A loaded cement mixer drives onto an old drawbridge, where it stalls with its ce

Question

A loaded cement mixer drives onto an old drawbridge, where it stalls with its center of gravity three-quarters of the way across the span. The truck driver radios for help, sets the handbrake, and waits. Meanwhile, a boat approaches, so the drawbridge is raised by means of a cable attached to its end opposite the hinge (see figure). The drawbridge span is 40.0 m long and has a mass of 11,400 kg; its center of gravity is at its midpoint. The cement mixer, with driver, has a mass of 30,300 kg. When the drawbridge has been raised to an angle of 30

A loaded cement mixer drives onto an old drawbridge, where it stalls with its center of gravity three-quarters of the way across the span. The truck driver radios for help, sets the handbrake, and waits. Meanwhile, a boat approaches, so the drawbridge is raised by means of a cable attached to its end opposite the hinge (see figure). The drawbridge span is 40.0 m long and has a mass of 11,400 kg; its center of gravity is at its midpoint. The cement mixer, with driver, has a mass of 30,300 kg. When the drawbridge has been raised to an angle of 30 degree above the horizontal, the cable makes an angle of 70 degree with the surface of the bridge.

Explanation / Answer

Let the reactions at the hinge be S (vertical), R (horizontal).

Resolving vertically and horizontally:
S + T sin(40) = 42g * 10^3 ...(1)
R = T cos(40) ...(2)

Taking moments about the hinge:
12g * 20 cos(30) * 10^3 + 30g * 30 cos(30) * 10^3 = 40T sin(70)
1140g cos(30) * 10^3 = 40T sin(70)

1.
T = [ 1140 * 9.81 cos(30) * 10^3 ] / [ 40 sin(70) ]
= 2.58 * 10^5 N.

2.
From (2):
R = 1.97 * 10^5 N.

3.
From (1):
S = 42g * 10^3 - T sin(40)
= 2.46 * 10^5 N.

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