Question 14.SP.7: Grain falls from a hopper onto a chute CB at the rate of 240......
Grain falls from a hopper onto a chute CB at the rate of 240 lb/s. It hits the chute at A with a velocity of 20 ft/s and leaves at B with a velocity of 15 ft/s, forming an angle of 10° with the horizontal. Knowing that the combined weight of the chute and of the grain it supports is a force W with a magnitude of 600 lb applied at G, determine the reaction at the roller support B and the components of the reaction at the hinge C.
STRATEGY: Because we have a steady stream of particles, apply the principle of impulse and momentum for the time interval Δt.
MODELING: Choose a system that consists of the chute, the grain it supports, and the amount of grain that hits the chute in the interval Δt. The impulse–momentum diagram for this system is shown in Fig. 1. Because the chute does not move, it has no momentum. Note that the sum Σm_{i}v_{i} of the momenta of the particles supported by the chute is the same at t and t + Δt and thus can be omitted.
ANALYSIS: You can use the impulse-momentum diagram to obtain scalar equations for the x and y directions and for moments about point C.
\underrightarrow{+} x components: C_{x}\,\Delta t=(\Delta m)\nu_{B}\,\cos\,10^{\circ} (1)
+ ↑ y components: -(\Delta m)\nu_{A}+C_{y}\,\Delta t-W\,\Delta t+B\,\Delta t=-(\Delta m)\,\nu_{B}\,\mathrm{sin}\,\,10^{\circ} (2)
+↺ moments about C: \qquad-3(\Delta m)\,\nu_{A}-7(W\Delta t)+12(B\Delta t)=6(\Delta m)\,\nu_{B}\cos\,10^{\circ}-12(\Delta m)\,\nu_{B}\sin\,10^{\circ} (3)
Using the given data, W = 600 lb, \nu_{A} = 20 ft/s, \nu_{B} = 15 ft/s, and Δm/Δt = 240/32.2 = 7.45 slug/s, and solving Eq. (3) for B and Eq. (1) for C_{x}, you obtain
12B = 7(600) + 3(7.45)(20) + 6(7.45)(15)(cos 10° − 2 sin 10°)
12B = 5075 B = 423 lb B = 423 lb ↑ ◂
C_{x}=(7.45)(15)\cos10^{\circ}=110.1\mathrm{~lb}\qquad\qquad\mathrm{C}_{x}=110.1\mathrm{~lb}\rightarrow ◂
Substituting for B and solving Eq. (2) for C_{y}, you end up with
C_{y} = 600 − 423 + (7.45)(20 − 15 sin 10°) = 307 lb
C_{y} = 307 lb ↑ ◂
REFLECT AND THINK: This kind of situation is common in factory and storage settings. Being able to determine the reactions is essential for designing a proper chute that will support the stream safely. We can compare this situation to the case when there is no mass flow, which results in reactions of B_{y} = 350 lb,C_{y} = 250 lb, and C_{x} = 0 lb.
