A Trolley That Will Not Accelerate as Predicted
A dynamics trolley of mass 0.800 kg is on a bench, connected by a string over a pulley to a hanging mass of 0.200 kg. The string is intended to be parallel to the bench. A card of width 0.100 m on the trolley passes through two light gates 0.500 m apart; the card interrupts the first gate for 0.167 s and the second for 0.091 s. Using Newton's second law, the class predicts a = 0.200 x 9.81/(0.800 + 0.200) = 1.96 m/s^2. The measured value is 0.85 m/s^2. A learner says the experiment proves Newton's second law is wrong. The teacher suspects the bench is not level, but the learner argues that friction is the obvious cause. The pulley is known to have negligible friction.
1. Show how the measured acceleration is obtained from the gate data and calculate the percentage difference from the prediction.
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v1 = 0.100/0.167 = 0.599 m/s; v2 = 0.100/0.091 = 1.099 m/s. Assuming uniform acceleration over 0.500 m, a = (v2^2 - v1^2)/(2s) = (1.208 - 0.359)/1.000 = 0.849, approximately 0.85 m/s^2. Predicted a = 0.200 x 9.81/1.000 = 1.962 m/s^2. Difference = 1.962 - 0.849 = 1.113 m/s^2; percentage = 1.113/1.962 x 100 = 56.7%, about 57% low.
2. Distinguish between friction and a tilted runway as explanations. Describe one measurement that would separate them.
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The missing force on the 1.000 kg system is F = ma = 1.000 x 1.113 = 1.11 N. Friction on the trolley could supply this: normal reaction = 0.800 x 9.81 = 7.85 N, so the coefficient would be 1.11/7.85 = 0.14, which is high for a dynamics trolley but not impossible. A tilted bench is equally plausible: to add a 1.11 N component along the bench, sin(theta) = 1.11/(0.800 x 9.81) = 0.141, theta is about 8 degrees. That is large enough to see, but a smaller tilt over part of the bench could still matter. Separate them by removing the string and giving the trolley a push with no hanging mass: if it decelerates at about 1.4 m/s^2, friction is sufficient; if it decelerates much less, or the acceleration changes when the trolley is run the other way, the bench is tilted. A spirit level checks the bench directly.
3. What should the class conclude, and what single improvement would most strengthen the investigation?
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The class should conclude that an unaccounted external force is acting, not that Newton's second law has failed. The law applies to the net force; the prediction omitted friction and any component of weight along the bench. The most strengthening single change is to level the bench and verify it with a spirit level, then repeat with the same masses and gates. Also check that the string is parallel, the pulley turns freely, and the card times are not affected by the trolley rocking. If the measured acceleration then agrees with 1.96 m/s^2 within about 5%, the original discrepancy is explained by the apparatus.