How Do You Calculate Uncertainty in Physics Equations?

[~Sam~]

Homework Statement

What are the uncertainty propagation formulas for:
Area of a rectangle
Density of a sphere
Height of an opposite side wall calculated by tan(θ)=o/a
Resistance

Homework Equations

Area of rectangle= side A *side B
Density of a sphere= mass/(4/3piR^3)
tan(θ)=opp/adj
Resistance= R=V/I

The Attempt at a Solution

Rectangle I have: A=LW
dA=dlW+dwL
dl and dw are the relative uncertainty of L and W by instrument.

Sphere: I have dp=(3 dm/4piR^3)+(3m/12piR^2dr)
dm and dr are uncertaintities of mass and radius

For heigh: do= tan(θ)da+sec²(θ)a*dtan(θ)
da and dtan(θ) are the relative uncertainties

For resistance: dR= 1dV/I+ VdI/1

If these are correct would I use a sum formula like u(c)= sum of all squares? or would I separate the partial derivatives from the uncertainties and then square and sum like (dl²)W²+(dw²)L²? I'm kinda confused which formula is the general formula for uncertainty propagation.

Homework Statement

Homework Equations

The Attempt at a Solution

 

[jbsteele]

For the area of a rectangle, the uncertainty propagation formula is: dA = dL * W + dW * Lwhere dL and dW are the relative uncertainties of the length and width of the rectangle, respectively. For the density of a sphere, the uncertainty propagation formula is: dp = (3 dm / 4πR3) + (3m / 12πR2 dr)where dm and dr are the relative uncertainties of the mass and radius of the sphere, respectively.For the height of an opposite side wall calculated by tan(θ) = o/a, the uncertainty propagation formula is: do = tan(θ) * da + sec2(θ) * a * dtan(θ)where da and dtan(θ) are the relative uncertainties.For the resistance, the uncertainty propagation formula is: dR = (1 dV / I) + (V dI / 1)where dV and dI are the relative uncertainties of the voltage and current, respectively. In all cases, the general formula for uncertainty propagation is to square and sum each individual uncertainty term, i.e. u(c) = Σi(di2).

 

[kerimek]

The uncertainty propagation formulas for the given quantities are as follows:

1. Area of a rectangle:
The uncertainty in the area of a rectangle (dA) can be calculated using the formula:
dA = L*dW + W*dL
where L and W are the lengths of the sides of the rectangle and dL and dW are the relative uncertainties in the measurements of L and W, respectively.

2. Density of a sphere:
The uncertainty in the density of a sphere (dp) can be calculated using the formula:
dp = (3*dm)/(4*pi*R^3) + (3*m*dR)/(12*pi*R^2)
where m is the mass of the sphere, R is the radius of the sphere, and dm and dR are the relative uncertainties in the measurements of m and R, respectively.

3. Height of an opposite side wall calculated by tan(θ)=o/a:
The uncertainty in the height of the opposite side wall (do) can be calculated using the formula:
do = tan(θ)*da + sec^2(θ)*a*dθ
where a is the length of the adjacent side and dθ and da are the relative uncertainties in the measurements of θ and a, respectively.

4. Resistance:
The uncertainty in the resistance (dR) can be calculated using the formula:
dR = (1/I)*dV + (V/I^2)*dI
where V is the voltage and I is the current, and dV and dI are the relative uncertainties in the measurements of V and I, respectively.

To calculate the overall uncertainty for each quantity, you can use the sum formula:
u(c) = sqrt(dL^2 + dW^2 + dR^2 + dθ^2 + dV^2 + dI^2)
where each d represents the individual uncertainties calculated above. This formula takes into account all the partial derivatives and uncertainties, and provides a more accurate estimation of the overall uncertainty.