I'm trying to solve a Physics problem. I already got the answer correct but I'm unable to do the uncertainties. Can someone please explain how to do in this case?
These are the variables used.
g = 9.802 m/s2 +/- 0.0005 m/s2
m2 = 951.0 g +/- 2.3 g
θ = 31.90º +/- 0.07º
And this is what I need to do (only thing left is uncertainties):
total weight on hook 2:
w2 = 9.32 N +/- ____ N
(9.32 = 0.9510 * 9.802)
magnitude of the force on hook 1:
F1 = 7.91 N +/- ____ N
(7.91 = 9.32 * cos(31.90))
magnitude of the total force on hook 3: F3 = 4.93 N +/- ___ N
(4.93 = 9.32 * sin(31.90))
The uncertainties need to have 3 significant digits
To calculate the uncertainties in this case, you can use the concept of error propagation. The idea is to propagate the uncertainties of the variables through the mathematical operations involved in the calculations. Here are the steps to calculate the uncertainties for each case:
1. Total weight on hook 2 (w2):
w2 = m2 * g
To calculate the uncertainty in w2, we need to account for the uncertainties in m2 and g. The formula for error propagation in multiplication is:
∂w2 = |w2| * sqrt((∂m2/m2)^2 + (∂g/g)^2)
Using the given values:
∂m2 = 2.3 g
∂g = 0.0005 m/s^2
Plug these values into the formula:
∂w2 = |0.951 * 9.802| * sqrt((2.3/951)^2 + (0.0005/9.802)^2)
Evaluate the expression to find the uncertainty ∂w2.
2. Magnitude of the force on hook 1 (F1):
F1 = w2 * cos(θ)
Again, we need to account for the uncertainties in w2 and cos(θ) using the error propagation formula:
∂F1 = |F1| * sqrt((∂w2/w2)^2 + (∂(cos(θ))/cos(θ))^2)
Using the given values and the previously calculated ∂w2:
∂(cos(θ)) = ∂(cos(31.90º)) = 0.07º
Plug these values into the formula:
∂F1 = |7.91| * sqrt((∂w2/9.32)^2 + (0.07/(cos(31.90º)))^2)
Evaluate the expression to find the uncertainty ∂F1.
3. Magnitude of the total force on hook 3 (F3):
F3 = w2 * sin(θ)
Using the error propagation formula:
∂F3 = |F3| * sqrt((∂w2/w2)^2 + (∂(sin(θ))/sin(θ))^2)
Using the given values and the previously calculated ∂w2:
∂(sin(θ)) = ∂(sin(31.90º)) = 0.07º
Plug these values into the formula:
∂F3 = |4.93| * sqrt((∂w2/9.32)^2 + (0.07/(sin(31.90º)))^2)
Evaluate the expression to find the uncertainty ∂F3.
By following these steps and plugging in the appropriate values, you will be able to calculate the uncertainties for each of the given cases.