The plank, the drum and the truck bed
Two workers in a warehouse in Rourkela must get a 120 kg drum onto a truck whose floor is 1.2 m above the ground. They have a straight wooden plank 4.0 m long, which they lean against the truck floor. A student watching them measures with a spring balance that a steady pull of 520 N along the plank is needed to slide the drum up at constant speed. If the drum is laid on its side and rolled, the pull needed falls to about 400 N. The plank bends visibly when the drum is halfway up, and its foot has slipped once on the smooth concrete floor. A second, shorter plank of 2.5 m is also lying in the yard.
1. Calculate the ideal effort, the efficiency of the plank as used, and state where the missing energy goes.
Read the worked answer
Treat the plank as an inclined plane. The useful work is lifting the drum through the height of the truck bed:
W_useful = mgh = 120 × 9.8 × 1.2 = 1411 J
Ideal effort along a frictionless plank of length L = 4.0 m:
F_ideal = mgh ÷ L = 1411 ÷ 4.0 ≈ 353 N
The actual effort is 520 N, so:
Efficiency = 353 ÷ 520 × 100 ≈ 68%
Work put in = 520 × 4.0 = 2080 J. The difference, 2080 − 1411 = 669 J, is not lost from the universe; it appears as heat and a little sound at the sliding contact between drum and plank, and at the drum's rim where it drags rather than rolls. The surfaces warm slightly.
The point of the machine is confirmed: the plank does not reduce the work needed, it only spreads the same work over a longer distance so that the force is smaller. The force falls from the full weight of 1176 N to about 353 N, at the cost of pushing 4 m instead of lifting 1.2 m.
2. Explain why rolling the drum lowers the effort, and why the plank's foot slipping is the more dangerous problem.
Read the worked answer
When the drum slides, the retarding force is kinetic friction, f = μN, with μ typically 0.4–0.6 for steel on wood. The normal component here is about mg cosθ, roughly 1120 N, so friction alone adds somewhere between 450 and 670 N of resistance; this is why the measured 520 N is so much larger than the ideal 353 N.
When the drum rolls, the contact point is instantaneously at rest relative to the plank, so sliding friction is replaced by rolling resistance, which comes mainly from deformation of the surfaces and is characterised by a coefficient nearer 0.05–0.1. The effort drops to 400 N, giving an efficiency of 353 ÷ 400 ≈ 88%. Rolling also avoids scraping the drum and reduces the chance of it skidding sideways.
The slipping foot is the serious hazard. The plank pushes down and back on the floor; if the horizontal component exceeds the static friction available between plank and concrete, the foot shoots backwards, the plank drops and the drum falls. On smooth concrete that limit is easily crossed. The fix is mechanical, not hopeful: butt the foot against a kerb or wall, nail a cleat, or have a third person stand firmly on the foot; and keep the plank angle low, because a steeper plank throws more of the push backwards. The visible bending is a second warning: an overloaded plank stores elastic energy and can snap, so a thicker plank or a prop under the middle is needed.
3. Compare the 4.0 m plank with the 2.5 m plank for this job, using numbers.
Read the worked answer
For the 2.5 m plank the ideal effort is:
F_ideal = mgh ÷ L = 1411 ÷ 2.5 ≈ 565 N
That is already more than the 520 N that was actually needed on the 4.0 m plank, and friction has not yet been counted - the real pull would be well over 600 N, close to what two people can manage on a sloped, bending surface, and the drum would be far more likely to slip back.
The geometry changes too. sinθ = 1.2 ÷ 4.0 = 0.30, so θ ≈ 17.5°; for the short plank sinθ = 1.2 ÷ 2.5 = 0.48, so θ ≈ 29°. The steeper angle increases the tendency of the drum to slide back and throws a larger horizontal force at the foot of the plank.
The trade-off is distance: the 4 m plank needs a 4 m pull, the 2.5 m plank a 2.5 m pull. The gravitational work is identical, about 1411 J, and the extra friction work on the longer plank is small once the drum is rolled. So the long plank is the better choice for force and safety, provided it is strong enough and its foot is secured; the short plank is only suitable if the load is reduced or more people push.