Plank on Two Trestles: Where Can the Box Go?
A 3.0 m uniform plank of mass 12 kg rests horizontally on two trestles. Trestle A is 0.50 m from the left end and trestle B is 2.50 m from the left end. Each trestle is rated to carry a maximum load of 250 N. A 20 kg box is to be placed on the plank. A student says: 'The box is safe anywhere between the two trestles, because that is where the supports are.' The plank is stiff enough not to bend noticeably, and the box is small compared with the plank. Take g = 10 N/kg and measure positions from the left end of the plank.
1. Determine the range of positions along the plank where the box can be placed without overloading either trestle. Take the left end as x = 0.
Read the worked answer
Take moments about trestle A. The plank's weight is 120 N at 1.50 m; the box is 200 N at x.
R_B × 2.00 = 120 × 1.00 + 200(x − 0.50), so R_B = 60 + 100(x − 0.50) and R_A = 320 − R_B.
Overloading A: R_A ≤ 250 gives R_B ≥ 70, so 60 + 100(x − 0.50) ≥ 70, i.e. x ≥ 0.60 m. Overloading B: R_B ≤ 250 gives 100(x − 0.50) ≤ 190, i.e. x ≤ 2.40 m.
Allowed range: 0.60 m ≤ x ≤ 2.40 m.
Both reactions stay positive for all x on the plank, so the plank never tips here - the 250 N rating is the limiting condition, not balance. Check x = 0.50 m: R_B = 60 N and R_A = 260 N, so A is already overloaded by 10 N.
2. Evaluate the student's claim. Include the case x = 0.30 m and propose one change to the apparatus that makes that position safe.
Read the worked answer
The claim is false. At x = 0.30 m: R_B = 60 + 100(−0.20) = 40 N, so R_A = 280 N, which is 30 N above the rating.
The physical reason is lever arm. A load placed left of A has a short arm about A, and it also produces an anticlockwise moment about A that reduces R_B; both effects throw load onto A. Being 'between the supports' prevents tipping but does not prevent overloading.
A workable change is to move A left to 0.20 m, keeping B at 2.50 m. The supports are then 2.30 m apart: R_B × 2.30 = 120 × 1.30 + 200 × 0.10 = 176, so R_B = 76.5 N and R_A = 243.5 N. Both are below 250 N, so the box at 0.30 m is now safe.
3. The technician moves trestle B to the right end of the plank (x = 3.00 m) instead. Find the new allowed range and explain, in terms of moments, why moving B outward helps.
Read the worked answer
With A at 0.50 m and B at 3.00 m the separation is 2.50 m, so taking moments about A: R_B = [120 × 1.00 + 200(x − 0.50)] / 2.50, and R_A = 320 − R_B.
R_A ≤ 250 requires R_B ≥ 70: 120 + 200(x − 0.50) ≥ 175, giving x ≥ 0.775 m. R_B ≤ 250 requires 120 + 200(x − 0.50) ≤ 625, giving x ≤ 3.03 m, which is beyond the plank, so the right limit is the end at 3.00 m.
New range: about 0.78 m ≤ x ≤ 3.00 m.
Moving B outward increases its moment arm about A, so B supplies the same moment with a smaller force. That relieves A and extends the allowed region to the right. It does not help near the left end, because the left limit is set by A being overloaded, which only moving A can fix.