Mechanical Equivalent of Heat lab report

1. Title and aim

Determination of the Mechanical Equivalent of Heat (Joule’s Constant) by Friction Calorimetry. Determine Joule’s mechanical equivalent of heat (J) using a rotating friction calorimeter with suspended load and revolution counter, measuring the mechanical work done against friction (W) and the corresponding thermal heat absorbed (Q) by the calorimeter water system.

2. Abstract

Five observations were generated from the ideal teaching model over the settings listed below. A straight-line fit gave an estimate of Mechanical equivalent of heat J of 4.185 J cal⁻¹, differing by 0.023% from the model value. The reported scatter comes from scale rounding and any stated model approximation, not measured laboratory noise.

3. Apparatus and fixed settings

• Precision Rotating Friction Drum Calorimeter with Mechanical Drive: A thin-walled, specularly turned electrolytic copper cylinder (outer diameter 100.0 mm, radius 50.0 mm) mounted upon a low-friction horizontal axle with thermal insulating phenolic couplings, driven at steady rotational speeds (100–200 RPM) by a variable-speed motor or geared hand crank.
• Calibrated Canvas Friction Belt with Suspended Counter-Load and Spring Balance: A wide, flexible woven friction band wrapped 1.5 turns around the circumference of the drum, supporting a calibrated suspended load mass (M_load = 3.000 kg) on the descending side and anchored through an adjustable helical spring balance (measuring back-tension T_2) on the ascending side to maintain steady floating equilibrium.
• Mechanical Digital Revolution Counter: A precision five-digit mechanical cyclometer directly geared to the rotating drum drive shaft to record the exact total number of revolutions N traversed during each experimental run with single-rotation resolution.
• Calibrated Precision Immersion Thermometer (0.05°C Subdivisions): A sensitive mercury-in-glass thermometer graduated from 15 °C to 45 °C in 0.05 °C increments, inserted axially into the hollow interior of the drum through a water-tight seal to monitor water temperature before and after friction cycling.
• Cylindrical copper friction drum (outer radius r = 0.0500 m, mass m_cal = 200.00 g, c_cu = 0.0924 cal g⁻¹ K⁻¹; water equivalent W_cal = 18.48 g).
• Distilled water charge m_w = 100.00 g (water equivalent 100.00 g); aggregate calorimeter system capacity C_sys = 118.48 cal K⁻¹ (495.96 J K⁻¹).
• Suspended counter-load mass M_load = 3.000 kg suspended by a flexible canvas friction belt wrapped 1.5 turns over the drum.
• Initial baseline ambient and calorimeter temperature T_0 = 20.00 °C; local gravitational acceleration g = 9.80665 m s⁻².
• Reference mechanical equivalent of heat: J = 4.1860 J cal⁻¹ (4186.0 J kcal⁻¹ = 4.1860 kJ kcal⁻¹).

4. Theory and graph choice

The establishment of the mechanical equivalent of heat represents one of the monumental conceptual revolutions in nineteenth-century physics, dismantling Antoine Lavoisier's caloric theory—which treated heat as an indestructible, weightless fluid conserved in all physical processes—and laying the cornerstone for the first law of thermodynamics. In 1798, Benjamin Thompson (Count Rumford) observed during the boring of brass cannons in Munich that a dull drill bit produced an inexhaustible quantity of heat as long as horses continued to turn the shaft, concluding that heat could not be a material substance but was a mode of microscopic motion excited by mechanical work. Between 1843 and 1850, James Prescott Joule performed a landmark series of quantitative experiments—measuring the temperature rise of water agitated by falling-weight paddle wheels, fluid friction in mercury, and electrical dissipation in resistive coils—demonstrating that regardless of the mechanism, the expenditure of a fixed quantity of mechanical work W always produces a strictly proportional quantity of thermal energy Q: W = J · Q, where J is Joule's mechanical equivalent of heat. In the classic Callendar and Rowland friction calorimeter apparatus, a hollow copper cylinder of radius r containing a known mass of water is rotated continuously about its horizontal axis. A flexible woven friction belt is wrapped around the drum; one end supports a suspended load M_load, while the opposite end is restrained by a light spring balance. When rotated at steady speed, the frictional torque exerted by the drum lifts the suspended weight until it floats freely in dynamic equilibrium, while the mechanical work dissipated against sliding friction is completely absorbed by the drum and water. For N complete rotations, the relative sliding distance traversed by the drum surface beneath the belt is s = 2π r N, giving a mechanical work input of W = 2π r M_load g N (in joules). The heat absorbed by the calorimeter and water is evaluated as Q = C_sys (T_f − T_0) (in calories), where C_sys is the aggregate heat capacity of the water and vessel metal. Ordinary least-squares regression of W against Q across serial revolution settings yields an exceptionally robust measurement of Joule's constant J ≈ 4.186 J cal⁻¹.

The calculation fits y = mx + b in SI units. The graph may display scaled axes for readability; those display factors are removed before calculating the physical result. The gradient has units J cal⁻¹.

