A box of negligible mass rests at the left end of a 1.90-m, 25.9-kg plank (see f

Question

A box of negligible mass rests at the left end of a 1.90-m, 25.9-kg plank (see figure). The width of the box is w = 80.8 cm, and sand is to be distributed uniformly throughout it. The center of gravity of the nonuniform plank is x = 51.0 cm from the right end. What mass of sand should be put into the box so that the plank balances horizontally on a fulcrum placed just below its midpoint? (The origin is at the fulcrum and +x to the right.)
1 kg

A box of negligible mass rests at the left end of a 1.90-m, 25.9-kg plank (see figure). The width of the box is w = 80.8 cm, and sand is to be distributed uniformly throughout it. The center of gravity of the nonuniform plank is x = 51.0 cm from the right end. What mass of sand should be put into the box so that the plank balances horizontally on a fulcrum placed just below its midpoint? (The origin is at the fulcrum and +x to the right.) 1 kg

Explanation / Answer

What you have is the 25.9 kg mass of the plank acting at the centre of mass,
which is; (1.9/2) - .51 = 0.44 m from the pivot,
which gives a force of 25.9*0.44 = 11.396 N
So, to balance the force on the other side must equal 11.396 N as well.
So first work out the distance from the centre of mass of the sand to the pivot, which would be
(1.9/2) - (0.808/2) = 0.546 m.
Then divide the force by the distance to get the mass, 11.396 / 0.546 = 20.87 kg of sand

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