Engineering Mechanics: Dynamics Dynamics

1 Engineering mechanics : Dynamics Dynamics Branch of mechanics that deals with the motion of bodies under the action of forces (Accelerated motion ) Two distinct parts: kinematics study of motion without reference to the forces that cause motion or are generated as a result of motion Kinetics relates the action of forces on bodies to their resulting motionsME101 -Division IIIK austubh Dasgupta1 Engineering mechanics : Dynamics Basis of rigid body Dynamics Newton s 2ndlaw of motion A particleof mass m acted upon by an unbalanced force F experiences an acceleration a that has the same direction as the forceand a magnitudethat is directly proportional to the force ais the resulting accelerationmeasured in a non-accelerating frame of referenceME101 -Division IIIK austubh Dasgupta2 Engineering mechanics .

2 Dynamics Space Geometric region occupied by bodies Reference system Linear or angular measurements Primary reference systemor astronomical frame of reference Imaginary set of rectangular axes fixed in space Validity for measurements for velocity < speed of light Absolute measurements Reference frame attached to the earth?? Time MassME101 -Division IIIK austubh Dasgupta3 Engineering mechanics : Dynamics Newton s law of gravitation F:: mutual force of attraction between two particles G:: universal gravitational constant m3/( )221rmmGF 4ME101 -Division IIIK austubh DasguptaEngineering mechanics : Dynamics Weight Only significant gravitational force between the earth and a particle located near the surface g= GMe/r2.

3 Acceleration due to gravity ( ) Variation of gwith altitude2rmMGWe mgW 5ME101 -Division IIIK austubh Dasgupta 220hRRgg gis the absolute acceleration due to gravity at altitude hg0is the absolute acceleration due to gravity at sea levelRis the radius of the earthEngineering mechanics : DynamicsEffect of Altitude on Gravitation Forceof gravitational attraction of the earth on a body depends on the position of the body relative to the earth Assumingthe earthto be a perfect homogeneous sphere, a massof 1 kgwould be attracted to the earth by a force of.

4 Nif the mass is on the surface of the earth Nif the mass is at an altitude of 1 km Nif the mass is at an altitude of 100 km Nif the mass is at an altitude of 1000 km Nif the mass is at an altitude of equal to the mean radius of the earth, 6371 km6ME101 -Division IIIK austubh DasguptaEngineering mechanics : Dynamics Effect of earth s rotation gfrom law of gravitation Fixed set of axes at the centre of the earth Absolute value of g Earth s rotation Actual accelerationof a freely falling bodyis less thanabsolute g Measuredfrom a position attached to the surface of the earth7ME101 -Division IIIK austubh DasguptaEngineering mechanics : Dynamics Effect of earth s rotation Engineering applications:: variation of g is ignored8ME101 -Division IIIK austubh Dasguptasea-level conditionsKinematics of particles motion Constrained.

5 Confined to a specific path Unconstrained :: not confined to a specific path Choice of coordinates Position of P at any time t rectangular ( , Cartesian) coordinates x, y, z cylindrical coordinates r, , z spherical coordinates R, , Path variables Measurements along the tangent tand normal nto the curve9ME101 -Division IIIK austubh DasguptaKinematics of particles Choice of coordinates10ME101 -Division IIIK austubh DasguptaKinematics of ParticlesRectilinear motion motion along a straight line11tt+ tME101 -Division IIIK austubh DasguptaKinematics of particles .

6 Rectilinear motion motion along a straight linePosition at any instance of time t:: specified by its distance smeasured from some convenient reference point Ofixed on the line :: (disp. is negative if the particle moves in the negative s-direction). Velocity of the particle:Acceleration of the particle:12tt+ tsdtdsv +ve or ve depending on +ve or ve displacementsdtsdavdtdva 22orBoth are vector quantitiesdsssdsordsavdv +ve or ve depending on whether velocity is increasing or decreasingME101 -Division IIIK austubh DasguptaKinematics of ParticlesRectilinear motion .

7 Graphical InterpretationsUsing s-tcurve, v-t& a-tcurves can be under v-tcurve during time dt= vdt== ds Net dispfrom t1to t2= corresponding area under v-tcurve or s2 -s1= (area under v-tcurve) Area under a-tcurve during time dt= adt== dv Net change in velfrom t1to t2= corresponding area under a-tcurve or v2 -v1= (area under a-tcurve) 2121ttssvdtds 2121ttvvadtdvME101 -Division IIIK austubh DasguptaKinematics of ParticlesRectilinear motion :Graphical InterpretationsTwo additional graphical relations:14 Area under a-scurve during dispds= ads == vdv Net area under a-scurve betnposition coordinates s1and s2 or (v22 v12) = (area under a-scurve) Slope of v-scurve at any point A= dv/ds Construct a normal ABto the curve at A.

8 From similar triangles: Veland posncoordinate axes should have the same numerical scales so that the acclnread on the x-axis in meters will represent the actual acclnin m/s2 2121ssvvadsvdvdsdvvCB )(onacceleratiadsdvvCB ME101 -Division IIIK austubh DasguptaKinematics of particles ::Rectilinear Motion15 Analytical Integration to find the position coordinateAcceleration may be specified as a function of time, velocity, or position coordinate, or as a combined function of these. (a)Constant AccelerationAt the beginning of the interval t = 0, s = s0, v = v0 For a time interval t: integrating the following two equationsSubstituting in the following equation and integrating will give the position coordinate:Equations applicable for Constant Acceleration and for time interval 0 to tdtdsv dtdva dsavdv ME101 -Division IIIK austubh DasguptaKinematics of particles .

9 Rectilinear Motion16 Analytical Integration to find the position coordinate(b) Acceleration given as a function of time, a= f(t)At the beginning of the interval t = 0, s = s0, v = v0 For a time interval t: integrating the following equation Substituting in the following equation and integrating will give the position coordinate:Alternatively, following second order differential equation may be solved to get the position coordinate: dtdsv dtdva dtdvtf )(sdtsda 22)(tfs ME101 -Division IIIK austubh DasguptaKinematics of particles ::Rectilinear Motion17 Analytical Integration to find the position coordinate(c) Acceleration given as a function of velocity, a= f(v)At the beginning of the interval t = 0, s = s0, v = v0 For a time interval t: Substituting a and integrating the following equation Solve for vas a function of tand integrate the following equation to get the position coordinate:Alternatively, substitute a= f(v)in the following equation and integrate to get the position coordinate.

10 Dtdsv dtdva dtdvvf )(dsavdv ME101 -Division IIIK austubh DasguptaKinematics of particles : Rectilinear Motion18 Analytical Integration to find the position coordinate(d) Acceleration given as a function of displacement, a= f(s)At the beginning of the interval t = 0, s = s0, v = v0 For a time interval t: substituting aand integrating the following equationSolve for vas a function of s: v= g(s), substitute in the following equation and integrate to get the position coordinate:It gives tas a function of s. Rearrange to obtain sas a function of tto get the position all these cases, if integration is difficult, graphical, analytical, or computer methods can be dsavdv ME101 -Division IIIK austubh DasguptaKinematics of particles : Rectilinear Motion19 ExamplePosition coordinate of a particle confined to move along a straight line is given by s= 2t3 24t+ 6, where sis measured in meters from a convenient origin and tis in seconds.