**30. The drawing shows an A-shaped stepladder. Both sides of the ladder are equal in length. This ladder is standing on a frictionless horizontal surface, and only the crossbar (which has a negligible mass) of the “A” keeps the ladder from collapsing. The ladder is uniform and has a mass of 20.0 kg. Determine the tension in the crossbar of the ladder.

 

w0436

 

To solve this problem we consider the ladder as having two symmetric halves.  The free body diagram for one half is shown below.

Take sum of torques at the floor.

 

Again since the ladder is not moving.  For the hinge force creating a torque I am going to use perpendicular distance.  Since the angle with the vertical is 15°, the angle with the horizontal is 75°

 

 

 

 

Note since we are only considering half the ladder we need half the mass for the weight.  The cos(75°) is the perpendicular distance the weight acts through so it is the distance from where the ladder touches the floor to where the line of force due to the weight acts.

 

Which can be simplified to

 

 

We want Tension T, but we have the unknown Hinge force H.  So do sum of forces in the x direction

Since we are in static equilibrium so

Plug that in

 

Now solve for T

 

 

 

 

 

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This page last updated on January 11, 2020