5. Method

1. Weigh the clean, dry copper drum to ±0.01 g on a laboratory analytical balance, recording its empty mass m_cal = 200.00 g. Add exactly 100.00 g of distilled water at room temperature. Seal the drum cap securely and remeasure the total mass to confirm the water charge.
2. Mount the filled drum onto the insulated drive axle. Wrap the woven friction belt 1.5 times around the copper drum circumference. Attach the 3.000 kg load mass to the free hanging tail and hook the opposing end to the upper spring balance. Verify that the belt tracks centrally without wandering off the drum edges.
3. Insert the precision thermometer axially into the drum cavity. Rotate the drum slowly by hand for several turns to mix the internal water, allow the reading to stabilize, and record the initial baseline temperature T_0 = 20.00 °C. Reset the mechanical revolution counter to 00000.
4. Start the drive motor or turn the hand crank smoothly, accelerating the drum to a steady speed of approximately 120 to 150 RPM. Observe that the friction between the spinning drum and belt lifts the 3.000 kg load slightly off the bench rest so that it floats in free dynamic equilibrium. Monitor the small residual tension T_2 on the spring balance.
5. Continue rotating smoothly until the target number of revolutions N is attained (e.g., N = 200 revs). Immediately halt the rotation, disengage the belt, and gently agitate the water inside the drum by rocking the axle. Watch the thermometer meniscus rise through an optical magnifier and record the maximum equilibrium temperature T_f attained before cooling commences.
6. Allow the drum to cool back to ambient baseline temperature. Repeat the measurement across five distinct revolution settings (N = 200, 400, 600, 800, and 1000 revolutions). For each run, compute the mechanical work W = 2π r M_load g N in joules and the heat absorbed Q = C_sys (T_f − T_0) in calories. Plot W against Q and determine Joule’s constant J from the slope of the regression line.

6. Observations and calculation table

All observations use fixed model settings. Values are rounded only at the instrument reading stage; further calculations use full precision. Repeated physical trials are not claimed.

Download the sample observations (CSV) ↓

7. Worked calculation for one observation

8. Graph, gradient and final calculation

Dots show the sample observations. The line is the ordinary least-squares fit.

9. Uncertainty and comparison

Percentage difference = |estimate − model value| / |model value| × 100 = 0.0227%. Percentage difference is not experimental uncertainty.

For n = 5 observations, Σresidual² = 9.83449; s² = Σresidual²/(n − 2). Gradient standard error = √[s²/Σ(x − x̄)²] = 0.00130. Intercept standard error = √[s²(1/n + x̄²/Σ(x − x̄)²)] = 1.89866.

Propagating only the relevant fit coefficient gives a fit-only standard error of 0.00130 J cal⁻¹. This is a one-standard-error quantity under the regression assumptions. Deterministic rounded model data do not supply a complete physical uncertainty budget.

Frictional heat dissipation to the ambient air and support bearings during rotation (~3–8 min), temperature rise non-uniformity across the rotating drum cylinder and internal water, tension fluctuations in the friction belt/cord, spring balance calibration offset, thermometer thermal lag and stem conduction, revolution counter indexing precision (±1 rev), and water evaporation loss contribute experimental uncertainty. The OLS regression standard error reflects random scatter across revolution counts, but does not capture unmodelled convective boundary losses or belt stiffness variations.

10. Result and discussion

The observations follow the stated model relationship. Check whether the intercept and residual pattern suggest a systematic offset or a missing approximation. Agreement with the model does not independently verify its input constants.

11. Precautions and limitations

• Ensure the drum is rotated at a steady, uniform speed so that the suspended load floats stably in mid-air without bouncing or bottoming out on the bench.
• Record the small residual tension on the spring balance and subtract it from the suspended load to evaluate the true net frictional force.
• Gently rock or agitate the drum after rotation ceases to thoroughly mix the internal water before recording the maximum equilibrium temperature.
• Inspect the canvas friction belt and copper drum surface to ensure they remain completely clean and free of oil or grease, which could cause erratic slipping.

The teaching simulation models friction calorimetry under idealized thermal conditions with constant initial baseline temperature T_0 = 20.00 °C, constant drum radius r = 0.0500 m, suspended load mass M_load = 3.000 kg, and invariant system heat capacity C_sys = 118.48 cal K⁻¹. In physical experiments, convective and radiative heat losses to the ambient room during rotation (~3–8 min) cause the observed final temperature to fall below the adiabatic value, belt tension fluctuations and spring balance vibrations cause small work variations, friction between the axle bearings and support pillars dissipates a small fraction of work outside the water chamber, water evaporation through drum seals can cause minor latent heat cooling, and thermometer thermal response lag introduces slight peak temperature reading delays. The simulation evaluates mechanical work using W = 2π r M_load g N, where r = 0.0500 m is the drum radius, M_load = 3.000 kg is the suspended load, g = 9.80665 m s⁻² is gravitational acceleration, and N is the revolution count. Heat absorbed is evaluated using Q = C_sys (T_f − T_0) in calories, where C_sys = 118.48 cal K⁻¹ is the total thermal capacity (100.00 g water + 18.48 g water equivalent of the copper drum). Equilibrium temperatures are calculated using T_f = T_0 + W / (C_sys · J), with J = 4.1860 J cal⁻¹ and rounded to 0.01 °C to simulate precision laboratory thermometry. Ordinary least-squares regression of W against Q across five revolution settings yields the reported mechanical equivalent of heat.

12. Conclusion

The worked example illustrates how to obtain Mechanical equivalent of heat J from measurements, graph variables and a fitted coefficient. The estimate is 4.185 J cal⁻¹. A real experiment must additionally evaluate the apparatus-specific uncertainties and corrections described above.