Ramp Energy Budget with a Spinning Ball
A student releases a 0.20 kg steel ball from rest at the top of a 1.20 m ramp inclined at 30.0° to the bench. A light gate at the bottom records the ball interrupting the beam for 0.012 s; the ball's diameter is 0.030 m. Using the vertical drop, the student predicts a centre speed of about 3.4 m/s at the bottom, but the gate gives 2.5 m/s. The ramp is smooth but slightly bowed, and the ball is sometimes seen spinning as it leaves. The student's partner argues that the missing energy must be friction, and that repeating with a heavier steel ball will make the measured speed match the prediction because gravitational potential energy increases with mass. The teacher asks the pair to decide whether the data support the partner's claim and to identify the measurement that would most improve confidence in the result.
1. Calculate the gravitational potential energy lost and the kinetic energy implied by the gate. Does the discrepancy prove that friction is the main loss? Explain.
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Use the vertical drop:
- Ramp length 1.20 m at 30.0°, so h = 1.20 sin 30.0° = 0.600 m.
- ΔPE = mgh = 0.20 × 9.81 × 0.600 = 1.18 J.
- Gate speed: v = diameter / time = 0.030 m / 0.012 s = 2.50 m/s.
- Translational KE = ½mv² = 0.5 × 0.20 × 2.50² = 0.625 J.
About 0.55 J is unaccounted for. This does not prove friction is the main loss. The gate measures the centre's translational speed; if the ball rolls, rotational KE is omitted. For a solid sphere rolling without slipping, I = (2/5)mr² and ω = v/r, so rotational KE = ½Iω² = 0.2mv² = 0.25 J at 2.50 m/s. That leaves roughly 0.30 J for friction, air resistance, and the bowed ramp. Timing uncertainty is also large: ±0.001 s changes v from 2.31 to 2.73 m/s, so the energy budget is not precise.
2. The partner claims a heavier ball will close the gap because gravitational potential energy grows with mass. Evaluate this claim quantitatively.
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For smooth conversion, predicted speed is v = √(2gh), which is independent of mass. Both ΔPE and translational KE scale linearly with m, so their ratio stays the same. The rolling fraction is also mass-independent for a given shape: rotational KE / total KE = 2/7 for a solid sphere. Friction work along the ramp is μmg cosθ × L, which scales with m as well, so the fractional loss is roughly unchanged. A heavier ball may reduce the relative effect of air resistance and timing scatter slightly, but it will not remove a systematic 30–40% gap. The claim is not supported by the energy budget.
3. Which change would most improve confidence: timing over a longer distance, using a heavier ball, or measuring the ramp angle more precisely? Justify with uncertainty estimates.
Read the worked answer
Timing uncertainty dominates. A 0.012 ± 0.001 s gate time is about ±8% in speed, which becomes about ±17% in kinetic energy. A 0.5° error in angle changes h by about 1.7% (sin 30.5° vs sin 29.5°), and thus predicted speed by less than 1%. A heavier ball does not fix the systematic rotational and friction issue. The best change is to time the ball over a longer distance, or use two gates a measured distance apart, so the fractional timing error falls. A sliding block instead of a rolling ball would also remove rotational KE and make the energy comparison cleaner.