I've tried measuring acceleration due to gravity, using a photogate with free falling picket fence.
I've also tried photogate with picket fence on a low-friction cart and inclined plane.
So far, the latter has provided slightly less varied results, but in both cases I've found it very difficult to reduce the variation in my results.
What other experiment can be setup at home, to produce more consistent results?
Using this pendulum and a tape measure plus cell phone stopwatch we calculated $g=9.7\pm 0.4 m/s^2$ using the formula for period length
$T=2\pi \sqrt{ l/g}$ to calculate $g=l(2\pi/T)^2$.
With $l=1.38m$ and $T=(23.72/10)s$
Hints: period length is independent of starting point so just count several periods and stop the time for those cumulatively. Also this formula only holds for small angles so keep your starting angle below 10° or so.
A good thing about this pendulum is that the rock is pretty heavy which mitigates air resistance a bit (the wind does not help though...).
Improvement ideas: using a smaller starting angle both helps reduce air resistance, because it's proportional to velocity (to some power...), as well as improve accuracy because of the small angle approximation. Downside of smaller angle is more out of plain of swing motion though, so that is bad.
(Error estimate sponsored by @Paul T assuming bad reaction time of 0.3s on both ends of measuring $T$ and 2cm inaccuracy on measuring $l$)
As suggested by @Kulambo's answer and @Farcher's comment a pendulum is a great method. There's a long history of pendulum gravimeters dating back to the 1600's.
The standard period of a pendulum $T = 2\pi \sqrt{L/g}$ depends on the small angle approximation, but is independent of the mass of the pendulum bob. The uncertainty in your determination of $g$ will depend on how well you can measure $L$ and $T$.
$$\delta g = g \sqrt{ \left(\frac{\delta L}{L}\right)^2 + 2\left(\frac{\delta T}{T}\right)^2}$$.
The small angle approximation $\sin\theta \approx \theta$ is good to about $2$% for $\theta = 20^\circ = 0.35$ rad. So keep your swing angle smaller than that.
Remember $L$ is the length from the pivot to the center of oscillation of the pendulum. With a light string and a small, heavy bob this will be very close to the center of mass of the bob. It is better to fix the string in place with a clamp, instead of tying it. This way it won't slide, so there won't be much frictional loss at the pivot. It's also easier to precisely locate the true pivot.
To better determine $T$, you can take the average of many swings. The biggest uncertainty will be your ability to press start and stop on your stopwatch at the correct times. If you time the pendulum making five complete swings then divide by five, that effectively cuts the start/stop error by a factor of five.
A $1$ m pendulum should have a period of about $2$ sec. Lets say you determine the pivot to center of oscillation length, $L$, to $1$ cm. Then you time five swings ($5T = 10$ sec) with $0.5$ sec of start/stop uncert. This will get you a determination of $g$ with better than $10$% precision.
$$\delta g = (9.8\,\mathrm{m/s}^2) \, \sqrt{\left( \frac{0.01\,\mathrm{m}}{1.00\,\mathrm{m}} \right)^2 + 2 \left( \frac{0.1\,\mathrm{s}}{2.0\,\mathrm{s}} \right)^2} \approx 0.7 \,\mathrm{m/s}^2$$
The limiting factor will likely be your ability to use the stop watch. You could do much better using your photogate. If you can get down to $0.01$ sec of start/stop precision on one swing, the measurement uncertainty will get close to $1$%. This would make the systematic error from the small angle approximation in the pendulum model take over as the dominate source of uncertainty.
The lack of friction and air resistance in the model may also be a factor if you are trying to get to the $1$% level. Historically, precision pendulums were operated in evacuated, low pressure chambers to reduce air resistance. Candela et al 2000 claim that one can measure $g$ to parts in $10^4$ using a Bessel pendulum. They also discuss how combining data from both pivots of a reversible pendulum causes the air resistance errors to cancel when calculating $g$. (The article is paywalled, but you can probably get it with the help of your local library.)
Put up a 10m (measured precisely) length of cheap pipe. Fill it with distilled water. Measure the pressure at the bottom. (Amazon has sufficiently accurate gauges for about $13) For extra accuracy, measure the temperature and correct the water density.
$$p_{gauge} = (\rho_{water} - \rho_{air}) g h$$
Error estimate:
The pressure measurement dominates by far, so 1% accuracy seems within reach for a reasonable cost.
One can use various smartphone sensors to measure $g$. See, e.g.,The Physics Teacher 59, 584 (2021); https://doi.org/10.1119/5.0056573
Abstract:
The use of sensors in smartphones to do physical experiments is a boom in the past decade, such as acceleration sensor,1,2 light sensor,3,4 magnetometer,5,6 camera,7,8 and gyroscope. However, few people study the application of proximity sensors in physical experiments, although, in our view, the employment of the proximity sensor is more accurate than other sensors (see Table I). For further research, this paper proposes a method to measure the oscillation period of a simple pendulum based on the proximity sensor integrated in the smartphone and determines the experimental value of the gravitational acceleration.
EDIT (8/6/2022): The cited article provides data on errors for $g$ as measured by 1. acceleration sensor; 2. light sensor; 3. magnetometer; 4. gyroscope: 1.40%; 0.71%; 0.20%; 0.94%, respectively ($g$= 9,66; 9.72; 9.77; 9.70 m/s^2, respectively).
