Jump to first page Chapter 4 Kinetics of a Particle xx f(x)f(x) x ff max min

Jump to first page Integration: the reverse of differentiation xoxo x+  x xx x f(x)f(x) x

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4 Newton’s 2 nd law Newton’s 1 st law

Jump to first page 5 Newton’s 3 rd law action = reaction

Jump to first page 6 Work done where Total work done Example 1 What is the work done by a force on a article: (a)in circular motion? (b)horizontal motion? (c)from A to B? A B h

Jump to first page 7 Kinetic energy K.E. Work done by an external force

Jump to first page 8 Power P Energy dissipated per unit time

Jump to first page 9 Dissipative force (e.g. friction): work done from one point to another point depends on the path. A B path 1 path 2

Jump to first page 10 Non-dissipative force (conservative force): work done from one point to another point is independent on the path. A B path 1 path 2  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 =.

Jump to first page 11 Example 2 ( gravitational potential) R M r m X=0 X X Example 3 Find V of a spring. Ans. kx 2 /2 Example 4 Potential energy of a mass m, positioned at h from the ground. Ans. mgh

Jump to first page 12 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.

Jump to first page 13 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 . 2m2m r m r (a) At  = 45 o, v = (gr) 1/2  P.E.=-2mgr sin  +mg(r-r cos  )  K.E. = (2m+m)v 2 /2  (P.E.+ K.E.) = 0 3mv 2 /2 – mgr(2 sin  + cos  -1) = 0 v = [2gr(2 sin  + cos  -1)/3] 1/2

Jump to first page 14 B A From definition of potential energy: dV(x) = -Fdx From the concept of differential dV =

Jump to first page 15 With defined as the linear momentum Linear momentum When (i) the total (external) force is zero, or (ii) the collision time t 1  t 2 is extremely short. 1. In a motion, linear momentum can be conserved, 2. Define impulse = change in linear momentum: F time

Jump to first page 16 A B Collision between systems A and B.

Jump to first page 17 Angular Momentum Take moment about O Angular momentum about O is : m O

Jump to first page 18 Torque = Moment of force about O is defined as :

Jump to first page 19 Example: Prove that the angular momentum of a particle under a central force is conserved. m r In polar coordination system :