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# 503 Review Sheet for ME 27400 at Purdue

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Fundamentals Exam review material l ME2 74 ME 274 Basic Mechanics ll Dynamics Fundamentals Mathematics Vectors Point Kinematics and Free Body Diagrams 1 Scalar kinematics a Differentiation b Integration c Sample problems 2 Vector operations a Addition and subtraction b Dot scalar products c Cross vector products d Moments about points e Projections of vectors f Writing a vector in terms of a new set of unit vectors g Sample problems 3 Vector kinematics a Cartesian components b Path components c Polar components d Sample problems 4 Free body diagrams 5 Solutions of sample problems provided at the end of the notes Fundamentals Exam review material 2 ME2 74 Scalar kinematics differentiation Suppose that a point P travels on a straight line path v gt rectilinear motion whose position on the path is O iven by the distance variable s The speed V of P is g gt given by v dsdt S Case 1 On this path we are given the speed V of point P in terms of time t v vt From this we want to determine the acceleration of the point The acceleration can be found by directly differentiating v with respect to t that is dv a dt Case 2 On this path we are given the speed v of point P in terms of its position 3 v vs Here we need to employ the chain rule of differentiation see below Chain rule of di erentiation Suppose y yx where x x The derivative of y with respect to t can be found from the chain rule of differentiation to be 2611 dt dx dt Using the chain rule of differentiation we see that dv dv ds dv a v dt ds dt ds where v dsdt Fundamentals Exam review material 3 ME2 74 Scalar kinematics integration Again suppose that a point P travels on a straight line given by the distance variable s The speed V of P is v gt path rectilinear motion whose position on the path is 0 given by v dsdt S gt Case 1 On this path we are given the acceleration of point P in terms of time t a at From this we want to determine the speed of the point at some instant in time t The acceleration can be found by directly integrating a with respect to t that is a s jatdtidv s vtv0jatdt dt 0 120 Case 2 On this path we are given the acceleration a of point P in terms of its position 3 v vs Here prior to using separation of variables we need to use the chain rule of differentiation to produce dv dv ds dv a V dt ds dt ds where v dsdt Now we can write V S vdvads gt J vdvJads gt Ev Ev 0Jads vo 0 0 Fundamentals Exam review material 4 ME2 74 Scalar kinematics sample problems Example 1 Suppose that the speed ofP is given by vt 3t2 Find the acceleration of P Example 2 Suppose that the speed ofP is given by vs 5sin3s Find the acceleration of P Example 3 Suppose that the acceleration of P traveling on a straightline path is given by at 6 sin2t with v0 4 Find the speed ofP as a function oftime Example 4 Suppose that the acceleration of P traveling on a straightline path is given by as 8S3 with vs 0 5 Find the speed ofP as a function ofposition s Fundamentals Exam review material 5 ME2 74 Vector operations Addition The addition of two vectors Q and Q is found through the parallelogram rule move the tail of b to the head of Q with Q b being the vector extending from the tail of Q to the new position of the head of b as shown in the figure to the right If Q and b are given in terms of sets of xyz components as then the components of the vector Q b are found from the scalar sum of the components of Q and b Ql2axbxgayby1azbzk S ubtruction The subtraction of two vectors Q and b is found through the parallelogram rule on Q and Q move the tail of b to the head of a with Q 2 being the vector extending from the tail of Q to the new position of the head of b as shown in the figure to the right If Q and b are given in terms of sets of xyz components as then the components of the vector Q b are found from the scalar difference of the components of Q and b Q llaX bxiay by1a bzg Fundamentals Exam review material 6 ME2 74 Dot scalar product The dot product of two vectors Q and b is the following SCALAR Ll l7LZICOS9 9 where 6 is the smallest angle between vectors g and b If g and b are given in terms of sets of xyz components as gax aylazlg llbxgby1bzk then the scalar quantity g o 7 is found byl 213 axgmyzngbxgby1bzg mbxzbyzbzkayzbxzbyzbzkazkbxibyzbzk k axbe39Zdxbyi 1axbzi aybx139Zaybyl laybz1 k aszk azbyk 1azbzk axbx 1 axby0 axbz 0 aybx0ayby1aybz0 asz0azby0azbz1 axbx ayby azbz HINT In finding the dot product you do not need to go through all of above steps simply recall that the dot product of g and b is found by the sum of the products of the like components of Q and b 1 From the above dot product definition note that gquot o gquot j 0 j k 0 k 1 and Z39jj kk ij ik ji k0 Fmdnmmml Exam review manor 7 mm Cmss vector product a X b A The cross product of two vectors g and b is given by a VECTOR whose magnitude is given by C W1 0f gxll gH sin9 ngers and whose direoion is found through the