d1 = 2.53 cm +/- .05 cm
d2 = 1.753 m +/- .001 m
0 = 23.5 degrees +/- .5 degrees
v1 = 1.55 m/s +/- .15 m/s
Using the measured quantities above, calculate the following. Express the uncertainty calculated value.
Z = 4d1 (cos (0))^ 2
http://idol.union.edu/vineyarm/teaching/phy17/uncertainties_intro.pdf
start with cosTheta
uncertainity is diff between Cos(23.5)and cos(.5)= .5sin23.5=.199
now it is squared, so double it.
error is appx 0.4
now in the expression, it is multiplied by 4d, so add the error in 4d or4*.05
now adding all the errors... +-.6
To calculate Z = 4d1 (cos(θ))^2, we need to substitute the given values and uncertainties into the formula. Let's break down the process step by step.
Step 1: Calculate the value of cos(θ).
θ = 23.5 degrees +/- 0.5 degrees
To calculate the value of cos(θ), we need to use the cosine function. However, most calculators use radians as the default unit, so we need to convert degrees to radians first. Since one complete revolution is 360 degrees or 2π radians, we have:
θ_radians = θ_degrees * π / 180
θ_radians = 23.5 degrees * π / 180 = 0.410 radians +/- (0.5 degrees * π / 180) = 0.009 radians
Now we can calculate the value of cos(θ):
cos(θ_radians) = cos(0.410 radians) = 0.921 +/- 0.009
Step 2: Substitute the values of d1 and cos(θ) into the formula Z = 4d1 (cos(θ))^2.
d1 = 2.53 cm +/- 0.05 cm
Z = 4 * (2.53 cm +/- 0.05 cm) * (0.921 +/- 0.009)^2
Step 3: Perform the calculations.
To calculate the uncertainty of Z, we need to consider both the uncertainties in d1 and cos(θ) separately.
Uncertainty of Z = |4 * d1 * cos(θ)^2| * √((Δd1/d1)^2 + (2Δcos(θ)/cos(θ))^2)
Using the given uncertainties:
Δd1/d1 = 0.05 cm / 2.53 cm = 0.0198
Δcos(θ)/cos(θ) = 0.009 / 0.921 = 0.0098
Now plug in these values to calculate the uncertainty of Z:
Uncertainty of Z = |4 * 2.53 cm * (0.921)^2| * √((0.0198)^2 + (2 * 0.0098)^2)
Finally, calculate the numerical value of Z with uncertainty:
Z = 4 * 2.53 cm * (0.921)^2 +/- Uncertainty of Z
Remember to round the final result to an appropriate number of significant figures based on the given uncertainties.