Rolling ball

You have first to obtain a relationship between the linear and angular speeds of the ball. If the ball were simply rolling, it would advance 2pR for each rotation. Suppose one rotation takes T. We have w=2p/T and v=2pR/T. Eliminating T will yield the required relation. But the ball is rolling on a smaller circle than its circumference.

The diagram shows the real situation (although it shows it horizontal and rectangular for convenience). You should be able to use the orange measurements + Pythagorus to calculate the green one, which is the rolling radius r as opposed to the actual radius R used above. Hence you can work out your relation between the linear and angular speeds.

The basic equation is

PE lost = translational KE gained + rotational KE gained + energy lost to frictional heating

mgh = ½mv² + ½Iw²  + Fl

where l is the down-the-slope distance rolled by the ball. In measuring l and h, you need to be careful about the various radius offsets. You may remember that we argued that, for uniform acceleration down the slope, the average velocity (l/t) is half the final velocity v. You will also remember that we planned to use 2mR²/5 for the moment of inertia I. I suggest that, at first at any rate, you treat F as a constant, albeit unknown (it could ultimately be obtained, perhaps, from the graph's intercept).

What you have to do is to massage all these ideas algebraically to obtain an equation of the form

p(h) = A ´ q(t) +B

where A and B are constants (or collections of constants), while p() and q() are functions of the two measured variables (I call these pseudo-variables). Then you plot a graph of p(h) against q(t). This will be a straight line whose gradient is A. If you have done your calculations properly, A will contain g and you will be able to obtain its value.

If you would like to see a worked example of the pseudo-variable idea, click here.

Your table should have uncertainties attached both to the raw variables and the pseudo-variables. These get translated into error bars when you plot the graph. Then you are able to draw steepest and shallowest tenable lines and these, in turn, enable you to attach uncertainty limits to the gradient and hence the final value of g.

Dealing with Uncertainty on my Physics home page gives more information on these ideas, although it is not yet finished.