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Kinematics of particles ppt

kinematics of particles problems and solutions ppt and kinematics of particles chapter 11 solutions kinematics of particles problems and solutions
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Published Date:19-07-2017
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Kinematics of Particles Space Curvilinear Motion Three-dimensional motion of a particle along a space curve. Three commonly used coordinate systems to describe this motion: 1. Rectangular Coordinate System (x-y-z) 2. Cylindrical Coordinate System (r-θ-z) 3. Spherical Coordinate System (R-θ-Φ) ME101 - Division III Kaustubh Dasgupta 1Kinematics of particles :: motion in space • Example 2 ME101 - Division III Kaustubh Dasgupta 2Kinematics of particles :: motion in space • Example 3 ME101 - Division III Kaustubh Dasgupta 3Kinematics of particles :: motion in space n- and t-coordinates for plane curvilinear motion can also be used for space curvilinear motion of a particle :: Considering a plane containing the curve and the n- and t-axes at a particular location (instance) •This plane will continuously shift its location and orientation in case of space curvilinear motion  difficult to use. ME101 - Division III Kaustubh Dasgupta 4Kinematics of particles :: motion in space Rectangular Coordinates (x-y-z) •Simply extend the previously derived equations to include third dimension. Plane Curvilinear Motion (2-D) Space Curvilinear Motion (3-D) In three dimensions, R is used in place of r for the position vector ME101 - Division III Kaustubh Dasgupta 5Kinematics of Particles Cylindrical Coordinates (r-θ-z) •Extension of the Polar coordinate system. •Addition of z-coordinate and its two time derivatives Position vector R to the particle for cylindrical coordinates: R = r e + zk r Velocity: Acceleration: Polar Cylindrical Polar Cylindrical 2    vrerezk arrer2re  vrere r r r 2   v r  r arrer2rezk r  v r r  v r 2 2     a rr a rr  r r v r   v z  z a r2r a r2r   2 2 v vv 2 2 2 r  a z 2 2 z v vvv a aa r z r 2 2 2 a aaa r z Unit vector k remains fixed in direction  has a zero time derivative ME101 - Division III Kaustubh Dasgupta 6Kinematics of Particles Spherical Coordinates (R-θ-Φ) •Utilized when a radial distance and two angles are utilized to specify the position of a particle. •The unit vector e is in the direction in which the particle P R would move if R increases keeping θ and Φ constant. •The unit vector e is in the direction in which the particle P θ would move if θ increases keeping R and Φ constant. •The unit vector e is in the direction in which the particle P Φ would move if Φ increases keeping R and θ constant. Resulting expressions for v and a: vv ev ev e aa ea ea e R R R R 2 2 2   a RRR cos v R R R cos d 2  aR2Rsin v R cos   R dt  1 d v R 2 2  aRR sincos  R dt ME101 - Division III Kaustubh Dasgupta 7Example (1) on cylindrical/spherical coordinate ME101 - Division III Kaustubh Dasgupta 8Example (1) on cylindrical/spherical coordinate ME101 - Division III Kaustubh Dasgupta 9Example (1) on cylindrical/spherical coordinate ME101 - Division III Kaustubh Dasgupta 10Example (1) on cylindrical/spherical coordinate ME101 - Division III Kaustubh Dasgupta 11Kinematics of Particles Relative Motion (Translating Axes) • Till now particle motion described using fixed reference axes  Absolute Displacements, Velocities, and Accelerations • Relative motion analysis is extremely important for some cases  measurements made wrt a moving reference system Motion of a moving coordinate system is specified wrt a fixed coordinate system (whose absolute motion is negligible for the problem at hand). Current Discussion: • Moving reference systems that translate but do not rotate • Relative motion analysis for plane motion Relative Motion Analysis is critical even if aircrafts ME101 - Division III Kaustubh Dasgupta 12 are not rotatingKinematics of Particles Relative Motion (Translating Axes) Vector Representation Two particles A and B have separate curvilinear motions in a given plane or in parallel planes. • Attaching the origin of translating (non-rotating) axes x-y to B. • Observing the motion of A from moving position on B. • Position vector of A measured relative to the frame x-y is r = xi + yj. Here x and y are the coordinates of A A/B measured