The Young’s modulus of steel is twice that of brass. Two wires of same length and of same area of cross section, one of steel and another of brass are suspended from the same roof. If we want the lower ends of the wires to be at the same level, then the weights added to the steel and brass wires must be in the ratio of:
(2015 Re)
1. $1 : 1$
2. $1 : 2$
3. $2 : 1$
4. $4 : 1$
View Answer
We know $\Delta L = \frac{FL}{AY}$. Since $L$, $A$, and $\Delta L$ are the same for both wires, $F \propto Y$. Therefore, $$\frac{F_s}{F_b} = \frac{Y_s}{Y_b} = \frac{2}{1} = 2:1$$.
Copper of fixed volume $V$ is drawn into wire of length $l$. When this wire is subjected to a constant force $F$, the extension produced in the wire is $\Delta l$. Which of the following graphs is a straight line?
(2014)
1. $\Delta l \text{ versus } 1/l$
2. $\Delta l \text{ versus } l^2$
3. $\Delta l \text{ versus } 1/l^2$
4. $\Delta l \text{ versus } l$
View Answer
Extension $\Delta l = \frac{Fl}{AY}$. Since volume $V = Al$, we have $A = V/l$. Substituting this, $$\Delta l = \frac{Fl}{(V/l)Y} = \frac{Fl^2}{VY}$$. Thus, $\Delta l \propto l^2$, meaning the graph of $\Delta l$ versus $l^2$ is a straight line.
The following four wires are made of the same material. Which of these will have the largest extension when the same tension is applied?
(2013)
1. $\text{Length } = 300 \text{ cm, diameter } = 3 \text{ mm}$
2. $\text{Length } = 50 \text{ cm, diameter } = 0.5 \text{ mm}$
3. $\text{Length } = 100 \text{ cm, diameter } = 1 \text{ mm}$
4. $\text{Length } = 200 \text{ cm, diameter } = 2 \text{ mm}$
View Answer
Extension $\Delta L = \frac{FL}{AY} = \frac{4FL}{\pi d^2 Y}$. For a given force and material, $$\Delta L \propto \frac{L}{d^2}$$. Calculating $L/d^2$ for the options reveals that option (b) gives the maximum value ($50 / 0.5^2 = 200$).
Given below are two statements : One is labelled as Assertion (A) and the other is labelled as Reason (R)
Assertion (A): The stretching of a spring is determined by the shear modulus of the material of the spring
Reason (R): A coil spring of copper has more tensile strength than a steel spring of same dimensions.
In the light of the above statements, choose the most appropriate answer from the options given below :
(2022)
1. (A) is false but (R) is true
2. Both (A) and (R) are true and (R) is the correct explanation of (A)
3. Both (A) and (R) are true and (R) is not the correct explanation of (A)
4. (A) is true but (R) is false
View Answer
When a spring is stretched, the wire itself undergoes torsion, which is governed by the shear modulus of the material, making the assertion true. Steel has a higher tensile strength and elasticity than copper, so the reason is false.