A plank, two supports and a child. Who holds up what?

Resolving forces tells you the total the supports carry. It can't tell you how they share it. For that, you need the turning effect of each force.

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A 6 m plank of mass 20 kg rests on supports 1 m from each end, and a 30 kg child stands 2 m from $B$. The supports push up with $R_C$ and $R_D$. Resolving vertically gives only

One equation, two unknowns.

Take moments about $C$. $R_C$ acts at $C$ itself, so its moment is zero: it drops out. The two weights, 2 m and 3 m from $C$, turn the plank clockwise; $R_D$, 4 m away, turns it anticlockwise. In equilibrium they balance.

Moments about $D$ give the same answer another way. $R_C$, 4 m to the left, turns the plank clockwise; the weights, 2 m and 1 m to the left, turn it anticlockwise:

so $R_C = 17.5g$ again.

As the child walks towards $B$, $D$ takes more and more of the load. At the very end, $R_D = 47.5g$ and $R_C = 2.5g$: still pushing up, so the plank stays level. Had $R_C$ reached zero, the plank would have been on the point of tipping about $D$.

Choose the point. Lose an unknown.

Moments of forces at an angle, centres of mass, the point of tilting, hinged rods held by strings, and twelve graded exam questions with full worked solutions are waiting.

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