The method and the results are on the centre of mass lab. Do the analysis yourself first, then check the answers below.
Quick answer: The centre of mass is the mass-weighted average position, xcm = (1/M)∫x dm. These answers go with the AP Physics C lab on centre of mass: the graph, the calculations, the uncertainties and the conclusion explained, for all 7 analysis questions.
Calculated values use masses in kilograms and positions in centimetres. Your own differences will vary, but they should be a few tenths of a centimetre and close to the reading uncertainty. Also, compare your explanation with the model answers below.
Level: AP Physics C (Mechanics and E&M, calculus-based) · Curriculum: AP PC:M Unit 4
xcm = (0.100 × 50.0 + 0.050 × 90.0)/(0.150) = 9.5/0.150 = 63.3 cm, compared with 63.0 cm measured.
Question 2 Complete the table
Trial 2: 9.0/0.300 = 30.0 cm. Then trial 3: (5.0 + 8.0 + 1.0)/0.300 = 46.7 cm. Trial 4: (5.0 + 10.5)/0.250 = 62.0 cm. Trial 5: (5.0 + 7.5)/0.350 = 35.7 cm.
Question 3 Differences
Measured minus calculated: −0.3, +0.3, −0.2, +0.2, −0.2 cm, which are 0.5 %, 1.0 %, 0.4 %, 0.3 % and 0.6 % of the calculated positions. The largest is 1.0 %.
Question 4 Pattern of errors
The differences have mixed signs and no trend, so they are random scatter rather than a systematic error.
Question 5 Why balancing works
Moreover, the weight acts as if at the centre of mass. With the support there, the weight and the support force have no turning effect, so the rule stays horizontal.
Question 6 Uncertainty
Also, the largest difference is 0.3 cm, which is equal to the ±0.3 cm reading uncertainty. In addition, the results are consistent with the formula within error.
Question 7 Better precision
Mark the rule with a thin pencil line and use a very narrow support, or find the balance point three times and average. Repeat and average.
