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).
To find the resistance of the hollow cylinder, we need to consider its dimensions and the resistivity of the material it is made of.
First, let's analyze the resistances of both the solid and hollow cylinders separately.
1. Solid Cylinder (Object I):
Given: Diameter of the solid cylinder = d
Resistance of the solid cylinder = R
The resistance of the solid cylinder can be calculated using the formula:
R = (ρ * L) / A
Where:
- ρ is the resistivity of the material,
- L is the length of the cylinder, and
- A is the cross-sectional area of the cylinder.
Since the solid cylinder is hollow with no inner diameter, its cross-sectional area can be found using the formula:
A = π * (d/2)^2
Substituting the values, we get:
R = (ρ * L) / (π * (d/2)^2)
2. Hollow Cylinder (Object II):
Given: Inner diameter of the hollow cylinder = d
Outer diameter of the hollow cylinder = 2d (from the cross-section)
To find the resistance of the hollow cylinder, we need to calculate the cross-sectional area of the annular ring. The area of the annular ring can be found by subtracting the area of the inner circle from the area of the outer circle.
Area of annular ring = π * [(2d/2)^2 - (d/2)^2]
= π * [(d^2) - (d^2/4)]
= π * (3d^2/4)
Now, using the resistance formula mentioned earlier,
R = (ρ * L) / A
Substituting the values, we get:
R = (ρ * L) / [π * (3d^2/4)]
So, the resistance of the hollow cylinder is (ρ * L) / [π * (3d^2/4)].
Note: To further simplify the equations, you may need additional information like the length of the cylinder or the resistivity of the material.