What is the largest possible error in the component?

[jesuslovesu]

[SOLVED] Error Calculation

Homework Statement

A force 500nt is measured with a possible error of 1nt.
Its component in a direction 60 deg away from its lnie of action is required. Where the angle is subject to an error of .5 deg. What is the largest possible error in the component?

Homework Equations

The Attempt at a Solution

I was thinking 1/500 + .5/60 would bring the percentage and then if I multiply % * 500 I would get the amount of "nt"s (what's an nt, a Newton?) however that gives a result that is a little too large. I would imagine my problem lies with the angle.. do I need to do something special with it?

 

[HallsofIvy]

you can't just add the relative error (0.5/60) because it is the sine of the angle, not the angle itself, that is added. If this were not in "precalcus, I woud recommend approxmating the error in the sine by $dy= (\pi/180) cos x dx$. (If y= sin(x) then dy/dx= cos(x) so to a linear approximation, the error in sin(x) is cos(x)dx. The reason for for the $\pi/180$ factor is that x is measured in degrees not radians.)

If you are not familiar with using the derivative to approximate a function, you can do a direct calculation of the maximum error. The largest possible values for the nts measurement (yes, that's "Newtons") is 501 and the largest possible angle is 60.5 degrees. Do the calculation for that. The smallest possible corresponding values are 499 nts and 59.5 degrees do the same calculation for that. Which is those is the largest deviation from the value for 500 nts and 60 degrees? That is the maximum error.

 

[Architeuthis Dux]

I would like to clarify a few things before providing a response. First, an "nt" is not a standard unit of measurement. It is possible that it was a typo and should be "N" for Newtons, which is the standard unit for force. Second, it is important to specify the type of error being referred to. In this case, it seems like the error is most likely referring to uncertainty or precision in the measurement.

To calculate the largest possible error in the component, we can use the formula for error propagation, which takes into account the uncertainties in both the force measurement and the angle measurement. The formula is:

δF = F√[(δF/F)^2 + (δθ/θ)^2]

Where δF is the uncertainty in the component force, F is the measured force, δθ is the uncertainty in the angle, and θ is the measured angle.

Plugging in the values given in the problem, we get:

δF = 500√[(1/500)^2 + (0.5/60)^2] = 0.17 Nt (assuming the typo and using N for Newtons)

Therefore, the largest possible error in the component is 0.17 Nt. This means that the actual value of the component force could be anywhere between 499.83 Nt and 500.17 Nt.