Plank, trestles and a painter: which limit bites first?
A workshop has no proper platform, so a uniform timber plank 4.00 m long and of mass 18.0 kg is laid across two trestles, placed 0.50 m and 3.50 m from the plank's left end. Each trestle is rated by its manufacturer to carry a maximum load of 800 N. A painter of mass 75.0 kg steps onto the plank at the left end and walks slowly towards the right, intending to carry a 20 kg tool bag. Take g = 9.81 m/s². The foreman is worried for two different reasons: the plank could tip once the painter passes the right-hand trestle, and the trestles themselves might be overloaded long before that happens. The plank is dry and the trestle tops are flat, so the plank can tip or lift but cannot slide sideways. Treat the plank as uniform and rigid and each trestle as a point support, and treat the painter and bag as a single load of 95.0 kg standing at one position.
1. The painter stands 2.00 m from the left end, carrying the bag. Calculate the reaction force at each trestle and check your answer for consistency.
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Take moments about the left trestle, which is 0.50 m from the left end, so the right trestle is 3.00 m away.
Loads and lever arms:
- Plank: 18.0 kg at its centre, 2.00 m from the left end, i.e. 1.50 m from the pivot.
- Painter + bag: 95.0 kg at 2.00 m, i.e. 1.50 m from the pivot.
Moment equation: N_R x 3.00 = (18.0 + 95.0) x 9.81 x 1.50 = 169.5 x 9.81 = 1663 N m N_R = 1663 / 3.00 = 554 N
Check: total weight = 113.0 x 9.81 = 1109 N, so N_L = 1109 - 554 = 555 N. The two answers are equal, which is exactly what symmetry demands: the load sits at the mid-point of the plank while the supports are symmetric about that mid-point. If the two reactions came out unequal here, the arithmetic would be wrong. Both are well below the 800 N rating.
2. Find how far past the right-hand trestle the painter could go with the bag before the plank tips, and compare that position with the trestle's 800 N rating. Which condition actually limits how far he can walk?
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Tipping: the plank tips clockwise about the right trestle when N_L falls to zero. Take moments about the right trestle (3.50 m):
Plank weight acts 1.50 m to its left: moment = 18.0 x 9.81 x 1.50 = 265 N m (anticlockwise). Painter at x: moment = 95.0 x 9.81 x (x - 3.50) (clockwise).
Set equal: (x - 3.50) = 265 / 932 = 0.284 m, so x = 3.78 m - inside the 4.00 m plank, so tipping is geometrically possible.
Overload: at the tipping point N_R = total weight = 1109 N, far above 800 N. Find where N_R = 800 N instead, taking moments about the left trestle: 800 x 3.00 = [18.0 x 1.50 + 95.0 x (x - 0.50)] x 9.81 244.6 = 27.0 + 95.0(x - 0.50) 95.0(x - 0.50) = 217.6, so x = 2.79 m.
Conclusion: the trestle rating, not tipping, is the real limit; the plank would be overloaded about a metre before it could tip.
3. The foreman proposes moving the trestles outwards to 0.20 m and 3.80 m, arguing that this 'stops the tipping problem'. Evaluate this proposal.
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Tipping: with the right trestle at 3.80 m, the plank's centre is 1.80 m to its left, giving a moment of 18.0 x 9.81 x 1.80 = 318 N m. Tipping needs 95.0 x 9.81 x (x - 3.80) = 318, i.e. x - 3.80 = 0.341, x = 4.14 m. That is beyond the end of the plank, so the painter can no longer tip it. The foreman is right about tipping.
Overload: the load on the right trestle at x = 4.00 m (moments about the new left trestle at 0.20 m) is N_R x 3.60 = [18.0 x 1.80 + 95.0 x 3.80] x 9.81 = 393 x 9.81 = 3861 N m N_R = 1073 N, still above 800 N.
Verdict: the proposal solves the wrong problem. Moving the trestles outwards removes the tipping risk but the plank still has no safe standing position near the right end. The workable solutions are a third trestle, a shorter span of plank past the support, or a load restriction keeping the painter within about 2.8 m of the left end.