Q4. A non-uniform bar 2 m long hangs at rest, horizontal, from two light strings tied to its ends. The left string makes 36.9° with the vertical and the right string 53.1°. Find the distance d of the bar’s centre of gravity from its left end. (NCERT Exercise 6.8.)
Let the tensions be T1 (left) and T2 (right); sin 36.9° ≈ 0.6, cos 36.9° ≈ 0.8, and the reverse for 53.1°.
Horizontal forces: 0.6T1 = 0.8T2, so T1 = (4/3)T2.
Moments about the centre of gravity (the horizontal parts of the tensions act along the bar, so they have no moment): 0.8T1 × d = 0.6T2 × (2 − d). With T1 = (4/3)T2: (16/15)d = 0.6(2 − d), so (5/3)d = 1.2 and d = 0.72 m.
Q5. A uniform beam 4 m long and of weight 200 N rests on supports at its two ends. A load of 300 N hangs 1 m from the left end. Find the force on each support.
Moments about the left support: Rright × 4 = 200 × 2 + 300 × 1 = 700, so Rright = 175 N. Forces: Rleft = 500 − 175 = 325 N. The support nearer the load carries more.
Q6. A uniform plank 6 m long and of weight 300 N lies on a flat roof with 2 m of it sticking out beyond the edge. How far beyond the edge can a person of weight 600 N walk before the plank starts to tip?
On the point of tipping, the plank pivots on the edge of the roof and the roof pushes only there. The plank’s weight acts at its middle, 3 m from either end, which is 1 m back from the edge. Moments about the edge: 600 × x = 300 × 1, so x = 0.5 m. Beyond 0.5 m the plank tips.