
Two wires A and B of same length and of the same material have the respective radii ${r_1}$ and ${r_2}$ . Their one end is fixed with a rigid support, and at the other end equal twisting couple is applied. Then the ratio of the angle of twist at the end of A and the angle of twist at the end of B will be
A. $\dfrac{{{r_1}^2}}{{{r_2}^2}}$
B. $\dfrac{{{r_2}^2}}{{{r_1}^2}}$
C. $\dfrac{{{r_2}^4}}{{{r_1}^4}}$
D. $\dfrac{{{r_1}^4}}{{{r_2}^4}}$
Answer
164.1k+ views
Hint: In this case, when a problem is based on Shear and Bulk Modulus in Mechanical properties of solids, we know that twisting couple is defined as $\dfrac{{\pi \eta {r^4}\theta }}{{2l}}$ plays a significant role in establishing a relationship between couple and angle of twist hence, use this formula to calculate the ratio in order to provide an accurate solution.
Formula Used:
Couple = $C = \dfrac{{\pi \eta {r^4}\theta }}{{2l}}$
Complete answer:
We know that Couple per unit angle of twist is $\dfrac{{\pi \eta {r^4}}}{{2l}}$.
If angle of twist is denoted by $\theta $.
Then, Couple = $C = \dfrac{{\pi \eta {r^4}\theta }}{{2l}}$
According to the given question, the lengths of wires A and B are equal. Also, the material of A and B are the same and an equal twisting couple is applied on both the wires.
Since $\eta ,C\;and{\text{ }}l$ are same.
Therefore, ${r^4}\theta $ = constant.
i.e., the angle of twist varies inversely with the radius of the wire.
As the radii of two wires A and B are ${r_1}$ and ${r_2}$ respectively (given)
$ \Rightarrow \dfrac{{{\theta _A}}}{{{\theta _B}}} = \dfrac{{{r_2}^4}}{{{r_1}^4}}$
Thus, the ratio of the angle of twist at the end of A and the angle of twist at the end of B will be $\dfrac{{{r_2}^4}}{{{r_1}^4}}$.
Hence, the correct option is (C) $\dfrac{{{r_2}^4}}{{{r_1}^4}}$ .
Note: Since this is a problem related to stress-strain analysis in mechanical properties of solids hence, given conditions must be analyzed very carefully and quantities that are required to calculate the ratio of the angle of twist must be identified on a prior basis as it gives a better understanding of the problem.
Formula Used:
Couple = $C = \dfrac{{\pi \eta {r^4}\theta }}{{2l}}$
Complete answer:
We know that Couple per unit angle of twist is $\dfrac{{\pi \eta {r^4}}}{{2l}}$.
If angle of twist is denoted by $\theta $.
Then, Couple = $C = \dfrac{{\pi \eta {r^4}\theta }}{{2l}}$
According to the given question, the lengths of wires A and B are equal. Also, the material of A and B are the same and an equal twisting couple is applied on both the wires.
Since $\eta ,C\;and{\text{ }}l$ are same.
Therefore, ${r^4}\theta $ = constant.
i.e., the angle of twist varies inversely with the radius of the wire.
As the radii of two wires A and B are ${r_1}$ and ${r_2}$ respectively (given)
$ \Rightarrow \dfrac{{{\theta _A}}}{{{\theta _B}}} = \dfrac{{{r_2}^4}}{{{r_1}^4}}$
Thus, the ratio of the angle of twist at the end of A and the angle of twist at the end of B will be $\dfrac{{{r_2}^4}}{{{r_1}^4}}$.
Hence, the correct option is (C) $\dfrac{{{r_2}^4}}{{{r_1}^4}}$ .
Note: Since this is a problem related to stress-strain analysis in mechanical properties of solids hence, given conditions must be analyzed very carefully and quantities that are required to calculate the ratio of the angle of twist must be identified on a prior basis as it gives a better understanding of the problem.
Recently Updated Pages
Uniform Acceleration - Definition, Equation, Examples, and FAQs

JEE Main 2021 July 25 Shift 1 Question Paper with Answer Key

JEE Main 2021 July 22 Shift 2 Question Paper with Answer Key

JEE Atomic Structure and Chemical Bonding important Concepts and Tips

JEE Amino Acids and Peptides Important Concepts and Tips for Exam Preparation

JEE Electricity and Magnetism Important Concepts and Tips for Exam Preparation

Trending doubts
JEE Main 2025 Session 2: Application Form (Out), Exam Dates (Released), Eligibility, & More

Atomic Structure - Electrons, Protons, Neutrons and Atomic Models

Displacement-Time Graph and Velocity-Time Graph for JEE

JEE Main 2025: Derivation of Equation of Trajectory in Physics

Learn About Angle Of Deviation In Prism: JEE Main Physics 2025

Electric Field Due to Uniformly Charged Ring for JEE Main 2025 - Formula and Derivation

Other Pages
JEE Advanced Marks vs Ranks 2025: Understanding Category-wise Qualifying Marks and Previous Year Cut-offs

NCERT Solutions for Class 11 Physics Chapter 1 Units and Measurements

Units and Measurements Class 11 Notes: CBSE Physics Chapter 1

NCERT Solutions for Class 11 Physics Chapter 2 Motion In A Straight Line

Motion in a Straight Line Class 11 Notes: CBSE Physics Chapter 2

Important Questions for CBSE Class 11 Physics Chapter 1 - Units and Measurement
