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 the end opposite the hinge (the figure (Figure 1) ). The drawbridge is 40.0 m long and has a mass of 18,000 kg; its center of gravity is at its midpoint. The cement mixer, with driver, has mass 30,000 kg. When the drawbridge has been raised to an angle of 30 above the horizontal, the cable makes an angle of 70 with the surface of the bridge.

Part A

What is the tension T in the cable when the drawbridge is held in this position?

2.58•105

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Part B

What is the horizontal component of the force the hinge exerts on the span?

1.97•105

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Part C

What is the vertical component of the force the hinge exerts on the span?

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Part C

What is the vertical component of the force the hinge exerts on the span?

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Figure 1 of 1

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 the end opposite the hinge (the figure (Figure 1) ). The drawbridge is 40.0 m long and has a mass of 18,000 kg; its center of gravity is at its midpoint. The cement mixer, with driver, has mass 30,000 kg. When the drawbridge has been raised to an angle of 30 above the horizontal, the cable makes an angle of 70 with the surface of the bridge.

Part A

What is the tension T in the cable when the drawbridge is held in this position?

T =

2.58•105

  N  

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Incorrect; Try Again; 6 attempts remaining

Part B

What is the horizontal component of the force the hinge exerts on the span?

NH =

1.97•105

  N  

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Incorrect; Try Again; 6 attempts remaining

Part C

What is the vertical component of the force the hinge exerts on the span?

NV =   N  

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Part C

What is the vertical component of the force the hinge exerts on the span?

NV =   N  

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Figure 1 of 1

Explanation / Answer

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

Resolving vertically and horizontally:
S + T sin(40) = 48g * 10^3 ...(1)
R = T cos(40) ...(2)

Taking moments about the hinge:
18g * 20 cos(30) * 10^3 + 30g * 30 cos(30) * 10^3 = 40T sin(70)
1260g cos(30) * 10^3 = 40T sin(70)

1.
T = [ 1140 * 9.81 cos(30) * 10^3 ] / [ 40 sin(70) ]
= 2.79 * 10^5 N.

2.
From (2):

R=T cos(40)

=>2.79 * 10^5*cos(40)
R = 2.25715 * 10^5 N.

3.
From (1):

S + T sin(40) = 48g * 10^3

S= 48g * 10^3 - T sin(40)

S=48*9.8*10^3 - 2.79 * 10^5 sin(40)
S = 42g * 10^3 - T sin(40)
= 3.06 * 10^5 N.

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