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Chapter 3 , kinetics of particles

Discuss all about kinetics of particles with some conceptual example

Gemechis1
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Chapter 3 , kinetics of particles

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  1. Mechanical Engineering Bule Hora University, 2023/24 By: Gemechis. E

  2. Kinetics of Particles Contents • Newton’s second law • Work Energy equation • Impulse and Momentum • Impact

  3. Introduction Newton’s Second Law of Motion • If the resultant force acting on a particle is not zero, the particle will have an acceleration proportional to the magnitude of resultant and in the direction of the resultant. • Must be expressed with respect to a Newtonian (or inertial) frame of reference, i.e., one that is not accelerating or rotating. • This form of the equation is for a constant mass system

  4. Linear Momentum of a Particle • Replacing the acceleration by the derivative of the velocity yields • Linear Momentum Conservation Principle: If the resultant force on a particle is zero, the linear momentum of the particle remains constant in both magnitude and direction.

  5. International System of Units(SI Units): base units are the units of length (m), mass (kg), and time (second). The unit of force is derived, • U.S. Customary Units: base units are the units of force (lb), length (m), and time (second). The unit of mass is derived, Systems of Units • System of the units for the four primary dimensions (force, mass, length, and time), three may be chosen arbitrarily. The fourth must be compatible with Newton’s 2nd Law.

  6. Equations of Motion Newton’s second law • Can use scalar component equations, e.g., for rectangular components,

  7. Alternate expression of Newton’s second law, Dynamic Equilibrium • With the inclusion of the inertial vector, the system of forces acting on the particle is equivalent to zero. The particle is in dynamic equilibrium. • Methods developed for particles in static equilibrium may be applied, e.g., coplanar forces may be represented with a closed vector polygon. • Inertia vectors are often called inertial forces as they measure the resistance that particles offer to changes in motion, i.e., changes in speed or direction. • Inertial forces may be conceptually useful but are not like the contact and gravitational forces found in statics.

  8. Free Body Diagrams and Kinetic Diagrams • The free body diagram is the same as you have done in statics; we will add the kinetic diagram in our dynamic analysis. 1. Isolate the body of interest (free body) y x 2. Draw your axis system (e.g., Cartesian, polar, path) 3. Add in applied forces (e.g., weight, 225 lb pulling force) 4. Replace supports with forces (e.g., normal force) 5. Draw appropriate dimensions (usually angles for particles) 225 N 25o Ff N mg

  9. Put the inertial terms for the body of interest on the kinetic diagram. 1. Isolate the body of interest (free body) 2. Draw in the mass times acceleration of the particle; if unknown, do this in the positive direction according to your chosen axes y x may 225 N max 25o Ff N mg

  10. Draw the FBD and KD for block A (note that the massless, frictionless pulleys are attached to block A and should be included in the system).

  11. 1. Isolate body 4. Replace supports with forces 5. Dimensions (already drawn) 2. Axes 6. Kinetic diagram 3. Applied forces y x T T NB T T may = 0 = Ff-B T max mg N1 Ff-1

  12. Draw the FBD and KD for the collar B. Assume there is friction acting between the rod and collar, motion is in the vertical plane, and q is increasing • Isolate body • Axes • Applied forces • Replace supports with forces • Dimensions • Kinetic diagram

  13. Example 3.1, A 200-lb block rests on a horizontal plane. Find the magnitude of the force P required to give the block an acceleration of 10 ft/s2 to the right. The coefficient of kinetic friction between the block and plane is mk = 0.25. • Unknowns consist of the applied force P and the normal reaction N from the plane. The two equations may be solved for these unknowns. • SOLUTION: • Resolve the equation of motion for the block into two rectangular component equations.

  14. y O x • Unknowns consist of the applied force P and the normal reaction N from the plane. The two equations may be solved for these unknowns. • Resolve the equation of motion for the block into two rectangular component equations.

  15. Example 3.2, The two blocks shown start from rest. The horizontal plane and the pulley are frictionless, and the pulley is assumed to be of negligible mass. Determine the acceleration of each block and the tension in the cord. • SOLUTION: • Write the kinematic relationships for the dependent motions and accelerations of the blocks. • Write the equations of motion for the blocks and pulley. • Combine the kinematic relationships with the equations of motion to solve for the accelerations and cord tension.

  16. x y Write equations of motion for blocks and pulley. • Write the kinematic relationships for the dependent motions and accelerations of the blocks. O

  17. x y Combine kinematic relationships with equations of motion to solve for accelerations and cord tension. O

  18. Example 3.3, The 12-lb block B starts from rest and slides on the 30-lb wedge A, which is supported by a horizontal surface. Neglecting friction, determine (a) the acceleration of the wedge, and (b) the acceleration of the block relative to the wedge. • SOLUTION: • The block is constrained to slide down the wedge. Therefore, their motions are dependent. Express the acceleration of block as the acceleration of wedge plus the acceleration of the block relative to the wedge. • Write the equations of motion for the wedge and block. • Solve for the accelerations.

