Object I is a solid cylinder with diameter d has resistance R. What would be resistance of another object made of the same material and with the same mass as Object I but has the cross section shown below (it is a hollow cylinder with outer diameter 2d and inner diameter d).

Question

Object I is a solid cylinder with diameter d has resistance R. What would be resistance of another object made of the same material and with the same mass as Object I but has the cross section shown below (it is a hollow cylinder with outer diameter 2d and inner diameter d).

R/9
9R
R
R/3
3R

Answer 1

R=ρL/A

R =4ρL/πd²
R₁ =4ρL/ (π4d²- πd²) =4ρL/ 3πd²
R₁=R/3

Answer 2

To find the resistance of the hollow cylinder, we can start by comparing its dimensions to that of the solid cylinder. The key information is that the hollow cylinder has an outer diameter 2d and an inner diameter d.

Let's assume the resistivity of the material composing both cylinders is the same.

Now, resistance is directly proportional to the resistivity of the material and the length of the object, while inversely proportional to the cross-sectional area.

For the solid cylinder (Object I), we can calculate the resistance using the formula:

R = (resistivity * length) / (cross-sectional area)

Since both cylinders have the same mass, we can assume they have the same length.

The cross-sectional area of Object I is given by:

A1 = π * (diameter/2)^2
= π * (d/2)^2
= π * (d^2/4)

Now, let's calculate the cross-sectional area of the hollow cylinder.

The outer cross-sectional area (A2_outer) is given by:

A2_outer = π * (outer diameter/2)^2
= π * (2d/2)^2
= π * d^2

The inner cross-sectional area (A2_inner) is given by:

A2_inner = π * (inner diameter/2)^2
= π * (d/2)^2
= π * (d^2/4)

The total cross-sectional area of the hollow cylinder (A2) is the difference between the outer and inner cross-sectional areas:

A2 = A2_outer - A2_inner
= π * d^2 - π * (d^2/4)
= 3π * d^2/4

Now we can calculate the resistance of the hollow cylinder using the same formula as before:

R2 = (resistivity * length) / (cross-sectional area)

Since both cylinders are made of the same material and have the same length, let's call their resistivities and lengths as resistivity_1, resistivity_2, length_1, and length_2, respectively.

R1 = (resistivity_1 * length_1) / (A1)
R2 = (resistivity_2 * length_2) / (A2)

We want to find the resistance of the hollow cylinder relative to the resistance of Object I. So, we can express R2 in terms of R1:

R2 = R1 * (A1 / A2)

Substituting the values of A1 and A2:

R2 = R1 * ((π * d^2/4) / (3π * d^2/4))
= R1 * (1/3)

Therefore, the resistance of the hollow cylinder (R2) is R1/3.

So, the correct answer is R/3.