Rolling cylinder

Measure the angular and linear acceleration of a cylinder rolling on an inclined plane to study the moment of inertia and energy conversion.

When a cylinder rolls down an inclined plane, it accelerates more slowly than a sliding object without friction. Why? Because some of the potential energy converts into rotational kinetic energy rather than translational kinetic energy. The ratio between these two forms of energy depends on the moment of inertia, which characterizes how mass is distributed relative to the rotation axis. Using the smartphone's gyroscope and accelerometer simultaneously, you can measure both angular and linear acceleration and determine the moment of inertia directly.

Activity overview:

The student attaches their smartphone to a cylinder and lets it roll on an inclined plane. The gyroscope measures angular velocity and the accelerometer records linear acceleration. They calculate the moment of inertia from the data.

Level:

High School

FizziQ

Author:

Duration (minutes) :

30

What students will do :

- Measure simultaneously the linear and angular acceleration of a rolling object
- Verify the rolling-without-slipping condition (v = Rω)
- Calculate the moment of inertia from the measured acceleration
- Compare solid and hollow cylinders
- Understand the conversion between translational and rotational kinetic energy

Scientific concepts:

- Moment of inertia
- Rotational kinetic energy
- Rolling without slipping
- Angular acceleration
- Kinetic energy theorem
- Translational-rotational energy conversion

Sensors:

- Gyroscope (angular velocity)
- Accelerometer (linear acceleration)

Material needed:

- Smartphone or tablet with FizziQ
- A cylinder (bottle, roll, tube)
- An inclined plane with a groove or gutter
- Adhesive tape or rubber bands to fix the phone
- A protractor for the incline angle

Experimental procedure:

Build an inclined plane with a groove or gutter to guide the cylinder. An angle of 5° to 15° gives good results.
Firmly attach your smartphone to a cylinder (bottle, roll, tube) using adhesive tape or rubber bands. Ensure the phone's gyroscope axis is aligned with the cylinder's rotation axis.
Open FizziQ and select the Gyroscope (angular velocity) and Accelerometer (linear acceleration) instruments.
Place the cylinder at the top of the inclined plane. Start the recording.
Release the cylinder and let it roll freely to the bottom.
Stop the recording. Observe the angular velocity ω(t) and acceleration a(t) graphs.
The angular velocity should increase linearly with time (constant angular acceleration): ω = αt.
Verify the rolling-without-slipping condition: v = Rω, hence a = Rα, where R is the cylinder radius.
Calculate the moment of inertia I from the ratio between measured acceleration and g sin θ: a = g sin θ / (1 + I/mR²).
Compare the experimental moment of inertia with the theoretical value: I = ½mR² for a solid cylinder, I = mR² for a hollow cylinder.

Expected results:

The angular velocity increases linearly with time, confirming constant angular acceleration. The measured acceleration for a solid cylinder is approximately 2/3 of g sin θ. The ratio v/(Rω) is close to 1.0, confirming rolling without slipping. The moment of inertia calculated from the acceleration agrees with the theoretical value within 10-20%.

Scientific questions:

- Why does a solid cylinder descend faster than a hollow cylinder of the same mass and radius?
- What happens if friction is insufficient to maintain rolling without slipping?
- How does the total kinetic energy split between translation and rotation for a solid cylinder?
- Why does the mass cancel from the acceleration formula?
- What would happen if you used a sphere instead of a cylinder?
- How could you make the experiment more precise?

Scientific explanations:

The moment of inertia I characterizes a body's resistance to rotational acceleration, just as mass characterizes resistance to translational acceleration: τ = Iα is the rotational analogue of F = ma.

The total kinetic energy of a rolling object is the sum of translational energy (½mv²) and rotational energy (½Iω²): E_k = ½mv² + ½Iω².

The rolling-without-slipping condition v = Rω links translational and rotational velocities. It requires that the contact point has zero velocity relative to the surface.

This condition is only possible thanks to static friction. Without friction, the cylinder would slide instead of roll. Paradoxically, friction enables rolling but does no work.

Applying the kinetic energy theorem with rolling gives the acceleration a = g sin θ / (1 + I/(mR²)). For a solid cylinder (I = ½mR²), a = (2/3)g sin θ.

For a hollow cylinder (I = mR²), a = (1/2)g sin θ. For a solid sphere (I = (2/5)mR²), a = (5/7)g sin θ. The higher the moment of inertia, the slower the object descends.

This experiment is remarkable because it allows measuring the moment of inertia of an object without laboratory equipment — just a smartphone, a cylinder, and a ramp.

The comparison between solid and hollow cylinders strikingly illustrates the role of mass distribution in rotational dynamics: the hollow cylinder, with all its mass at the rim, descends more slowly.

Extension activities:

Frequently asked questions:

Q: The phone does not rotate around the correct axis.
R: The measured angular velocity depends on the phone orientation on the cylinder. The rotation axis should correspond to one of the phone's principal axes. Try different mounting orientations.

Q: The rolling-without-slipping condition is not satisfied.
R: The surface may be too smooth. Use a textured surface or add rubber bands around the cylinder to increase friction.

Q: My moment of inertia does not match the theoretical value.
R: Account for the phone's mass and its distance from the rotation axis, which adds to the cylinder's moment of inertia. The phone acts as a point mass at distance R.

Q: The angular velocity graph is not perfectly linear.
R: This can be due to the phone shifting on the cylinder or the cylinder wobbling. Ensure the phone is firmly and symmetrically attached.