Propagating uncertainty when calculating acceleration due to gravity?

[Zane]

Hello, I'm having trouble with a lab report. The experiment conducted was we used an angled air-track and a timer to determine the speed at which an object slid down the track and its acceleration.

The final average acceleration we calculated was (61.034 +- 2.227)(cm/s²)

We're then given a formula to calculate gravitation acceleration from this figure: g=acceleration/(sinx)

Where x is the angle of the air-track, let's say 3.523 degrees.

How do I propagate uncertainty for this? I can calculate g easily, but I don't understand how I'm supposed to find a value for the +- bit. I don't know the uncertainty of the measured angle. My best guess would be that since I do not know the uncertainty of X, and thus I don't know the uncertainty of sin(x), I treat sin(x) like a precise number and divide acceleration's uncertainty by it to determine the uncertainty of g. Is this correct? If not, how do I do it?

 

[Simon Bridge]

In general, if $z=f(x)$ where the uncertainty on x is $\sigma_x$ then $$\sigma_z=\frac{df}{dx}\sigma_x$$
This means the error on sin(x) is the same as the error on x, if the angle is very small.

In general, for small angles $\sin\theta \approx \theta$ where the angle is in radians.

 

[Zane]

In general, if $z=f(x)$ where the uncertainty on x is $\sigma_x$ then $$\sigma_z=\frac{df}{dx}\sigma_x$$
This means the error on sin(x) is the same as the error on x, if the angle is very small.

In general, for small angles $\sin\theta \approx \theta$ where the angle is in radians.

Interesting. For the experiment, we used a meter stick to determine the length/height of the device, so we were only able to measure to the nearest milimeter. Does this mean there is an implied uncertainty of .05cm? From the length/height we used trig functions to calculate the angle of 3ish degrees, so do I then propagate that error as I would with multiplication/division to find the uncertainty in the angle?

 

[Simon Bridge]

Oh I get you.

From your kinematics coursework you should know that the acceleration of a block on a frictionless ramp inclined angle $\theta$ to the horizontal is given by $a=g\sin\theta$. Therefore $$g=\frac{a}{\sin\theta}$$Why did you calculate the angle? You don't need it.

If you measured the length of the track L and the height you lifted the end above the table h, then your trigonometry tells you that $$\sin\theta=\frac{h}{L}\implies g=\frac{aL}{h}$$... you should be able to propagate those errors.

If you measured a length along the table x and a height to the ramp y, the $$\tan\theta = \frac{y}{x}$$
But for small angles, $\tan\theta\approx\sin\theta$ ... it is likely that the difference between the tangent and the sine of the angle is smaller than the uncertainties involved.