A uniform drawbridge must be held at a 38.0 angle above the horizontal to allow

Question

A uniform drawbridge must be held at a 38.0 angle above the horizontal to allow ships to pass underneath. The drawbridge weighs 4.80×104 N , is 11.0 m long, and pivots about a hinge at its lower end. A cable is connected 3.75 m from the hinge, as measured along the bridge, and pulls horizontally on the bridge to hold it in place.

What is the tension in the cable?

Find the magnitude of the force the hinge exerts on the bridge.

Find the direction of the force the hinge exerts on the bridge.

If the cable suddenly breaks, what is the initial angular acceleration of the bridge?

Explanation / Answer

a) Apply the first and second conditions of equilibrium to the drawbridge

T = 2*w * cos38/sin38
= 2*48000 / tan38
= 1.228 * 10^5 N

b) Hv & Hh are the components of the force the hinge exerts on the bridge

Fx = 0 gives Hh = T = 1.228 * 10^5 N

Fy = 0 gives Hv = w = 4.5 * 10^4 N

H = sqrt(Hh^2 + Hv^2) = 1.32 * 10^5 N

c) tanX = Hv/Hh = 21.35 deg

So, the hinge force has a magnitude of 1.32 * 10^5 and direction 21.35 deg above the horizontal.

d) a = 3g(cos38) / 2*L

= (3*9.81*(cos38)) / (2*11)

= 1.05 rad/s^2

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