapter 5 Problem 5.5. A bag of cement whose weight is hangs in equilibrium from three wires shown in igure P5.4. wo of the wires make angles θ = 60.0 and θ = 40.0 with the horizontal.
More informationLecture 6. Weight. Tension. Normal Force. Static Friction. Cutnell+Johnson: 4.8-4.12, second half of section 4.7
Lecture 6 Weight Tension Normal Force Static Friction Cutnell+Johnson: 4.8-4.12, second half of section 4.7 In this lecture, I m going to discuss four different kinds of forces: weight, tension, the normal
More informationMechanics lecture 7 Moment of a force, torque, equilibrium of a body
G.1 EE1.el3 (EEE1023): Electronics III Mechanics lecture 7 Moment of a force, torque, equilibrium of a body Dr Philip Jackson http://www.ee.surrey.ac.uk/teaching/courses/ee1.el3/ G.2 Moments, torque and
More informationMechanics 1: Conservation of Energy and Momentum
Mechanics : Conservation of Energy and Momentum If a certain quantity associated with a system does not change in time. We say that it is conserved, and the system possesses a conservation law. Conservation
More informationPhysics 9e/Cutnell. correlated to the. College Board AP Physics 1 Course Objectives
Physics 9e/Cutnell correlated to the College Board AP Physics 1 Course Objectives Big Idea 1: Objects and systems have properties such as mass and charge. Systems may have internal structure. Enduring
More informationEDEXCEL NATIONAL CERTIFICATE/DIPLOMA MECHANICAL PRINCIPLES AND APPLICATIONS NQF LEVEL 3 OUTCOME 1 - LOADING SYSTEMS
EDEXCEL NATIONAL CERTIFICATE/DIPLOMA MECHANICAL PRINCIPLES AND APPLICATIONS NQF LEVEL 3 OUTCOME 1 - LOADING SYSTEMS TUTORIAL 1 NON-CONCURRENT COPLANAR FORCE SYSTEMS 1. Be able to determine the effects
More informationSOLID MECHANICS BALANCING TUTORIAL BALANCING OF ROTATING BODIES
SOLID MECHANICS BALANCING TUTORIAL BALANCING OF ROTATING BODIES This work covers elements of the syllabus for the Edexcel module 21722P HNC/D Mechanical Principles OUTCOME 4. On completion of this tutorial
More informationLecture L6 - Intrinsic Coordinates
S. Widnall, J. Peraire 16.07 Dynamics Fall 2009 Version 2.0 Lecture L6 - Intrinsic Coordinates In lecture L4, we introduced the position, velocity and acceleration vectors and referred them to a fixed
More informationLecture 16. Newton s Second Law for Rotation. Moment of Inertia. Angular momentum. Cutnell+Johnson: 9.4, 9.6
Lecture 16 Newton s Second Law for Rotation Moment of Inertia Angular momentum Cutnell+Johnson: 9.4, 9.6 Newton s Second Law for Rotation Newton s second law says how a net force causes an acceleration.
More informationAnswer, Key { Homework 6 { Rubin H Landau 1 This print-out should have 24 questions. Check that it is complete before leaving the printer. Also, multiple-choice questions may continue on the next column
More informationAP Physics - Chapter 8 Practice Test
AP Physics - Chapter 8 Practice Test Multiple Choice Identify the choice that best completes the statement or answers the question. 1. A single conservative force F x = (6.0x 12) N (x is in m) acts on
More informationSTATICS. Introduction VECTOR MECHANICS FOR ENGINEERS: Eighth Edition CHAPTER. Ferdinand P. Beer E. Russell Johnston, Jr.
Eighth E CHAPTER VECTOR MECHANICS FOR ENGINEERS: STATICS Ferdinand P. Beer E. Russell Johnston, Jr. Introduction Lecture Notes: J. Walt Oler Texas Tech University Contents What is Mechanics? Fundamental
More information5. Forces and Motion-I. Force is an interaction that causes the acceleration of a body. A vector quantity.
5. Forces and Motion-I 1 Force is an interaction that causes the acceleration of a body. A vector quantity. Newton's First Law: Consider a body on which no net force acts. If the body is at rest, it will
More informationTorque Analyses of a Sliding Ladder
Torque Analyses of a Sliding Ladder 1 Problem Kirk T. McDonald Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544 (May 6, 2007) The problem of a ladder that slides without friction while
More information226 Chapter 15: OSCILLATIONS
Chapter 15: OSCILLATIONS 1. In simple harmonic motion, the restoring force must be proportional to the: A. amplitude B. frequency C. velocity D. displacement E. displacement squared 2. An oscillatory motion
More informationAP1 Oscillations. 1. Which of the following statements about a spring-block oscillator in simple harmonic motion about its equilibrium point is false?
1. Which of the following statements about a spring-block oscillator in simple harmonic motion about its equilibrium point is false? (A) The displacement is directly related to the acceleration. (B) The
More information3 Work, Power and Energy
3 Work, Power and Energy At the end of this section you should be able to: a. describe potential energy as energy due to position and derive potential energy as mgh b. describe kinetic energy as energy
More informationAPPLIED MATHEMATICS ADVANCED LEVEL
APPLIED MATHEMATICS ADVANCED LEVEL INTRODUCTION This syllabus serves to examine candidates knowledge and skills in introductory mathematical and statistical methods, and their applications. For applications
More informationSHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Exam Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 1) A person on a sled coasts down a hill and then goes over a slight rise with speed 2.7 m/s.
More informationFigure 1.1 Vector A and Vector F
CHAPTER I VECTOR QUANTITIES Quantities are anything which can be measured, and stated with number. Quantities in physics are divided into two types; scalar and vector quantities. Scalar quantities have
More information