Physics · Mechanical Properties Of Solids · NEET
When a beam bends, its top layers get compressed and bottom layers get stretched. Layers far from the centre line resist bending most strongly. Increasing depth d moves material farther from this neutral line, and the resistance grows as d^3. Breadth b only adds more parallel material at the same distances, so it helps only as b^1. That is why delta = Wl^3/(4bd^3 Y): doubling depth cuts sag by 8 times, but doubling breadth cuts it only by 2 times.
Treat the beam as many thin layers. The bending moment from load W at the centre is balanced by the internal elastic (restoring) moment, which involves Young's modulus Y and the geometric moment of the cross-section (proportional to bd^3/12). Solving the beam bending equation for a beam of length l supported at both ends with central load W gives delta = Wl^3/(4bd^3 Y). NEET does not ask the full calculus; it asks how delta scales with l, b, d and Y.
delta is proportional to l^3, so a longer beam sags dramatically more. A beam twice as long between supports sags 8 times more for the same load. This is why bridges use closely spaced pillars: shorter spans mean far less depression, keeping the structure rigid and safe.
delta is inversely proportional to Y, so a material with a large Young's modulus (like steel, Y about 2 x 10^11 Pa) bends least. NCERT states directly: to reduce bending for a given load, use a material with a large Young's modulus.
No. A cantilever (fixed at one end, load at free end) has depression delta = Wl^3/(3 Y I) where I is the geometric moment. The beam supported at both ends with a central load gives delta = Wl^3/(4bd^3 Y). For NEET, remember this two-ends-supported version from NCERT; do not mix up the constant.
Try the real previous-year questions from this chapter — each with the answer and a full solution.
For a beam of length l, breadth b and depth d supported at both ends with weight W at the centre, the depression is delta = Wl^3 / (4 b d^3 Y), where Y is Young's modulus. This is the standard NCERT relation.
Because depression delta is proportional to 1/d^3 but only to 1/b. Increasing depth reduces sag as the cube, so a deep beam resists bending much better than a wide one for the same amount of material.
Yes, strongly. delta is proportional to l^3, so a longer span sags much more. Doubling the length increases the depression by a factor of 8 for the same load.
delta is inversely proportional to Y. A stiffer material with larger Young's modulus (like steel) bends less, so it is preferred for load-bearing beams and girders.
Since delta is proportional to 1/d^3, doubling the depth reduces the depression to one-eighth of its original value, keeping all other quantities the same.
