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Reconcile falling-mass rotation with a separate stopping receiver

This cumulative mathematical packet joins motion, force, rotation, work and thermal accounting. A 2 kg mass descends from rest while unwinding a massless no-slip cord from a drum of radius 0.1 m and inertia 0.02 kg m². Gravity is 10 m/s² and a constant axle torque 0.2 N m opposes rotation. These are fictional low-energy parameters, not instructions to build or load a machine.

Learning outcome: Predict the acceleration and tension from separate recipient diagrams before checking energy. Compare two heights and two capture fractions, preserve the two time intervals, and retain a rejected 10 N cord-capacity alternative. Explain what an independent timing or temperature calibration could challenge.

Starting recap

Choose recipient, system, time interval and calibration reference before combining equations. Preserve alternative explanations and distinguish predicted, generated and independently observed evidence.

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Try

At h=1 m and f=0.8, speed before braking is 3 m/s and angular speed 30 rad/s. The 20 J release becomes 2 J axle heating plus 18 J kinetic energy; stopping sends 14.4 J to the receiver and 3.6 J elsewhere, giving receiver rise 0.24 K.

Open the experiment

Predict

For the stated model, receiver heat from the terminal brake is:

Your prediction and reason

Experiment

Predict the acceleration and tension from separate recipient diagrams before checking energy. Compare two heights and two capture fractions, preserve the two time intervals, and retain a rejected 10 N cord-capacity alternative. Explain what an independent timing or temperature calibration could challenge.

Connect falling-mass rotation with a separate thermal receiver

Worked teaching model · synthetic values · Fictional mass 2 kg, drum I=.02 kg m², R=.1 m, g=10, opposing axle torque .2 N m. Massless taut no-slip cord, released from rest. An idealized impulsive terminal brake stops both motions with negligible further descent, so additional gravitational work is neglected; selected fraction reaches receiver C=60 J/K. Earlier axle heat is outside that receiver. Virtual only, not construction or equipment rating.

Descent height: 1 m Brake energy fraction captured: 0.8 ratio

Reset keeps your prediction, captures and practice history.

Separate descent and stopping energy destinations

Unlike graph quantities rescale independently; read units and values, not screen angles. 1: gravitational release (solid).2: axle heating during descent (dashed).3: terminal receiver capture (dotted). Joined curves show the supplied model, not additional observations.

Equivalent numerical coordinates for Separate descent and stopping energy destinations

gravitational release

descent height (m)	energy (J)
0	0
1	20

At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.

axle heating during descent

descent height (m)	energy (J)
0	0
1	2

At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.

terminal receiver capture

descent height (m)	energy (J)
0	0
1	14.4

At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.

Calculated quantities under the declared measurement model
Quantity	Value	Unit
Downward acceleration	4.5	m/s²
Required cord tension	11	N
Descent duration	0.666667	s
Speed before brake	3	m/s
Angular speed before brake	30	rad/s
Gravitational energy released	20	J
Axle thermal energy during descent	2	J
Translational kinetic energy before brake	9	J
Rotational kinetic energy before brake	9	J
Receiver gain from terminal brake	14.4	J
Other terminal-brake thermal gain	3.6	J
Receiver temperature rise	0.24	K
Whole endpoint energy residual	4.44089e-16	J

The receiver captures a specified part of stopping energy, not automatically all gravitational energy.

Combine mg−T=ma with TR−tau=I alpha and a=R alpha. Effective inertial mass m+I/R²=4 kg gives acceleration 4.5 m/s² and tension 11 N. Axle work is tau h/R=2h J; remaining 18h J splits equally into translation and rotation. Stopping is a separate interval with external brake forces: the falling mass momentum is not conserved through it. A cord-capacity premise below 11 N rejects this motion rather than lowering tension while retaining its acceleration.

Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.

The receiver captures a specified part of stopping energy, not automatically all gravitational energy.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

Freeze the system and measurement contract

Derive the coupled model

Choose downward motion positive for the mass and the corresponding unwinding rotation positive for the drum. The mass equation is mg−T=ma. The drum equation is TR−τ=Iα, with no-slip a=Rα. Combining gives a=(mg−τ/R)/(m+I/R²)=(20−2)/4=4.5 m/s² and tension T=20−2×4.5=11 N. The inertia contributes an effective mass I/R²; treating the drum as massless would change the prediction.

Check the numerical energy and power account

After a descent h, constant-acceleration motion gives v²=2ah=9h. Gravitational energy released is 20h J. Axle heating during that interval is τ(h/R)=2h J. Translational kinetic energy mv²/2 is 9h J, and rotational kinetic energy I(v/R)²/2 is another 9h J. Thus the motion and energy derivations independently agree under the same disclosed model assumptions.

Reject an unsupported inference

An idealized impulsive terminal brake then stops both motions with negligible further descent. Additional gravitational work during this stopping idealization is neglected; a finite stopping distance would require a revised energy budget. It converts their combined 18h J, of which the selected fraction f reaches a receiver of heat capacity 60 J/K. Receiver rise is f18h/60=0.3fh K. Earlier axle heat is outside that receiver; counting all gravitational energy as receiver heat would overstate its rise. The remaining brake energy reaches a separate environment. Momentum of the falling mass is not conserved during stopping because the brake/support supplies external impulse. If a newly stipulated cord can support at most 10 N, the required 11 N rejects the assumed taut no-slip motion rather than authorizing a clipped-tension calculation.

Worked original case

Cumulative evidence action

Changed-case numerical practice

Try the changed case before feedback

In the taught drum model, take h=2 m and receiver capture fraction 0.5 with capacity 60 J/K. Find receiver temperature rise in K.

Numerical result in the requested units

This practice is public and unsaved. A matching number does not verify understanding. Retain your explanation, assumptions and changed-case evidence in the separate course workspace.

Sources and completion boundaries

Original synthesis checked 30 October 2026 against OpenStax College Physics 2e and the course’s earlier source-backed force, energy and thermal models. All records are stipulated or generated; no machinery, electrical connection, heating, liquid handling or physical experiment is requested. Completing these authored pages does not establish independent understanding, supervised practical skill or institutional credit. Use the separate retained course assessment and project for actual evidence.

Apply

Public practice checks a specific claim. It does not establish independent mastery.

Why does a correct total energy balance not automatically establish each receiver’s thermal gain?

Your explanation and limitations

Continue

Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.

Skipping retains experiments and awards no mastery.

Previous: Trace lost oscillator energy through a finite thermal receiver Next: Audit a whole flow-power account before inferring efficiency

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