  19. The block is constrained to slide down the wedge. Therefore, their motions are dependent. • Write equations of motion for wedge and block. y x

  20. Solve for the accelerations.

  21. Example 3.4, The two blocks shown are originally at rest. Neglecting the masses of the pulleys and the effect of friction in the pulleys and between block A and the horizontal surface, determine (a) the acceleration of each block, (b) the tension in the cable. • SOLUTION: • Write the kinematic relationships for the dependent motions and accelerations of the blocks. • Draw the FBD and KD for each block • Write the equations of motion for the blocks and pulley. • Combine the kinematic relationships with the equations of motion to solve for the accelerations and cord tension.

  22. Write the kinematic relationships for the dependent motions and accelerations of the blocks. Differentiate this twice to get the acceleration relationship. xA yB (1)

  23. Draw the FBD and KD for each block 2T mAg T A +y B maAx T = = maBy +x mBg NA • Write the equation of motion for each block A (2) From Eqn(1) (3) • Solve the three equations, 3 unknowns (3) (2)

  24. Concept Question • The three systems are released from rest. Rank the accelerations, from highest to lowest. (1) (2) (3) • (1) > (2) > (3) d) (1) = (2) = (3) b) (1) = (2) > (3) e) (1) = (2) < (3) c) (2) > (1) > (3)

  25. Normal and Tangential Coordinates Aircraft and roller coasters can both experience large normal forces during turns.

  26. Equations of Motion Newton’s second law • For tangential and normal components,

  27. Example 3.5, The bob of a 2-m pendulum describes an arc of a circle in a vertical plane. If the tension in the cord is 2.5 times the weight of the bob for the position shown, find the velocity and acceleration of the bob in that position. • SOLUTION: • Resolve the equation of motion for the bob into tangential and normal components. • Solve the component equations for the normal and tangential accelerations. • Solve for the velocity in terms of the normal acceleration.

  28. Solve for velocity in terms of normal acceleration. • Resolve the equation of motion for the bob into tangential and normal components. • Solve the component equations for the normal and tangential accelerations.

  29. Example 3.6, Determine the rated speed of a highway curve of radius r= 400 ft banked through an angle q= 18o. The rated speed of a banked highway curve is the speed at which a car should travel if no lateral friction force is to be exerted at its wheels . • SOLUTION: • The car travels in a horizontal circular path with a normal component of acceleration directed toward the center of the path. The forces acting on the car are its weight and a normal reaction from the road surface. • Resolve the equation of motion for the car into vertical and normal components. • Solve for the vehicle speed.

  30. Solve for the vehicle speed. • The car travels in a horizontal circular path with a normal component of acceleration directed toward the center of the path. The forces acting on the car are its weight and a normal reaction from the road surface. • Resolve the equation of motion for the car into vertical and normal components.

  31. Example 3.7, The 3-kg collar B rests on the frictionless arm AA. The collar is held in place by the rope attached to drum D and rotates about O in a horizontal plane. The linear velocity of the collar B is increasing according to v= 0.2 t2 where v is in m/s and t is in sec. Find the tension in the rope and the force of the bar on the collar after 5 seconds if r = 0.4 m. • SOLUTION: • Draw the FBD and KD for the collar. • Write the equations of motion for the collar. • Determine kinematics of the collar. • Combine the equations of motion with kinematic relationships and solve. v

  32. Draw the FBD and KD of the collar • Given: v= 0.2 t2, r = 0.4 m • Find: T and N at t = 5 sec mat et = en T man N • Write the equations of motion

  33. Kinematics : find vt, an, at et mat en = T man q N Substitute into equations of motion

  34. How would the problem change if motion was in the vertical plane? et You would add an mg term and would also need to calculate q mat en = T man q N mg • When is the tangential force greater than the normal force? Only at the very beginning, when starting to accelerate. In most applications, an >> at

  35. Concept Question B C • Draw the approximate direction of the car’s acceleration at each point. A D A car is driving from A to D on the curved path shown. The driver is doing the following at each point: A – going at a constant speed B – stepping on the accelerator C – stepping on the brake D – stepping on the accelerator

  36. Radial and Transverse Coordinates • Hydraulic actuators and extending robotic arms are often analyzed using radial and transverse coordinates.

