A homogeneous horizontally located rod with a mass of M and a length of 1.2 m with a point weight of 0.5 M

A homogeneous horizontally located rod with a mass of M and a length of 1.2 m with a point weight of 0.5 M fixed at its end is in equilibrium, resting on a support. Determine the distance from the fulcrum to the load.

Given:

M is the mass of the rod;

L = 1.2 meters – rod length;

m = 0.5 * M – weight of the cargo;

g = 10 m / s2 – acceleration of gravity.

It is required to determine l (meter) – the distance from the load to the fulcrum.

According to the problem statement, the rod is homogeneous. Then:

M1 * g * L1 = M2 * g * L2 + m * g * l where M1 is the mass of the bar on the left side of the fulcrum, M2 is the mass of the bar on the right side of the fulcrum, L1 and L2 are the distance from the centers of gravity of the sides of the bar to the point supports.

M1 * L1 = M2 * L2 + m * l;

M1 * (L – l) / 2 = M2 * l / 2 + m * l.

Taking into account that the mass of a rod of length x will be equal to M * x / L, we get:

M * (L – l) ^ 2 / (2 * L) = M * l ^ 2 / (2 * L) + 0.5 * M * l;

(L – l) ^ 2 / (2 * L) = l ^ 2 / (2 * L) + 0.5 * l;

(L – l) ^ 2 = l ^ 2 + l * L;

L ^ 2 – 2 * l * L + l ^ 2 = l ^ 2 + l * L;

L ^ 2 = l * L + 2 * l * L;

L ^ 2 = 3 * l * L;

L = 3 * l, hence:

l = L / 3 = 1.2 / 3 = 0.4 meters.

Answer: the distance from the load to the fulcrum will be equal to 0.4 meters.