Theory — Conservation of Energy in Rolling Motion
The Rolling Constraint
When an object rolls without slipping, its linear velocity (v) and angular velocity (ω) are linked by the radius R:
Energy Conservation — The Master Equation
As each object rolls from height h to the bottom of the incline, all gravitational potential energy converts into translational kinetic energy (motion of the center of mass) and rotational kinetic energy (spinning). Setting PE = KE_total:
Substitute ω = v/R:
Mgh = ½Mv² + ½I(v/R)²
Mgh = ½v²(M + I/R²)
Mgh = ½Mv²(1 + I/MR²)
Solve for v:
The critical insight: the factor (1 + I/MR²) controls the final speed. A smaller I/MR² gives a higher final velocity. Since M and R cancel completely, the finish order depends only on the shape — not the mass or size.
Moment of Inertia for Each Object
Step-by-Step Derivations
Solid Sphere — I = 2MR²/5
gh = ½v² + (1/5)v²
gh = v²(5/10 + 2/10) = 7v²/10
Solid Cylinder — I = MR²/2
gh = ½v² + ¼v²
gh = 3v²/4
Hollow Cylinder — I = ½M(R² + R₁²), where R₁ = 2R/3
I = ½M(R² + 4R²/9) = ½M(13R²/9) = 13MR²/18
I/MR² = 13/18 ≈ 0.7222
Mgh = ½Mv² + ½(13MR²/18)(v/R)²
gh = ½v² + (13/36)v² = (18/36 + 13/36)v² = 31v²/36
Predicted Finish Order
Higher coefficient → higher final speed → wins the race. Sphere has the largest coefficient:
Key result: Mass and radius cancel out entirely. A heavier or larger sphere still beats a lighter cylinder — shape is everything. The hollow cylinder loses because more of its mass is far from the center, storing more energy as rotation and leaving less as translational speed.
Instructions — Running the Virtual Experiment
Simulation — The Rolling Race
Ramp Parameters
Object Properties
| Object | Formula | Calculated v (m/s) | Simulated v (m/s) | % Difference | Your Calculated v |
|---|---|---|---|---|---|
| Solid Sphere | √(10gh/7) | — | — | — | |
| Solid Cylinder | √(4gh/3) | — | — | — | |
| Hollow Cylinder | √(36gh/31) | — | — | — |
Example Lab Report
The following is a sample lab report demonstrating the expected format, level of detail, and equation work required. Use this as a guide when writing your own report.
Rotational Motion: The Rolling Race
Physics 171 | Section: [Your Section] | Date: [Date]
Lab Members: [Names of all members present]
Purpose
To calculate the final velocity of three rolling objects — a solid sphere, solid cylinder, and hollow cylinder — as they roll without slipping down an inclined plane, and to verify these predictions experimentally. This lab demonstrates conservation of energy when gravitational potential energy converts to both translational and rotational kinetic energy.
Theory
When an object rolls without slipping down an incline of height h, energy conservation gives:
Mgh = ½Mv² + ½Iω²
Using the rolling constraint ω = v/R and solving for v:
v = √[ 2gh / (1 + I/MR²) ]
Note that mass M and radius R cancel, so the result depends only on the shape factor I/MR².
Calculations — h = 0.50 m, g = 9.81 m/s²
Solid Sphere (I = 2MR²/5 → I/MR² = 0.400):
v = √[2(9.81)(0.50) / (1 + 0.400)] = √[9.81 / 1.400] = √7.007 = 2.647 m/s
Solid Cylinder (I = MR²/2 → I/MR² = 0.500):
v = √[2(9.81)(0.50) / (1 + 0.500)] = √[9.81 / 1.500] = √6.540 = 2.557 m/s
Hollow Cylinder (R₁ = 2R/3 → I = 13MR²/18 → I/MR² = 13/18 = 0.722):
v = √[2(9.81)(0.50) / (1 + 0.722)] = √[9.81 / 1.722] = √5.697 = 2.387 m/s
Results Table
| Object | Simulation v (m/s) | Your Calculated v (m/s) | % Difference | Finish Position |
|---|---|---|---|---|
| Solid Sphere | 2.647 | 2.63 | 0.6% | 1st |
| Solid Cylinder | 2.557 | 2.54 | 0.7% | 2nd |
| Hollow Cylinder | 2.387 | 2.37 | 0.7% | 3rd |
Discussion
The student-calculated velocities (using energy conservation equations) agreed with the simulation values to within 1%, confirming that the theory accurately predicts rolling motion. Any small differences are due to rounding in the calculations.
The most significant finding is that mass and radius have no effect on the finish order — only the distribution of mass (captured by I/MR²) determines which object wins. This is a powerful demonstration of how the moment of inertia governs rotational dynamics.
The hollow cylinder stored the greatest proportion of its energy as rotation (41.9%) because its mass is concentrated far from the axis, giving it a large moment of inertia relative to MR². This explains why it loses the race despite having the same mass as the solid cylinder.
Conclusion
The rolling race experiment successfully demonstrated conservation of energy in rotational motion. The predicted finish order (sphere first, solid cylinder second, hollow cylinder third) was confirmed by the simulation. Student-calculated velocities matched simulation values within 1%. The key conclusion is that for rolling objects, the shape — not the mass or size — determines the outcome.
Practice Questions
Use these questions to prepare for exams. Show all work and include units.