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.

 

[HalfLostAndSearching]

As a scientist, it is important to acknowledge and account for uncertainties in any experimental data. In this case, the uncertainty in the acceleration due to gravity calculation should be propagated from the uncertainties in the measurements of the object's speed and the angle of the air-track.

To do this, you can use the formula for propagating uncertainties in a division operation:

δg = g * √[(δa/a)^2 + (δsin(x)/sin(x))^2]

Where δg is the uncertainty in the calculated value of g, g is the calculated value of g, δa is the uncertainty in the measured acceleration, and δsin(x) is the uncertainty in the measured angle.

Since you do not have the uncertainty for the measured angle, you can use a conservative estimate of ±0.5 degrees for δsin(x). This value can be adjusted based on the precision of your measurement equipment and any potential sources of error in the measurement process.

Using this formula, you can calculate the uncertainty in g as (61.034 cm/s2) * √[(2.227/61.034)^2 + (0.5/3.523)^2] = ±0.227 cm/s2. This means that the final value for acceleration due to gravity would be (61.034 ± 0.227) cm/s2.

It is important to note that the uncertainty in the angle measurement has a larger impact on the overall uncertainty in the calculation of g, highlighting the importance of precise measurements in scientific experiments.

In conclusion, to properly propagate uncertainty in the calculation of acceleration due to gravity, it is necessary to consider the uncertainties in both the measured acceleration and the angle of the air-track. Using the formula provided and a conservative estimate for the uncertainty in the angle measurement, you can determine the uncertainty in g and report it along with the calculated value.