  37. Equations of Motion in Radial & Transverse Components • Consider particle at r and q, in polar coordinates,

  38. Example 3.8 , A block B of mass m can slide freely on a frictionless arm OA which rotates in a horizontal plane at a constant rate • Knowing that B is released at a distance r0 from O, express as a function of r • the component vr of the velocity of B along OA, and • the magnitude of the horizontal force exerted on B by the arm OA. • SOLUTION: • Write the radial and transverse equations of motion for the block. • Integrate the radial equation to find an expression for the radial velocity. • Substitute known information into the transverse equation to find an expression for the force on the block.

  39. Integrate the radial equation to find an expression for the radial velocity. • Substitute known information into the transverse equation to find an expression for the force on the block. • Write the radial and transverse equations of motion for the block.

  40. Example 3.9, The 3-kg collar B slides on the frictionless arm AA. The arm is attached to drum D and rotates about O in a horizontal plane at the rate where and t are expressed in rad/s and seconds, respectively. As the arm-drum assembly rotates, a mechanism within the drum releases the cord so that the collar moves outward from O with a constant speed of 0.5 m/s. Knowing that at t = 0, r = 0, determine the time at which the tension in the cord is equal to the magnitude of the horizontal force exerted on B by arm AA. • SOLUTION: • Draw the FBD and KD for the collar. • Write the equations of motion for the collar. • Determine kinematics of the collar. • Combine the equations of motion with kinematic relationships and solve.

  41. Given: • Find: time when T = N • Draw the FBD and KD of the collar maq eq = mar er T N • Write the equations of motion

  42. Kinematics : find expressions for r and q Substitute values into ar, aq Set T = N Substitute into equation of motion

  43. Concept Question The girl starts walking towards the outside of the spinning platform, as shown in the figure. She is walking at a constant rate with respect to the platform, and the platform rotates at a constant rate. In which direction(s) will the forces act on her? e2 e1 Top View v a) +e1 b) - e1 c) +e2 d) - e2 e) The forces are zero in the e1 and e2 directions

  44. WORK ENERGY EQUATION • Previously, problems dealing with the motion of particles were solved through the fundamental equation of motion, • The current portion introduces two additional methods of analysis. • Method of work and energy: directly relates force, mass, velocity and displacement. • Method of impulse and momentum: directly relates force, mass, velocity, and time. Forces and Accelerations Velocities and Displacements Velocities and Time Impulse-Momentum Newton’s Second Law Work-Energy

  45. Work of a force during a finite displacement, • Work is represented by the area under the curve of Ft plotted against s. Work of a Force • Work is a scalarquantity, i.e., it has magnitude and sign but not direction. Differential vector is the particle displacement. Work of the force is • Dimensions of work are force length. Units are1J(joule) = (1N)(1m) = 1ft.lb = 1.356J • Ft is the force in the direction of the displacement ds

  46. What is the work of a constant force in rectilinear motion? a) b) c) d) • Work of the force of gravity, • Work of the weight is equal to product of weight W and vertical displacement Dy. • In the figure above, when is the work done by the weight positive? a) Moving from y1 to y2 b) Moving from y2 to y1 c) Never

  47. Magnitude of the force exerted by a spring is proportional to deflection, • Work of the force exerted by spring, Displacement is in the opposite direction of the force • Work of the force exerted by spring is positive when x2 < x1, i.e., when the spring is returning to its undeformed position. • Work of the force exerted by the spring is equal to negative of area under curve of F plotted against x, As the block moves from A0 to A1, is the work positive or negative? As the block moves from A2 to Ao, is the work positive or negative?

  48. Work of a gravitational force (assume particle M occupies fixed position O while particle m follows path shown), Does the normal force do work as the block slides from B to A? Does the weight do work as the block slides from B to A? Positive or Negative work? • Forces which do not do work (ds = 0 or cosa = 0): • Reaction at frictionless pin supporting rotating body, • Reaction at frictionless surface when body in contact moves along surface, • Reaction at a roller moving along its track, and • Weight of a body when its center of gravity moves horizontally.

  49. Integrating from A1 to A2 , Particle Kinetic Energy: Principle of Work & Energy • Consider a particle of mass m acted upon by force • The work of the force is equal to the change in kinetic energy of the particle. • Units of work and kinetic energy are the same:

  50. Applications of the Principle of Work and Energy • The bob is released from rest at position A1. Determine the velocity of the pendulum bob at A2 using work & kinetic energy. • Force acts normal to path and does no work. • Velocity is found without determining expression for acceleration and integrating. • All quantities are scalars and can be added directly. • Forces which do no work are eliminated from the problem. • Principle of work and energy cannot be applied to directly determine the acceleration of the pendulum bob. • Calculating the tension in the cord requires supplementing the method of work and energy with an application of Newton’s second law. • As the bob passes through A2 , • If you designed the rope to hold twice the weight of the bob, what would happen?

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