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58 PHYSICS The total momentum of an isolated system of interacting particles is conserved. An important example of the application of the law of conservation of momentum is the collision of two bodies. Consider two bodies A and B, with initial momenta p and p . The bodies collide, A B get apart, with final momenta p′ and p′A B Fig. 4.7Equilibrium under concurrent forces. respectively. By the Second Law In other words, the resultant of any two forces say F and F , obtained by the parallelogram FAB ∆t = pA− p A and 1 2 law of forces must be equal and opposite to the F ∆t= p ′ − p third force, F3. As seen in Fig. 4.7, the three BA B B forces in equilibrium can be represented by the (where we have taken a common interval of time sides of a triangle with the vector arrows taken for both forces i.e. the time for which the two in the same sense. The result can be bodies are in contact.) generalised to any number of forces. A particle Since F = −F by the third law, is in equilibrium under the action of forces F ,1 AB BA F2,... n if they can be represented by the sides of a closed n-sided polygon with arrows directed p′A p A = −(pB− p B ) in the same sense. i.e. p′A p B = pA+ p B (4.9) Equation (4.11) implies that which shows that the total final momentum of F 1xF +2x = 3x the isolated system equals its initial momentum. F 1yF 2y+ F 3y= 0 Notice that this is true whether the collision is F + F + F = 0 (4.12) elastic or inelastic. In elastic collisions, there is 1z 2z 3z where F 1xF 1yd F are1zhe components of F 1 a second condition that the total initial kinetic along x, y and z directions respectively. energy of the system equals the total final kinetic ⊳ energy (See Chapter 5). Example 4.6 See Fig. 4.8. A mass of 6 kg is suspended by a rope of length 2 m from the ceiling. A force of 50 N in the 4.8 EQUILIBRIUM OF A PARTICLE horizontal direction is applied at the mid- Equilibrium of a particle in mechanics refers to point P of the rope, as shown. What is the angle the rope makes with the vertical in the situation when the net external force on the -2 particle is zer. Accordingtothefirstlaw,this equilibrium ? (Take g = 10 m s ). Neglect the mass of the rope. means that, the particle is either at rest or in uniform motion. If two forces F 1nd F , a2t on a particle, equilibrium requires F1= − F 2 (4.10) i.e. the two forces on the particle must be equal and opposite. Equilibrium under three concurrent forces F ,1F 2 and F 3equires that the vector sum of the three forces is zero. (a) (b) (c) Fig. 4.8 F + F + F = 0 (4.11) 1 2 3 * Equilibrium of a body requires not only translational equilibrium (zero net external force) but also rotational equilibrium (zero net external torque), as we shall see in Chapter 6. Reprint 2026-27