In the experiment, the centripetal force was provided by the horizontal component of the tension, T sin(θ). The actual radius of the stopper's circle was not the length of the string, but L sin(θ). So it appears as though the angle should be critically important to the experimental value of the string's tension but we did not make any attempt to measure its value. Let's now show you why the value of the angle was theoretially not important to our error calculations.
Using net F = ma, we can write T sin(θ) = mstopper ac = mstopper [4 π2 r f2 ]
In this equation T represents the tension in the string attached to the stopper and f represents frequency (rev/sec) at which the stopper moved along its circular path. But remember that the radius of the circular path, r, is equal to L sin(θ). Substituing in this value for the radius allows us to cancel the value of sin(θ) from both sides of the equation.
Our final equation then becomes Tension = mstopper [4 π2 L f2 ] which is the slope of your lines from the graphs of
v2 vs string length