Cart on a Ramp: Missing Energy
At a school laboratory, a student releases a 2.40 kg cart from rest at the top of a ramp that is 1.80 m long and inclined at 25° to the horizontal. Using conservation of energy, the student predicts the speed at the bottom will be 3.86 m/s. A motion sensor records the actual speed at the bottom as 3.10 m/s. Three repeat runs give 3.02 m/s, 3.10 m/s and 3.17 m/s. The ramp is smooth but not frictionless, and the cart's small wheels turn without slipping. The student has a metre ruler, a stopwatch and a spring balance. The student claims that the missing energy is stored in the cart's wheels as rotational kinetic energy, and that the discrepancy would disappear if the ramp were steeper. The student also says the average frictional force can be found by dividing the missing energy by the length of the ramp.
1. Calculate the loss of mechanical energy for the run with a final speed of 3.10 m/s, and the average frictional force along the ramp. State any assumption you make.
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Take the bottom of the ramp as the zero of gravitational potential energy. The vertical height fallen is h = 1.80 sin 25° = 0.761 m. Initial potential energy: Ep = mgh = 2.40 × 9.8 × 0.761 = 17.9 J. Final kinetic energy: Ek = ½mv² = 0.5 × 2.40 × 3.10² = 11.5 J. Energy lost = 17.9 − 11.5 = 6.4 J. If this loss is due only to a constant friction force f acting along the 1.80 m ramp, then f × 1.80 = 6.4 J, so f = 3.6 N. This assumes no rotational kinetic energy, no air resistance, and that friction acts along the whole ramp; if some energy goes into the wheels, 3.6 N is an upper limit for the friction force.
2. Evaluate the student's claim that the missing energy is stored in the wheels' rotation and that the discrepancy would disappear on a steeper ramp.
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The wheels could store rotational kinetic energy, but the effect is unlikely to explain 6.4 J out of 17.9 J (36%). For small cart wheels, the rotational energy is ½Iω²; with ω = v/r and I = k m_w r², this is a fraction k(m_w/m) of the cart's translational kinetic energy. Unless the wheels are a large fraction of the 2.40 kg mass, the fraction is small. A steeper ramp reduces the normal force N = mg cosθ, so if friction is proportional to N, the friction force and the energy lost to friction decrease; the measured speed should move closer to the prediction. But it will not necessarily match exactly, because axle friction, air resistance and rotational energy may remain. The claim is therefore only partly reasonable: the trend with a steeper ramp is plausible, but the exact disappearance is not justified.
3. Explain why dividing the missing energy by the ramp length gives only an average force, and suggest one improvement to test whether friction is the main cause.
Read the worked answer
The missing energy is the total non-conservative work over the whole run. Friction may vary along the ramp, and other losses such as air resistance, sound and wheel bearings may contribute. Dividing by the total length gives the average force component along the ramp, not the instantaneous friction at any point. A better test is to pull the cart up the ramp at constant speed with a spring balance; the reading then gives the friction force more directly, after allowing for the component of weight. Repeating on different surfaces or with different cart masses would show whether friction is the main loss.