Calculating Mass of a Balanced Beam with Two People Using Torque Equations

[EchoTheCat]

Homework Statement

Determine the mass of a 4.5 meter beam, if it is balanced on a fulcrum 2.04 meters from the left end. A 886 N man stands on the very left end, and a 393 N girl stands on the right end. [/B]

Homework Equations

T = Fr

The Attempt at a Solution

The torque applied by the man would be 886 N * 2.04 m = 1807.44 N*m.
Since the length of the beam on the right side is 4.5-2.04 = 2.46 m, the torque applied by the girl is 393 N * 2.46 m = 966.78 N*m.
Since the fulcrum is slightly closer to the left, the torque applied by the beam is (2.46-2.04 m)*9.8*mass of the beam.
Since the torques must cancel, 1807.44 = 966.78 + 4.116*m
m = 204.242 kg Not the right answer though.

 

[BvU]

How do you know ?
Make a sketch

 

[kuruman]

(2.46-2.04 m)*9.8*mass of the beam.

Why is this the torque exerted by gravity on the beam? Where is the CM of the beam?

 

[EchoTheCat]

BvU this is what my sketch looks like.

 

[EchoTheCat]

Why is this the torque exerted by gravity on the beam? Where is the CM of the beam?

The torque from 2.04 meters on the left side and 2.04 meters on the right side would cancel out. Then, we are left with 0.42 meters on the right side. 0.42 meters*9.8m/s/s*mass of the beam, since T = Fr. The center of mass of the beam would be at 2.25 meters.

 

[kuruman]

The center of mass of the beam would be at 2.25 meters.

Right. And the torque exerted by gravity is the weight of the plank times the distance from the CM of the plank to the fulcrum. This business of 2.04 m on the left cancelling 2.04 m to the right is nonsense. Just add the three torques about the fulcrum and set them equal to zero.