E-8

Pre-lab exercises

1. By trial and error with a calculator: (set q in radian), for example, q=0.3 rad yields sinq=0.29552 for a relative error of 0.0015=15%. Proceeding in this manner, one finds = 0.53 rad gives a 5% error. Another way to proceed is to plot (q - sinq)/sinq. The error is 5% at q = 30.7°
   
2. % error = 100[T(q_max) - T(0)]/T(q_max)
   where T(q_max) = 2pÖ(L/g)[1 + 1/4sin²(q_max/2] and T(0) = 2pÖ(l/g). If we neglect the correction term in the denominator which would contribute to order sin⁴q_max

Calculation

1. The length of the pendulum(L) = one half of the thickness , h of the bob + the length of the line
2. Compute mean values and standard diviations of the oscillation times and divide by 50 to get mean values and standard diviations of the period, T.
3. For small amplitudes, use eq 8-3 to determine values of g
4. For l near 95cm, compute the ratio of your periods at small and large amplitudes. Substitude 30° into eq 8-5 and compare with your result.

Error analysis: Do not forget to apply eq 1-10 to estimate error s_g

Presentation of the data

Re-write eq 8-3 as

The standard deviation in T² is 2Ts_T