right hand rulequot sweep the fingers ofyour right hand FROMETD b throughthe sma11estang1e between yeoors L1 and h and your right thumb points in the direction of g X b Ifg and b are given interms ofsets ofxyz components as iuyluzg Wu 7 azbyia1bz a azbtza1by 7 am x 1K le by oz by by bZ 2 From the above cross product de nition note that i zxie71xesexz mam Fundamentals Exam review material 8 ME2 74 Moments about points Recall that the moment about a point 0 due to a force E acting at point P is found through the cross product w Mo KPo XE where rPO is the position vector from O to P Vector projections The projection of vector k onto the line of action of a second b vector 2 is found by placing vectors g and b tail to tail and drawing a line from the head of b that is perpendicular to g as shown in the figure to the right From this figure we see that this projection is given by a quot projection of l onto Ltquot cos9 projection of g onto the line of action of g Say we determine a unit vector ea g I gl where lgal 1 that points in the same direction as g Using the above we see that quotprojection of 7 onto Ltquot lgalcos9 7 o g Fundamentals Exam review material 9 ME2 74 Writing a vector in terms ofu new set of unit vectors Suppose that you are given a vector Q in terms of its XY components unit vectors 1 and 1 IK gaxlayl ax and ay areKNOWN e and you desire to know the same vector 2 in terms of its xy components unit vectors 139 and j g axiay1 ax and ay are UNKNOWN where the coordinate axes are shown in the figure above One way to do this is to first find i and j in terms of their 1 and 1 components see projections in the above figure gquot cos9 sin91 139 sin9cos9 Now we can use dot products to determine 0LC and ay through a Q39E X ayl aXaylo cos9sin91 aX cos9ay sine and y 3 axlayl 1 aXaylo sin9cos91 aX sin9ay cos9 Fundamentals Exam review material 1 0 ME2 74 Vector operations 7 sample problems Given two vectors g and Q in terms of xyz components eZ3j5k Q 2g 4Z6Qft Example 5 Find a Q Example 6 Find a Q Example 7 Findg Q Example 8 Find a X Q Example 9 Find Q X g Example 10 Find the projection of g onto Q Example 11 Find the projection of Q onto a Example 12 Consider a vector Q given in XY components as g 2 I 41 ft sec Find the xy components of Q where the xy coordinate axes are oriented with respect to the XY axes as shown below Example 13 Consider a vector Q1 given in xy components as d 3g39 5 ft sec2 Find the XY components of a where the XY coordinate axes are oriented with respect to the xy axes as shown below IK 3687 Fundamentals Exam review material l l ME2 74 Vector kinematics The following are the three sets of vector kinematical equations for describing the motion velocity and acceleration of a point P moving in a plane using Cartesian path and polar coordinates path of P Cartesian coordinates x X W 2 36Lquot yj Path coordinates Z V2 path of P Note Since the rate of change of speed vis the projection of the acceleration vector a onto the unit tangent vector we can find 1 from path of P Polar coordinates v r39gr rege a r r92gr r 2r 9g9 Fundamentals Exam review material 1 2 ME2 74 Vector kinematics sample problems Example 14 A point P travels on a path given by yx 03 05x2 both X and y in meters The Xcomponent of velocity of P is known to be x 4 m sec constant Find the velocity and acceleration vectors for point P in Cartesian components when X 2 meters Example 15 At one instant in time the speed of a point P traveling on a plane is 10 msec with this speed decreasing at a rate of 3 msec2 The radius of curvature for the path of P at this instant is 50 meters Find the acceleration of P in terms of its path variables Example 16 Point P travels on a path given in polar coordinates as r 3sin9 where r is in feet 9 is in radians and 9 2 rad sec constant Find the velocity and acceleration of P in terms of its polar coordinates when 6 7272 Example 1 7 Make a sketch of the velocity and acceleration vectors for Example 14 Include the unit tangent and unit normal vectors in your sketch Is the speed increasing or decreasing Example 18 Make a sketch of the velocity and acceleration vectors for Example 16 Include the unit tangent and unit normal vectors in your sketch Is the speed increasing decreasing or constant Fundamentals Exam review material l 3 ME2 74 Free body diagrams FBD39s If we dissect the phrase free body diagram we see that it means a given body is cut free from its environment and a diagram is constructed representing the forces and couples that are acting on the free body One of the first questions that you should ask yourself before drawing an FBD is what body or system of bodies are to be represented by the FBD Later on in this course we will need to study the needs for different choices of systems in solving a problem Right now let s focus on given a system what is the FBD for this system Consider the following example Example 19 Construct the following four FBD s for the system shown below 0 one for A alone 0 one for Blower pulley alone 77 0 one