Physics 12 - Quiz #5 

Cantilever Beam 2

Solution:

a) Free Body Diagram

We begin by writing the 3 conditions of static equilibrium. We wisely choose the pivot point to be the pin since it has the unknown reaction forces acting through it. This make the torque calculations possible.

Solving the torque equation yields the value of tension, T.

b) Using the initial equations for static equilibrium and the above value for tension, we can now easily solve for Rx, Ry, and therefore the resultant force, R.

c) If the cable was replaced by one that could only withstand 85% of the required tensile force of the previous cable then we could not have the cable horizontal as before. If the cable would break at 85% (227 N) then it couldn't possibly provide the counter-clockwise torque we require for static equilibrium. 

We need to solve for an angle in which the cable could provide the required torque which is given by

Thus it follows that 

This is just one possible angle. Any less than 640 and the cable would snap. To find another angle, we make use of the trigonometric identity:  sin(180-q) = sin q. Thus another possible angle would be 1160. Conversely any angle greater than 1160 and the cable will snap. So our cable position would look like the following.

A keen observer will note that the value of Rx and Ry would necessarily vary for these two angles. A really keen observer would realize that 640 and 1160 are the limiting angles for the cable and that as long as the cable was at any angle between 640 and 1160 relative to the beam the cable will safely withstand the force. 

d) If the cable was at 900 to the beam, this would require the least amount of tensile strength from the cable. In fact, even at 85% tensile strength the cable would be more than adequate.

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