Analytical mechanics for students of physics and engineering . city u, then the change in themomentum of dm is (v — u) dm. Therefore _ dv . / dm F=,„-+(v-u1(// d f \ dm x ,,, 254 ANALYTICAL MECHANICS ILLUSTRATIVE EXAMPLES. 1. A jet of water strikes a concave vessel with a velocity of 80 feet persecond and then leaves it with a velocity which has the same magnitudeas the vetocity of impact but makes an angle of 120° with it. If thediameter of the jel is I inch find the force necessary to hold the concavevessel in position. The force experienced by the vessel equals the rate at which it receives
Analytical mechanics for students of physics and engineering . city u, then the change in themomentum of dm is (v — u) dm. Therefore _ dv . / dm F=,„-+(v-u1(// d f \ dm x ,,, 254 ANALYTICAL MECHANICS ILLUSTRATIVE EXAMPLES. 1. A jet of water strikes a concave vessel with a velocity of 80 feet persecond and then leaves it with a velocity which has the same magnitudeas the vetocity of impact but makes an angle of 120° with it. If thediameter of the jel is I inch find the force necessary to hold the concavevessel in position. The force experienced by the vessel equals the rate at which it receivesmomentuin. Suppose the vessel to be symmetrical with respect to theaxis of the jet, as in Fig. 119, then by symmetry there can be no resultantforce on the vessel in a direction perpendicular to the axis of the we need to consider only the change in momentum along theaxis. Let m be the mass of water delivered by the jet in the time t,v the velocity of impact, and a the change in the direction of flow. Thenthe force is a V — mv cos a i—-. Fig. 119. where .1 is the area of the cross-section of the jet and wi is the weight of:i cubic foot of water. Replacing the various magnitudes by their nu-merical values we obtain ?»g*(s<0*-(w=:)*( + cos 60°) 32 ? lb. I Mm i S8ION. It is evident from the general expression of F that itsvalue depends upon a and varies between zero for a = 0 and for Whe J IMPULSE AXD MOMENTUM 255 2. A uniform chain is hung from its upper end so thai its lower endjust touches an inelastic horizontal table, and then it is allowed to the force which the table will experience at any instanl during thefall of the chain. The force is partly due to the weight of that part of the chain which ison the table at the instant considered and partly due to the rate at whichthe table is receiving momentum. Let x be the height of the upper endof the chain above the table, / the total length, and p the mass per unitlength. Then pg {I - x)
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