in the x-y frame. (A/B  A relative to B) • Absolute position of B is defined by vector r B measured from the origin of the fixed axes X-Y. • Absolute position of A  r = r + r A B A/B • Differentiating wrt time  Velocity of A wrt B: Acceleration of A wrt B: Unit vector i and j have constant direction  zero derivatives ME101 - Division III Kaustubh Dasgupta 13Kinematics of Particles Relative Motion (Translating Axes) Vector Representation Velocity of A wrt B: Acceleration of A wrt B: Velocity or Absolute Velocity or Absolute Velocity  = + Acceleration of Acceleration of A or Acceleration of B A relative to B. • The relative motion terms can be expressed in any convenient coordinate system (rectangular, normal-tangential, or polar) • Already derived formulations can be used.  The appropriate fixed systems of the previous discussions becomes the moving system in this case. ME101 - Division III Kaustubh Dasgupta 14Kinematics of Particles Relative Motion (Translating Axes) Selection of Translating Axes Instead of B, if A is used for the attachment of the moving system:  r = r + r B A B/A v = v + v B A B/A a = a + a B A B/A  r = - r ; v = - v ; a = - a B/A A/B B/A A/B B/A A/B Relative Motion Analysis: • Acceleration of a particle in translating axes (x-y) will be the same as that observed in a fixed system (X-Y) if the moving system has a constant velocity  A set of axes which have a constant absolute velocity may be used in place of a fixed system for the determination of accelerations  Interesting applications of Newton’s Second law of motion in Kinetics A translating reference system that has no acceleration  Inertial System ME101 - Division III Kaustubh Dasgupta 15Kinematics of Particles Relative Motion (Translating Axes) Inertial Reference Frame Or Newtonian Reference Frame • When applying the eqn of motion (Newton’s Second Law of Motion), it is important that the acceleration of the particle be measured wrt a reference frame that is either fixed or translates with a constant velocity. • The reference frame should not rotate and should not accelerate. • In this way, the observer will not accelerate and measurements of particle’s acceleration will be the same from any reference of this type.  Inertial or Newtonian Reference Frame Study of motion of rockets and satellites: inertial reference frame may be considered to be fixed to the stars. Motion of bodies near the surface of the earth: inertial reference frame may be considered to be fixed to the earth. Though the earth rotates its own axis and revolves around the sun, the accelerations created by these motions of the earth are relatively small and can be neglected. ME101 - Division III Kaustubh Dasgupta 16Example on relative motion ME101 - Division III Kaustubh Dasgupta 17Example on relative motion Graphical method Trigonometric method Vector algebra ME101 - Division III Kaustubh Dasgupta 18Kinematics of Particles Constrained Motion of Connected Particles • Inter-related motion of particles One Degree of Freedom System Establishing the position coordinates x and y measured from a convenient fixed datum.  We know that horz motion of A is twice the vertical motion of B. Total length of the cable: L, r , r and b are constant. First and second time derivatives: 2 1  Signs of velocity and acceleration of A and B must be opposite  v is positive to the left. v is positive to the down A B Lower Pulley  Equations do not depend on lengths or pulley radii Alternatively, the velocity and acceleration magnitudes can be determined by inspection of lower pulley. SDOF: since only one variable (x or y) is needed to specify the positions of all parts of the system ME101 - Division III Kaustubh Dasgupta 19Kinematics of Particles Constrained Motion of Connected Particles One Degree of Freedom System Applying an infinitesimal motion of A’ in lower pulley. • A’ and A will have same motion magnitudes • B’ and B will have same motion magnitudes • From the triangle shown in lower figure, it is clear that B’ moves half as far as A’ because point C has no motion momentarily since it is on the fixed portion on the cable. • Using these observations, we can obtain the velocity and acceleration magnitude relationships by inspection. • The pulley is actually a wheel which rolls on the fixed cable. Lower Pulley ME101 - Division III Kaustubh Dasgupta 20