for the upper pulley alone A 0 one for the total system of ABpulleys WW Assume that A is moving to the right along a rough incline Assume the pulleys to be massless with frictionless bearings Fundamentals Exam review material 1 4 ME2 74 SOLUTIONS for sample problems Example 1 Suppose that the speed ofP is given by vt 3t2 Find the acceleration of P d The acceleration of P on its straightline path is found from a d V 6t 1 Example 2 Suppose that the speed ofP is given by vs 5sin3s Find the acceleration of P a g v 5sin3s1500s3s 75 sin3scos3s Example 3 Suppose that the acceleration of P traveling on a straightline path is given by at 6 sin2t with v0 4 Find the speed ofP as afunction oftime V t 6sin2t gt IdvI6sin2tdt gt dt 120 0 vt v0 316 sin2tdt v0 gcos2tlg 4 3cos2t 1 7 3cos2t vt v0 316 sin2tdt v0 gcos2tlg 4 3cos2t 1 7 3cos2t Example 4 Suppose that the acceleration of P traveling on a straightline path is given by as 8 s3 with vs 0 5 Find the speed ofP as a function ofposition s Fundamentals Exam review material 1 5 ME2 74 dv dv ds dv a v gt dt ds dt ds V S J vd Jads gt 120 0 S vz v20 ads Therefore Example 5 Findab glgg 3Z513 21 4l6l 1 7llllg Example 6 Finda Q 2112z3z52224z612mgW Example 7 Finda b a1223z51222416212345640 z Example 8 Find a X Q 16 Fundamentals Exam review material axlgZ 31SI gtlt 2 4Z6lg z 139 lg 2 1 3 5 25 161 1012ft 2 4 6 Example 9 Findbxa 13gtltg 21 4Z6lgxg 31513 z 139 lg 2 4 6 216l 10gft2 1 3 5 Example 10 Find the projection of a onto Q 2i 4396k 40 g2bz 3j51c ft2 I224262 3 Example 11 Find the projection of 2 onto a i 3j5k 40 2 M 21394j6 ft a 4123252 5 Example 12 Consider a vector g given in XY components as ME274 g 2 41 ft sec Find the xy components of Q where the xy coordinate axes are oriented with respect to the XY axes as shown to the right J From the gure we see that J quotquotquotquotquotquotquotquotquotquotquotquot quot 3687 g cos3687 sin3687 1 081 061 K VVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVV W Therefore 36870 ex 91 241081061 20840608 sec Also from the gure we see that 139 sin3687 cos3687 1 06081 Fundamentals Exam review material 1 7 ME2 74 Therefore cy Z 241o 06081 9 2 06 40844ftsec In summary gcxzcyj0844l sec Example 13 Consider a vector 51 given in xy components as 51 3g39 5 ft secz Find the XY components ofgl where the XY coordinate axes are oriented with respect to the xy axes as shown to the right From the gure we see that 1 cos3687 sin3687 1 081 061 Therefore Also from the gure we see that 1 sin3687 zcos3687 l 06g081 Therefore dY g13g 51 06081 306 508 22 ftsec2 In summary gdX dY154g 221 sec2 Fundamentals Exam review material 1 8 ME2 74 Example 14 A point P travels on a path given by yx 03 05x2 both X and y in meters The Xcomponent of velocity of P is known to be x 4 m sec constant Find the velocity and acceleration vectors for point P in Cartesian components when X 2 meters y xx248msec ymxxxx422016msec2 Therefore yx1y1lt4g81msec axzyll6lmsec2 Example 15 At one instant in time the speed of a point P traveling on a plane is 10 msec with this speed decreasing at a rate of 3 msec2 The radius of curvature for the path of P at this instant is 50 meters Find the acceleration of P in terms of its path variables 2 102 a vet V en 3et5en 3et 2enmsec2 Example 16 Point P travels on a path given in polar coordinates as r 3sin9 where r is in feet 9 is in radians and 9 2 rad sec constant Find the velocity and acceleration of P in terms of its polar coordinates when 9 152 g lt3c0s9gtelt3gtltogtlt2gto f 39cos9 3 cos9 392 sin9 0 322 l 12 ftsec Therefore 2 fer 929 09 3299 6gel sec 12 322jer 00e9 a f r92er r 2f9e9 243 ft sec2 Fundamentals Exam review material 1 9 ME2 74 Example 1 7 Make a sketch of the velocity and acceleration vectors for Example 14 Include the unit tangent and unit normal vectors in your sketch Is the speed increasing or decreasing 4Z81msecvgt 2 g 16139 m se02 9g V gn P The unit tangent vector gt must point in the same direction as the velocity vector The unit normal vector en is perpendicular to g and is directed in a way that the component of g in the en is positive Note that the projection of g onto gtis positive Therefore the rate of change of speed is positive and the speed is increasing Example 18 Make a sketch of the velocity and acceleration vectors for Example 16 Include the unit tangent and unit normal vectors in your sketch Is the speed increasing decreasing or constant 6 9 secvgt 2 g 243 ftsec2 vetV gn The unit tangent vector gt must point in the same direction as the velocity vector The unit normal vector gnis perpendicular to Q and is directed in a way that the component of g in the en is positive Note that the projection of g onto g is zero Therefore the rate of change of speed is zero and the speed is instantaneously constant Fundamentals Exam review material 20 Example 19 0y I my T A h T V OX N T T A Questions 0 Why is the friction force on A pointing to the left 0 Why does the tension force T not appear in the FBD of ABpulleys ME274

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