The couple per unit twist C (torque per unit angle of twist) of a solid wire of length L, … — JAMB UTME

The couple per unit twist $C$ (torque per unit angle of twist) of a solid wire of length $L$ , radius $r$ , and shear modulus $\eta$ is modeled by the equation:

C = \frac{\pi \eta r^x}{2 L}

Using dimensional analysis, determine the numerical value of the exponent $x$ .

Answer: 4

Answer

The numerical value of the exponent x is 4.

Applying the principle of dimensional homogeneity requires the dimensions of couple per unit twist

[\text{M L}^2 \text{T}^{-2}]

to equal the dimensions of

\frac{\eta r^x}{L}

, which simplifies to

[\text{M L}^{x-2} \text{T}^{-2}]

. Equating exponents of length gives

2 = x - 2

, yielding

x = 4

Step-by-Step Solution

Determine the dimensional formula of couple per unit twist

C

[C] = \text{M L}^2 \text{T}^{-2}

Couple (torque) is force multiplied by perpendicular distance, which has dimensions

[\text{M L T}^{-2}][\text{L}] = [\text{M L}^2 \text{T}^{-2}]

. The angle of twist (in radians) is dimensionless.

Determine the dimensional formula of shear modulus

\eta

[\eta] = \text{M L}^{-1} \text{T}^{-2}

Shear modulus is defined as shear stress divided by shear strain. Stress has dimensions of force per unit area

[\text{M L T}^{-2}]/[\text{L}^2] = [\text{M L}^{-1} \text{T}^{-2}]

, while strain is dimensionless.

Set up the dimensional equation for the relation

C = \frac{\pi \eta r^x}{2 L}

[\text{M L}^2 \text{T}^{-2}] = \frac{[\text{M L}^{-1} \text{T}^{-2}][\text{L}]^x}{[\text{L}]} = [\text{M L}^{x-2} \text{T}^{-2}]

Pure numerical constants such as

\pi

and

2

are dimensionless. Length

L

and radius

r

both have dimension

[\text{L}]

Equate the exponents of length

\text{L}

on both sides of the dimensional equation.

x = 4

Comparing powers of

\text{L}

on both sides gives

2 = x - 2

, which solves to

x = 4

Key Concept

Dimensional Homogeneity in Mechanics

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