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Can I measure the value of g using only a metre stick and a ball? I am not supposed to use a stopwatch and that has been the problem. NOTE: I do not know if a solution exists or not.

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As @Ron has answered we can not measure value of g using a stick and a ball; however if we take unit of length to be stick's length and define one "stick time" to be amount of time required for anything to fall from rest from a height equal to length of given stick, then value of g in these units would be 2. – user10001 Aug 31 '12 at 4:05
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No you can't. You can see this because you are only given things that can define a units of length and mass (the meterstick and the ball), so you need something that can define the unit of time. If there was another process, nongravitational, with which you could define a unit of time, then you can find g relative to this unit of time, but absent such a thing, you can only define the unit of time by dropping something or measuring something oscillate in gravity, and then you are stuck.

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Galileo used his heartbeat to measure the period of oscillations of a chandelier. You can also use your heartbeat as a kind of stopwatch. You make a pendulum from a stick and a ball and let it swing with a small amplitude. The period of oscillations is:

$$T = 2 \pi \sqrt{\frac{l}{g}}$$

Thus $g$:

$$g = \frac{4 \pi^2 l}{T^2}$$

To find $T$ you let the pendulum swing $n$ times and count the number of your heartbeats $N$. Then $T = N /n$ in heartbeats.

However note, that your $g$ would have dimensions of $\frac{\text{meter}}{\text{heart beat}^2}$. And you can't translate heart beats into seconds without a clock of some kind.

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