Question

A solid cylinder is radiating power. It has a length that is ten times its radius. It is cut into a number of smaller cylinders, each of which has the same radius. Each small cylinder has the same temperature as the original cylinder. The total radiant power emitted by the pieces is twice that emitted by the original cylinder. How many smaller cylinders are there? Give your answer as a number with no units.

Answer 1

Answer:

The total number of small cylinder = 7.

Explanation:

Lets take

Radius of the large cylinder = R

length = L

L = 10 R

The total area A = 2 π R² + π R L

The length of the small cylinder = l

The number of small cylinder = n

L = n l

The total area of small cylinders

A'=n (2 π R² + π R l)

As we know that emissive power given as

P = A ε σ T⁴

For large cylinder

P = A ε σ T⁴      -----------1

For small cylinders

P'=A' ε σ T⁴    ------2

From 1 and 2

Given that

P'= 2 P

A' ε σ T⁴ =2 A ε σ T⁴

A'=2 A       (All others are constant)

n (2 π R² + π R l) =(2 2 π R² + π R L)

n (2  R² +  R l) = (2  R² +  R L)

$n(2R^2+R\times \dfrac{L}{n}) = 2(2R^2+RL)$

L = 10 R

$n(2R^2+R\times \dfrac{10R}{n}) =2 (2R^2+R\times 10R)$

$n(2+\dfrac{10}{n}) =2( 2+ 10)$

2 n +10 = 2 x 12

2 n +10 = 24

2 n = 24 -10

2 n = 14

n = 7

The total number of small cylinder = 7.