Trolley Acceleration on an Incline: A Flawed Assumption
A student investigates the acceleration of a 0.500 kg trolley down an inclined track. The track is raised at one end to a height of 0.150 m over a length of 1.200 m. Two light gates are placed 0.800 m apart along the track. The trolley is released from rest from a point above the first gate. The times recorded for the trolley to travel between the gates are: 0.452 s, 0.448 s, 0.455 s, and 0.450 s. The student assumes the trolley starts from rest at the first gate and uses s = ½ a t² to calculate an acceleration of 7.86 m/s². The theoretical acceleration down a frictionless incline is g sin θ, where sin θ = 0.150/1.200. Taking g = 9.81 m/s², the theoretical acceleration is 1.23 m/s². The student is puzzled by the large discrepancy. They then add 0.200 kg to the trolley and repeat, obtaining a mean time of 0.470 s and an acceleration of 7.25 m/s². They wonder if friction is significant.
1. Calculate the theoretical acceleration down the frictionless incline and compare with the student's calculated value. Explain why the student's value is much larger.
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The theoretical acceleration is a_theory = g sin θ = 9.81 × (0.150 / 1.200) = 9.81 × 0.125 = 1.226 m/s². The student calculates a_student = 2s / t² = 2 × 0.800 / (0.45125)² = 7.86 m/s², using the mean time t = (0.452 + 0.448 + 0.455 + 0.450)/4 = 0.45125 s. The student's value is about 6.4 times larger than the theoretical value.
This large overestimate arises because the student incorrectly assumes the trolley starts from rest at the first gate (u = 0). In reality, the trolley is released from rest above the first gate, so it already has a significant velocity when it passes the first gate. The correct kinematic equation is s = u t + ½ a t², where u > 0. Setting u = 0 attributes the entire displacement to acceleration from rest, which overestimates the acceleration. With a positive initial velocity, the trolley covers the 0.800 m in a shorter time than if it started from rest, so using ½ a t² yields an artificially high value for a.
2. Identify the likely error in the student's method and propose a correct experimental procedure to measure the acceleration.
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The likely error is the assumption that the trolley is at rest at the first gate. A correct procedure would be to release the trolley from rest exactly at the first gate (e.g., using a release mechanism aligned with the gate), so that u = 0 and s = ½ a t² is valid. Alternatively, the student could measure the trolley's velocity at each gate by attaching a card of known length and using the gate's timing, then calculate acceleration using v² = u² + 2as. Another method is to use three gates and measure the time between successive gates to find the change in velocity and hence acceleration. The student should also ensure the track is clean and the trolley moves freely, and repeat the measurement several times to reduce random error.
3. Given the student's data, assess whether it is possible to determine the coefficient of friction. If not, explain what additional measurements are needed.
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It is not possible to determine the coefficient of friction reliably from the given data because the acceleration values are fundamentally flawed due to the incorrect assumption about initial velocity. The student's calculated accelerations (7.86 and 7.25 m/s²) are much larger than the theoretical frictionless value (1.23 m/s²), so they cannot be used to infer friction. Even the observed decrease in acceleration when mass is added (from 7.86 to 7.25 m/s²) is unreliable because it is based on the same flawed method.
To determine the coefficient of friction, the student must first correct the experimental procedure so that acceleration is measured accurately. Then they should measure the acceleration for at least two different masses (or two different incline angles) to check consistency. For a trolley on an incline with friction, the acceleration is a = g(sin θ − μ cos θ). Using the known angle θ (sin θ = 0.125, cos θ ≈ 0.992), they can solve for μ = (g sin θ − a) / (g cos θ). They would also need to ensure that other forces (e.g., air resistance) are negligible and that the trolley's wheels are rolling freely.