The twisting couple per unit twist for a solid cylinder of radius 4.9 cm is 0.1 N-m. The twisting couple per unit twist for a hollow cylinder of same material with outer and inner radii of 5 cm and 4 cm respectively, will be

The correct answer is 0.064 N-m.

1. Formula for Twisting Couple per Unit Twist:
The twisting couple per unit twist (or torsional rigidity) $C$ for a solid cylinder of radius $r$, length $l$, and modulus of rigidity $\eta$ is given by the formula:
$C=\frac{\pi \eta r^4}{2l}$
For a hollow cylinder of the same material and length, with outer radius $r_2$ and inner radius $r_1$, the twisting couple per unit twist is:
$C_{\text{hollow}}=\frac{\pi \eta (r_2^4-r_1^4)}{2l}$

2. Ratio of Twisting Couples:
Since both cylinders are made of the same material and have the same length, we can take the ratio of their twisting couples per unit twist:
$\frac{C_{\text{hollow}}}{C_{\text{solid}}}=\frac{r_2^4-r_1^4}{r^4}$

3. Calculation:
Given the following values:
• Radius of the solid cylinder, $r=4.9\text{cm}$
• Twisting couple per unit twist of the solid cylinder, $C_{\text{solid}}=0.1\text{N-m}$
• Outer radius of the hollow cylinder, $r_2=5\text{cm}$
• Inner radius of the hollow cylinder, $r_1=4\text{cm}$

Substituting these values into the ratio formula:
$r_2^4-r_1^4=5^4-4^4=625-256=369$
$r^4={4.9}^4=576.4801$

Now, calculate $C_{\text{hollow}}$:
$C_{\text{hollow}}=C_{\text{solid}}\times \frac{369}{576.4801}$
$C_{\text{hollow}}\approx 0.1\times 0.64009$
$C_{\text{hollow}}\approx 0.064\text{N-m}$