A Trolley on a Ramp: Choosing What to Measure
A student is investigating how the acceleration of a trolley depends on the angle of a ramp. The ramp is 1.20 m long and can be tilted using books. The trolley has a mass of 0.50 kg. A light gate at the bottom of the ramp measures the time for a 5.0 cm card on the trolley to pass. The student plans to release the trolley from rest at the top and measure the time to reach the bottom with a stopwatch, then calculate acceleration using s = ½at². Preliminary trials give times of 1.4 s, 1.9 s, 1.6 s and 2.4 s for the same ramp angle. The student also notices the trolley sometimes veers sideways and rubs the ramp edge.
1. Evaluate the student's plan for measuring acceleration and recommend improvements, explaining the reasoning.
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Using a stopwatch to time the run is not reliable here. The spread of preliminary times (1.4 s to 2.4 s) is far too large for a single ramp angle. Human reaction time is typically 0.2–0.3 s, which is a large fraction of a 1.6 s run, and the student must both release the trolley and start the watch, so the two actions cannot be perfectly synchronised.
A better approach uses the light gate. The card of length 5.0 cm passes the gate, and the gate records the time; the instantaneous speed at the gate is v = card length / time. If a second gate is placed near the top, the acceleration can be found from a = (v₂ - v₁)/t, where t is the time between gates. This removes reaction time from the measurement.
Alternatively, if only one gate is available, the student can still use s = ½at² provided the release is triggered electronically, or use a ticker-timer. The veering problem must also be fixed: the trolley rubbing the edge adds friction that varies between runs. The ramp should be levelled sideways and a straight guide used. Each angle should be repeated at least three times and a mean taken.
2. The student wants to test whether acceleration is proportional to the sine of the ramp angle. Describe how the data should be processed and what graph would show this, including how friction would affect the result.
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For each ramp angle θ, the student should measure the height h raised at one end and the ramp length L, then calculate sin θ = h/L. The acceleration for each angle is found from the light-gate speeds or from a reliable timing method. Repeating each angle and averaging reduces random error.
The test for proportionality is a graph of acceleration against sin θ. If acceleration is proportional to sin θ, the points should lie on a straight line through the origin. The gradient of the line should be close to g (about 9.8 m/s²) if friction is negligible.
Friction changes the result in a predictable way. The net force along the ramp is mg sin θ minus the friction force f, so a = g sin θ - f/m. On a graph of a against sin θ, friction shifts the line downward but does not change its gradient much, and it produces a small negative intercept. This means the line will not pass exactly through the origin: the trolley will not move until sin θ exceeds a small threshold value.
The student should compare the measured gradient with 9.8 m/s² and comment on the difference. A gradient noticeably below 9.8 m/s² suggests friction or an unresolved timing error; a gradient close to 9.8 m/s² with a small negative intercept is consistent with friction acting as expected.