Trolley on a Compressed Spring: Energy Bookkeeping
A 0.60 kg dynamics trolley sits against a compressed spring of spring constant 240 N/m, held so the spring is compressed by 0.080 m. The trolley is released on a horizontal track. A light gate measures the trolley's speed just after it leaves the spring as 1.4 m/s. The spring's own mass is negligible and the track is horizontal. A student predicts the speed using conservation of energy and is surprised the measured value is lower than predicted. The same apparatus is then tilted so the track rises at 8 degrees above the horizontal, and the trolley is released from rest again with the same compression. Friction is described by the teacher as 'small but not zero'.
1. Calculate the speed predicted by a frictionless energy model and explain, with reference to the measured value, whether the data are consistent with a small frictional loss.
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Elastic potential energy stored: E = ½kx² = ½ × 240 × (0.080)² = ½ × 240 × 0.0064 = 0.768 J.
If all of this becomes kinetic energy: ½mv² = 0.768, so v² = 2 × 0.768 / 0.60 = 2.56, giving v = 1.6 m/s.
The measured speed of 1.4 m/s corresponds to kinetic energy ½ × 0.60 × 1.4² = 0.588 J. The 'missing' energy is 0.768 − 0.588 = 0.180 J, about 23% of the stored energy.
A 23% loss is larger than one would normally call 'small', so the data are only loosely consistent with a small frictional loss. A better judgement is that the loss is significant and must be accounted for, or that another effect (for example the spring still pushing as the trolley accelerates, or the light gate reading taken before the spring fully extends) is present. The key exam skill is to quantify the discrepancy rather than simply assert 'friction'.
2. For the tilted track, the trolley rises a vertical height of 0.050 m before momentarily stopping. Using the measured launching speed of 1.4 m/s as the speed at the bottom, test whether energy is conserved and comment on the role of friction.
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At the bottom (using measured speed): KE = 0.588 J, gravitational PE = 0 (take this as the zero level).
At the top: KE = 0, gravitational PE = mgh = 0.60 × 9.8 × 0.050 = 0.294 J.
Total mechanical energy at bottom = 0.588 J; at top = 0.294 J. The loss is 0.294 J over the climb.
This loss is work done against friction (and any air resistance): W_f = 0.294 J. The average friction force along the slope is found from the distance travelled along the slope, s = h / sin 8° = 0.050 / 0.1392 = 0.359 m, so F_f ≈ 0.294 / 0.359 ≈ 0.82 N.
The comparison shows mechanical energy is not conserved; the decrease equals the work done by non-conservative forces. Note that the launch itself already lost energy relative to the spring model, so the full spring-to-top energy budget loses 0.768 − 0.294 = 0.474 J, of which part is lost on the horizontal section and part on the slope.
3. State one assumption in the model that, if false, would change the predicted launch speed, and describe how you would test it experimentally.
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Assumption: the spring delivers all its stored elastic energy to the trolley, i.e. the spring is massless and the trolley leaves it exactly at the natural length. If the spring has appreciable mass, part of the energy becomes kinetic energy of the spring itself and the trolley receives less; the predicted 1.6 m/s would be too high.
A second assumption is that the track is level and frictionless during the launch; any friction on the launch section reduces the speed.
Experimental test: use a spring of the same constant but different mass (or attach a known small mass to the spring) and compare launch speeds with the same compression. If the launch speed depends on spring mass, the massless-spring assumption is invalid. Alternatively, measure speed at several compressions and plot v² against x²; a straight line through the origin with gradient k/m supports the simple model, while a line with a lower gradient or a non-zero intercept reveals losses or spring-mass effects.