1. Two wire A and B are of the same material. Their lengths are in the ratio of 1:2 and the diameter are in the ratio 2:1. If they are pulled by the same force, then increase in length will be in the ratio of
(a) 2:1, (b) 1:4, (c) 1:8, (d) 8:1
As both A and B are of same material, their Young’ modulus (Y) will be same.
The ratio of their lengths
\frac{l_{A}}{l_{B}}=\frac{1}{2}
The ratio of their diameters
\frac{d_{A}}{d_{B}}=\frac{2}{1}
The applied force(F) is same for both A & B
Y=\frac{\frac{F}{A}}{\frac{\Delta l}{l}}
Y=\frac{Fl}{A\Delta l}
Y=\frac{Fl}{\frac{\pi d^{2}}{4}\Delta l}
Y=\frac{4Fl}{\pi d^{2}\Delta l}
\Delta l=\frac{4Fl}{\pi d^{2}Y}
Therefore, the ratio of the increase of the length is
\frac{\Delta l_{A}}{\Delta l_{B}}=\frac{4Fl_{A}}{\pi d_{A}^{2}Y} \times \frac{\pi d_{B}^{2}Y}{4Fl_{B}}
\frac{\Delta l_{A}}{\Delta l_{B}}=\frac{l_{A}}{l_{B}}\times \frac{d_{B}^{2}}{d_{A}^{2}}
\frac{\Delta l_{A}}{\Delta l_{B}}=\frac{l_{A}}{l_{B}}\times (\frac{d_{B}}{d_{A}})^{2}
\frac{\Delta l_{A}}{\Delta l_{B}}=\frac{1}{2} \times (\frac{1}{2})^{2}
\frac{\Delta l_{A}}{\Delta l_{B}}=\frac{1}{2} \times \frac{1}{4} = \frac{1}{8}
So, the ratio of the increase in the lengths is 1:8
2. When an elastic material with Young’s modulus Y is subjected to stretching stress S, elastic energy stored per unit volume of the material is
(a) \frac{YS}{2} (b) \frac{S^{2}Y}{2} (c) \frac{S^{2}}{2Y} (d) \frac{S}{2Y}
3. The following four wires are made of the same material. Which of these will have the largest extension when the same tension is applied?
Length = 100 cm, diameter = 1 mm, (b) Length = 200 cm, diameter = 2 mm, (c) Length = 300 cm, diameter = 3 mm, (d) Length = 50 cm, diameter = 0.5 m
4. The Young’s modulus of the material of a wire of length L and radius r is Y newton per m2. If the length of the wire is reduced to L/2 and the radius to r/2, then its Young’s modulus will be
(a) Y/2, (b) Y, (c) 2Y, (d) 4Y
5. The maximum load that a wire can sustain is W. If the wire is cut to half its value, the maximum load it can sustain is
(a) W, (b) W/2, (c) W/4, (d) 2W
6. The length of a steel wire is l1 when the stretching force is T1 and l2 when the stretching force is T2 The natural length of the wire is