[3] (20 pts) A landscape architect is planning an artificial waterfall in a city

Question

[3] (20 pts) A landscape architect is planning an artificial waterfall in a city park. Water flowing at 1.70 m/s will leave the end of a horizontal channel at the top of a vertical wall h-2.35 m high, and from there will fall into a pool, as shown in the figure. a) will the space behind the waterfall be wide enough for a pedestrian walkway? b) To sell her plan to the city council, the architect intends to build a model that is one-twelfth actual size. How fast should the water flow in the channel in the model? [4] (20 pts) An enterprising child wants to reach an apple in a tree without needing to climb. He arranges a chair connected to a rope that passes over a frictionless pulley, as shown in the figure. He then pulls on the end of the rope with such a force that the spring scale reads 250 N when his feet no longer touch the ground. His true weight is 320 N, and the chair weighs 160 N. a) Draw two free-body diagrams showing the forces on the child and the chair when considered as separate systems, and a third diagram for the combined system of the child and chairs b) Find the upward acceleration of the child and chall. c) Find the force the child exerts on the chair. [5) (20 pts) A single bead can slide with negligible friction on a stiff wire that has been bent into a circular loop of radius R=15.0 cm, as shown in the figure. The circle is always in a vertical plane and rotates steadily about a vertical axis with period T. As shown in the figure, the position of the bead on the wire is described by the angle 8 between the vertical and the radial dashed line from the center of the loop to the bead. a) For a rotation period T -0.45s at what angles a will the bead be motionless on (i.e. not slide along) the wire? Note that there are two solutions in this case, b) Now take the rotation period to be T=0.85s. For this period there is only a single angle e where the bead is motionless on the wire. Find this angle. ) Find the relationship between T and R that distinguishes between situations where there are two or only a single stable position for the bead. Hint: Note the similarity of this problem to that of a banking airplane. As in that case the bead here travels along a circle of radius r-R sind in the horizontal plane. You can resolve the force balance in the plane of the loop either in the lateral (P) and vertical (8) directions, or along and perpendicular to the wire (as usual, one choice is simpler).

Explanation / Answer

3)

time taken by water to fall down is

1/2gt^2 = h

t = sqrt(2h/g)

t = sqrt[(2.35 x 2) / 9.81]

t = 0.692 sec

d = v x t

d = 1.70 m/s x 0.692 sec

d = 1.177 m

Yes, it is enough

b)

for 1/12 th of total height

t = sqrt(2h / 12g)

t = sqrt[(2.35 x 2) / (12 x 9.81)]

t = 0.1998 sec

Now d = 1.177 m / 2 = 0.098 m

V = d/t

V = 0.098 / 0.1998

V = 0.49 m/s

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