Experimenting with pendulums, you attach a light string to the ceiling and attach a small metal sphere to the lower end of the string. When you displace the sphere 2.00 m to the left, it nearly touches a vertical wall; with the string taut, you release the sphere from rest. The sphere swings back and forth as a simple pendulum, and you measure its period T. You repeat this act for strings of various lengths L, each time starting the motion with the sphere displaced 2.00 m to the left of the vertical position of the string. In each case the sphere€™s radius is very small compared with L. Your results are given in the table:

(a) For the five largest values of L, graph T² versus L. Explain why the data points fall close to a straight line. Does the slope of this line have the value you expected?

(b) Add the remaining data to your graph. Explain why the data start to deviate from the straight-line fit as L decreases. To see this effect more clearly, plot T/T₀ versus L, where T₀ = 2Ï€ˆšL/g and g = 9.80 m/s².

(c) Use your graph of T/T₀ versus L to estimate the angular amplitude of the pendulum (in degrees) for which the equation T = 2Ï€ˆšL/g is in error by 5%.

12.00 10.00 8.00 6.00 5.00 4.00 3.00 2.50 2.30 L (m) 6.36 5.70 4.95 4.54 4.08 3.60 3.35 3.27 T (s) 6.96