On a board with a length of l = 4 m and a mass of M = 30 kg, two boys with masses of m1 = 30 kg and m2 = 40 kg swing.

On a board with a length of l = 4 m and a mass of M = 30 kg, two boys with masses of m1 = 30 kg and m2 = 40 kg swing. Where should the stop point at the board be if the boys are sitting at the ends of the board?

Given:

L = 4 meters – board length;

M = 30 kilograms – board weight;

m1 = 30 kilograms – the weight of the first boy;

m2 = 40 kilograms – the mass of the second boy.

It is required to determine the stop point at the board l (meter).

Suppose the first boy is sitting on the left and the second on the right. Then, in order for the board to be in balance, the following condition must be met:

F1 * l1 + l1 * F11 = F2 * l2 + l2 * F22, where F1, F2 are the forces acting on the boys, and F11 and F22 are the gravity forces acting on the sections of the board.

m1 * g * l1 + l1 * M * g * l1 / L = m2 * g * l2 + l2 * M * g * l2 / L;

m1 * l1 + l1 ^ 2 * M / L = m2 * g + l2 ^ 2 * M / L.

since l1 + l2 = L, then l2 = L – l1, then:

m1 * l1 + l1 ^ 2 * M / L = m2 * (L – l1) + (L – l1) ^ 2 * M / L;

m1 * l1 + l1 ^ 2 * M / L = m2 * (L – l1) + (L ^ 2 – 2 * l1 * L + l1 ^ 2) * M / L;

30 * l1 + l1 ^ 2 * 30/4 = 40 * (4 – l1) + (4 ^ 2 – 2 * 4 * l1 + l1 ^ 2) * 30/4;

30 * l1 + 7.5 * l1 ^ 2 = 160 – 40 * l1 + (16 – 8 * l1 + l1 ^ 2) * 7.5;

30 * l1 + 7.5 * l1 ^ 2 = 160 – 40 * l1 + 120 – 60 * l1 + 7.5 * l1 ^ 2

30 * l1 = 280 – 100 * l1;

130 * l1 = 280

l1 = 280/130 = 2.15 meters;

l2 = L – l1 = 4 – 2.15 = 1.85 meters.

Answer: the stop point should be at a distance of 2.15 meters from a boy with a lower mass or 1.85 meters from a boy with a higher mass.