The momentum of a body of constant mass m moving with velocity u is, by definition, equal to the product of mass and velocity, that is

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1 Newtons Lws 1

2 Newton s Lws There re three lws which ber Newton s nme nd they re the fundmentls lws upon which the study of dynmics is bsed. The lws re set of sttements tht we believe to be true in most circumstnces, since they re in very exct greement with the results obtined from experimenttion. Newton 1 This sttes tht every body continues in its stte of rest or of uniform motion in stright line, unless cted upon by some externl forces to chnge this stte. The lw relly defines force s something which chnges the stte of rest or uniform motion of body. This force my mke contct with the body nd therefore be in the form of push or pull or, it my hve no direct contct s in the cse of grvittionl, electricl nd mgnetic forces. Newton s first lw lso suggests tht mtter hs built-in resistnce to motion. This resistnce is clled inerti nd it is the mss of body which defines its inerti. To illustrte this, consider first tennis bll which will require only smll force to chnge its stte of rest. On the other hnd, lrge rticulted lorry will require considerbly more force to chnge its stte. Newton 2 This lw is concerned with how force is mesured. It sttes tht the rte of chnge of momentum is proportionl to the impressed force, nd tkes plce in the direction of the stright line in which the force cts. The momentum of body of constnt mss m moving with velocity u is, by definition, equl to the product of mss nd velocity, tht is Momentum = mss (m) x Velocity (u) If force cts on the body for period of time t nd chnges its velocity from u to v, then the chnge in momentum is given by Chnge in momentum = mv mu It follows tht the rte of chnge of momentum = mv mu t Hence by the second lw: Force (F) = km (v u) t Also v u t where k = constnt of proportionlity which in the SI system of units is unity. = ccelertion of the body Therefore, we end up with the fmilir eqution F = m This eqution is nother form of Newton 2 which enbles us to mesure force by finding the ccelertion it produces on known mss. An lterntive form of this eqution my be used when considering the effect of grvity on known mss nd this defines the weight of body. Tht is Weight (W) = mss (m) x ccelertion due to grvity (g) or W = mg 2

3 Close to the erth g is normlly tken s 9.81 or 9.8 m/s 2. Newton 3 Newton s third lw sttes tht to every ction there is n equl nd opposite rection. Wht this is relly stting is tht forces cn never occur singulrly, but lwys in pirs. This is something tht we considered in n erlier course nd will not be pursued further here. Inerti Force The methods used in sttics (e.g. equilibrium of forces) could be pplied in dynmics if we could reshpe the problem to void ccelertions, i.e. such tht the body is t rest or in uniform motion. We cn do this by substituting imginry inerti forces for ccelertions using Newton 2, so tht the body or system is in pprent equilibrium. This principle is known s D Alemberts Principle. When we dopt this principle P m P m m This Becomes this The inerti force corresponding to ccelertion is m (by Newton 2) cting opposite to to give pprent equilibrium (inerti force drwn s hollow rrow becuse it s imginry). Inerti force my be considered s resistnce to ccelertion. Similrly, frictionl force my be considered s resistnce to stedy motion. Such force cts in the opposite direction to velocity. If R = resistnce to stedy motion the body digrm becomes: P m R v For sttic blnce in ccelerted motion we my equte forces, thus: P = m + R While the ccelerting force is F = P R = m 3

4 Tutoril Problems From Applied Mechnics by Hnnh & Hillier 1. A mss of 1 kg is hung from spring blnce in lift. Wht is the spring blnce reding when the lift is () t rest; (b) ccelerting upwrds t 3 m/s 2 : (c) ccelerting downwrds t 3 m/s 2 ; (d) moving downwrds nd retrding t 3m/s 2? (9.8 N; 12.8 N; 6.8 N; 12.8 N) 2. A plnning mchine tble of mss 450 kg ttins speed of 0.6 m/s t distnce of 600 mm from rest. The coefficient of friction between tble nd bed is 0.1. Clculte the friction force nd effort required during this period. If during the cutting stroke the force of the tool is 950 N nd the speed is held constnt t the mximum vlue ttined, clculte the effort required to mintin the cutting stroke. (441 N; 576 N; 1390 N) 3. The tool force on shping mchine during the cutting stroke is 180 N nd the reciprocting prts re equivlent to moving mss of 45 kg. If power is suddenly shut off wht would be the furthest distnce cut by the tool if the cutting speed were initilly 1.2 m/s? Assume the cutting force is independent of the speed. (180 mm) Appliction to Connected Bodies Consider the exmple shown below where mss m 2 is being used to move mss m 1 long horizontl surfce. v P 1 m 1 P 2 Drwing the FBD m 1g m 2 v m 2 m 1 P 2 P 1 F = μn N m 2g From the FBD For m 1 For m 2 P 1 = m 1 + F m 2g = P 2 + m 2 Provided the pulley is light nd frictionless P 1 = P 2 If pulley hs inerti nd/or friction P 2 > P 1 4

5 Let consider simple hoist r α m 2 v P 1 P 2 m 1 m 1 m 2 m 2 m 1g m 2g = αr nd v = ωr (see lter) v m 1 FBD s For m 1 P 1 = m 1g + m 1 For m 2 m 2g = P 2 + m 2 Agin if we ssume pulley light nd frictionless then P 1 = P 2. Tutoril Problems From Applied Mechnics by Hnnh & Hillier 1. Two lods, ech of mss 2 kg re tied together by light inextensible cord. They re ccelerted long the level by pull of 18 N t one lod. Find the ccelertion of the system nd the tension in the cord. Resistnce to motion my be neglected. (4.5 m/s 2 ; 9 N) 2. A locomotive of mss 80 t pulls trin of mss 200 t with n ccelertion of 0.15 m/s 2 long the level. The resistnce to motion of both locomotive nd trin is 45 N/t. Clculte () the trctive effort required, (b) the pull in the coupling hook t the locomotive. (54.6 kn; 39 kn) 3. In n experiment, lod if 4.5 kg is pulled long level trck by mss of 0.5 kg ttched to it by light inextensible cord pssing over light frictionless pulley nd hnging verticlly. Clculte the distnce trvelled from rest in 2 s. (1.96 m) 4. A mine cge of mss 500 kg is returned to the surfce by wire cble pssing over loose pulley t the pit hed. The cble is fstened to counterweight of mss 600 kg. Find the ccelertion of the empty cge if llowed to move freely. (0.89 m/s 2 ) 5

6 5. A motor cr develops trctive force of 1.8 kn on the level, when towing nother exctly similr cr whose engine is out of ction. Find the tension in the tow rope nd the ccelertion. The resistnce to motion is 650 N on ech cr. The mss of ech cr is 1 t. (900 N; 0.25 m/s 2 ) 6. A mss if 15 kg is supported by light rope which psses over light smooth pulley, nd crries t its other end mss of 6 kg. The 6 kg mss is held fst by pwl. If the pwl is relesed find the tension in the rope nd the time tken for the 15 kg mss to rech the level of the pwl. The 15 kg mss is initilly 2 m bove the level of the pwl. (84 N; 0.98 s) 6

7 Blnk pge for you to mke notes 7

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