How Do You Calculate Uncertainty in the Volume of a Cylinder?

[Necrosity]

Homework Statement

9.The diameter of a cylindrical piece of copper rod was measured at a 24.8 +/- 0.5mm with a vernier caliper. Its length was measured at 135 +/- 0.5mm with a meter rule.

a. Calculate the percentage uncertainty in the diameter.
b. Calculate the percentage uncertainty in the length
c. Calculate the percentage uncertainty in the volume of the cylinder.
d. Calculate the absolute uncertainty in the volume of the cylinder in mm^3

Homework Equations

%U = AU/Measurement (atleast that's how i do it)
V = Pi x r^2 x l
Thats all i know.

The Attempt at a Solution

I figured out:
A. %U = 0.20%
B. %U = 0.37%
C. %U of V = 0.77%

But I've got no idea about D. Needs to show full working out.

 

[Redbelly98]

Welcome to PF

Before doing D, I'll suggest re-doing A and C. Post back here, then I can help with D.

(B is correct, by the way.)

 

[nlightner]

I would like to commend you on your efforts in calculating the uncertainties in this experiment. It is important to always consider uncertainties when reporting measurements, as it helps to provide a more accurate representation of the data.

To calculate the absolute uncertainty in the volume of the cylinder, we first need to calculate the absolute uncertainties in the diameter and length. This can be done by multiplying the percentage uncertainties by their respective measurements.

For the diameter, the absolute uncertainty is calculated as 0.20% of 24.8mm, which is 0.05mm. Similarly, the absolute uncertainty in the length is 0.37% of 135mm, which is 0.50mm.

To calculate the absolute uncertainty in the volume, we use the formula for the volume of a cylinder (V = πr^2l) and take the partial derivatives with respect to each variable (r and l). This gives us the following formula for the absolute uncertainty in the volume:

ΔV = V√((Δr/r)^2 + (Δl/l)^2)

Plugging in the values for V, Δr, and Δl, we get:

ΔV = π(24.8mm)^2(135mm)√((0.05mm/24.8mm)^2 + (0.50mm/135mm)^2)

Simplifying this equation, we get:

ΔV = 1.125mm^3

Therefore, the absolute uncertainty in the volume of the cylinder is 1.125mm^3. This means that the actual volume of the cylinder could range from (V-ΔV) to (V+ΔV), giving us a range of possible values for the volume. In this case, the volume of the cylinder could range from 4905.375mm^3 to 4907.625mm^3.

I hope this helps you in understanding how to calculate uncertainties in a more complex situation. Keep up the good work!