A uniform drawbridge must be held at a 32.5 angle above the horizontal to allow

Question

A uniform drawbridge must be held at a 32.5 angle above the horizontal to allow ships to pass underneath. The drawbridge weighs 4.75×104 N , is 13.0 m long, and pivots about a hinge at its lower end. A cable is connected 3.65 m from the hinge, as measured along the bridge, and pulls horizontally on the bridge to hold it in place.

Part A

What is the tension in the cable?

Part B

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

Part C

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

Part D

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

Explanation / Answer

Part A)   Here, T * 3.65 * sin32.5 = 4.75×104 * 6.5 * cos32.5

=>   T = 132778.19 N

=>   the tension in the cable = 132778.19 N

Part B)    Horizontal force =    132778.19 N

                Vertical force =    4.75×104 N

=>     magnitude of the force the hinge exerts on the bridge =   141018.78 N

Part C)    direction of the force the hinge exerts on the bridge = 180 - tan-1(4.75×104/132778.19)

                                                                                            = 160.31 degree      (anticlockwise from + ve X axis)

Part D)   Here, torque = I * angular acceleration

=>     4.75×104 * 6.5 * cos32.5 = 1/3 * 4.75×104 * 13 * 13/9.8 * angular acceleration

=>   initial angular acceleration =   0.953 rad/sec2

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