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Chapter 4 Kinetics of a Particle

max. f ( x ).  f. min. x.  x. Chapter 4 Kinetics of a Particle. f ( x ). x. x o.  x. x+  x. x. Integration: the reverse of differentiation. Newton’s 2 nd law. Newton’s 1 st law. Newton’s 3 rd law action = reaction. Work done. where. Total work done.

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Chapter 4 Kinetics of a Particle

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  1. max f(x) f min x x Chapter 4 Kinetics of a Particle

  2. f(x) x xo x x+x x Integration: the reverse of differentiation

  3. Newton’s 2nd law Newton’s 1st law

  4. Newton’s 3rd law action = reaction

  5. Work done where Total work done Example 1 What is the work done by a force on a article: • in circular motion? • horizontal motion? • from A to B? B h A

  6. Work done by an external force Kinetic energy K.E.

  7. Power P Energy dissipated per unit time

  8. B A Dissipative force (e.g. friction): work done from one point to another point depends on the path. path 1 path 2

  9. B A Non-dissipative force (conservative force): work done from one point to another point is independent on the path. path 1 P.E. between two points is equal to the work done by an external force against the field of a conservative force for bringing the particle from the starting point to the end point, with the external force = . path 2

  10. M r m R X X=0 X Example 2 (gravitational potential) Example 3 Find V of a spring. Ans. kx2/2 Example 4 Potential energy of a mass m, positioned at h from the ground. Ans. mgh

  11. In general, the two types of forces coexist: If there is no dissipative force, K.E. + P.E. = 0, i.e. conservation of mechanical energy.

  12. P.E.=-2mgr sin+mg(r-r cos ) • K.E. = (2m+m)v2/2 • (P.E.+ K.E.) = 0 • 3mv2/2 – mgr(2 sin + cos  -1) = 0 • v= [2gr(2 sin + cos  -1)/3]1/2 Example The rod is released at rest from  = 0, find : (a) velocity of m when the rod arrives at the horizontal position. (b) the max velocity of m. (c) the max. value of . r r 2m m (a) At  = 45o, v = 0.865 (gr)1/2

  13. B A From definition of potential energy: dV(x) = -Fdx From the concept of differential dV =

  14. 1. In a motion, linear momentum can be conserved, F time Linear momentum With defined as the linear momentum When (i) the total (external) force is zero, or (ii) the collision time t1t2 is extremely short. 2. Define impulse = change in linear momentum:

  15. B A Collision between systems A and B.

  16. m O Angular Momentum Take moment about O Angular momentum about O is :

  17. Torque = Moment of force about O is defined as :

  18. m r Example: Prove that the angular momentum of a particle under a central force is conserved. In polar coordination system :

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