Question 882: Thermodynamics & KTG

Question

Two uniform brass rods $$A$$ and $$B$$ of length '$$l$$' and '$$2 l$$' and their radii '$$2 r$$' and '$$r$$' respectively are heated to same temperature. The ratio of the increase in the volume of $$\operatorname{rod} \mathrm{A}$$ to that of $$\operatorname{rod} \mathrm{B}$$ is

Accepted Answer

To determine the ratio of the increase in the volume of rod $$A$$ to that of rod $$B$$, we need to consider the formula for the volumetric expansion due to heating. The change in volume ($$\Delta V$$) of a rod due to a change in temperature can be given by:
$$ \Delta V = \beta V \Delta T $$
where:
$$\beta$$ = coefficient of volumetric expansion (which is the same for both rods since they are made of brass),
$$V$$ = initial volume of the rod,
$$\Delta T$$ = change in temperature (which is the same for both rods).
First, let's calculate the volume of each rod. The volume $$V$$ of a cylindrical rod is:
$$ V = \pi r^2 l $$
For rod $$A$$:
Length = $$l$$,
Radius = $$2r$$.
Therefore, the initial volume of rod $$A$$ ($$V_A$$) is:
$$ V_A = \pi (2r)^2 l = \pi \cdot 4r^2 \cdot l = 4 \pi r^2 l $$
For rod $$B$$:
Length = $$2l$$,
Radius = $$r$$.
Therefore, the initial volume of rod $$B$$ ($$V_B$$) is:
$$ V_B = \pi r^2 (2l) = 2 \pi r^2 l $$
Now, the increase in volume ($$\Delta V$$) of each rod due to heating is given by:
For rod $$A$$:
$$ \Delta V_A = \beta V_A \Delta T = \beta (4 \pi r^2 l) \Delta T = 4 \beta \pi r^2 l \Delta T $$
For rod $$B$$:
$$ \Delta V_B = \beta V_B \Delta T = \beta (2 \pi r^2 l) \Delta T = 2 \beta \pi r^2 l \Delta T $$
To find the ratio of the increase in the volume of rod $$A$$ to that of rod $$B$$, we take:
$$ \frac{\Delta V_A}{\Delta V_B} = \frac{4 \beta \pi r^2 l \Delta T}{2 \beta \pi r^2 l \Delta T} = \frac{4}{2} = 2 $$
Thus, the ratio of the increase in volume of rod $$A$$ to that of rod $$B$$ is:
2:1
.
Therefore, the correct option is
Option C: 2:1
.

Answer

$$1: 1$$

Answer

$$1: 2$$

Answer

$$2: 1$$

Answer

$$1: 4$$

Thermodynamics & KTG

Two uniform brass rods $A and B of length 'l' and '2 l' and their radii '2 r' and 'r' respectively are heated to same temperature. The ratio of the increase in the volume of \operatorname{rod} \mathrm{A} to that of \operatorname{rod} \mathrm{B}$ is

Choose the correct answer: