Inclined plane
Study the decomposition of weight into parallel and perpendicular components on an inclined plane using the smartphone's accelerometer and inclinometer.
Activity overview:
The student places their smartphone on an inclined plane at different angles and measures the gravitational force components with the accelerometer. They verify P‖ = mg sin θ and P⊥ = mg cos θ.
Level:
Author:
High school
FizziQ
Duration (minutes) :
30
What students will do :
- Measure the weight components on an inclined plane with the accelerometer
- Verify the relationships P‖ = mg sin θ and P⊥ = mg cos θ
- Plot and interpret force versus angle graphs
- Determine the static friction coefficient from the critical sliding angle
- Verify that the weight magnitude remains constant regardless of inclination
Scientific concepts:
- Vector decomposition of forces
- Weight components (parallel and perpendicular)
- Inclined plane
- Trigonometry (sin θ, cos θ)
- Normal reaction
- Static friction coefficient
Sensors:
- Accelerometer (G-force in x, y, z)
- Inclinometer (tilt angle)
Material needed:
- Smartphone or tablet with FizziQ
- A board or rigid book
- Supports of varying heights (books, blocks)
- Optional: tape to prevent the phone from sliding
Experimental procedure:
Open FizziQ and select two instruments: the Accelerometer (G-force measurement in x, y, z) and the Inclinometer (angle).
Place your smartphone flat on a horizontal table. Note the G-force values in all three axes. You should read approximately 0 in x and y, and 1g in z.
Build an inclined plane by resting a rigid board on a support. Start with a small angle (about 10°).
Place the smartphone flat on the inclined plane, screen up, with the bottom edge toward the bottom of the slope.
Record the G-force values in all three axes along with the inclination angle. If the phone slides, use a small piece of tape to hold it.
Increase the plane angle in steps of about 5° (by adding books under the board). At each angle, record the forces and angle.
Plot the graphs: parallel component versus sin θ, and perpendicular component versus cos θ.
Verify that the parallel component is proportional to sin θ and the perpendicular component is proportional to cos θ.
For each angle, verify that the quadratic sum of the two components gives 1g (constant weight magnitude).
Note the critical angle at which the phone begins to slide. This angle gives the static friction coefficient: μs = tan θ_c.
Expected results:
The perpendicular component decreases with angle following cos θ: it equals 1g at 0° and approaches 0 at 90°. The parallel component increases following sin θ: it equals 0 at 0° and approaches 1g at 90°. The sum of squares always gives 1g². The critical sliding angle typically ranges from 15° to 35° depending on the surface, giving μs = 0.27 to 0.70.
Scientific questions:
- Why does the weight not change when the plane angle varies?
- How is the parallel weight component balanced when the phone does not slide?
- What happens to the normal reaction force as the angle increases?
- Why does the phone slide above a certain critical angle?
- How would you measure the kinetic friction coefficient (while the phone is sliding)?
- Can you predict the critical angle from knowledge of the surface materials?
Scientific explanations:
On an inclined plane, the weight P = mg decomposes into two orthogonal components in the plane's reference frame: P‖ = mg sin θ (along the plane, pulling the object down) and P⊥ = mg cos θ (perpendicular to the plane, balanced by the normal reaction).
The smartphone accelerometer actually measures the pseudo-force experienced by the internal test mass. When the phone is stationary on the plane, the accelerometer readings directly give the weight components normalized to g.
The experiment allows experimental verification of the fundamental trigonometric relationships and connects them to measurable physical quantities.
The critical sliding angle gives the static friction coefficient μs = tan θ_c. At this angle, the parallel component mg sin θ_c exactly equals the maximum friction force μs × mg cos θ_c.
Simplifying, we get μs = sin θ_c / cos θ_c = tan θ_c, which provides a simple but precise measurement of the friction coefficient.
The weight magnitude, calculated as √(P‖² + P⊥²) = mg√(sin²θ + cos²θ) = mg, remains constant regardless of the angle — a beautiful verification of the Pythagorean identity.
The G-forces measured by the accelerometer are normalized to g: a reading of 1g corresponds to an acceleration of 9.81 m/s². This makes the readings directly proportional to sin θ and cos θ.
This experiment is particularly useful for giving concrete meaning to trigonometric functions, which are often perceived as abstract mathematical objects.
Extension activities:
- Why does the weight not change when the plane angle varies?
- How is the parallel weight component balanced when the phone does not slide?
- What happens to the normal reaction force as the angle increases?
- Why does the phone slide above a certain critical angle?
- How would you measure the kinetic friction coefficient (while the phone is sliding)?
- Can you predict the critical angle from knowledge of the surface materials?
Frequently asked questions:
Q: Which accelerometer axis corresponds to which direction?
R: It depends on the phone orientation. The z-axis is usually perpendicular to the screen. Place the phone flat and identify which axis reads 1g — that is perpendicular.
Q: The readings fluctuate even when the phone is stationary.
R: Small vibrations and electronic noise cause fluctuations of ±0.01-0.02g. Take the average of several seconds of data.
Q: The sin/cos fit is not perfect.
R: Verify the angle measurement. The inclinometer may need calibration. Also ensure the phone is not sliding or vibrating during measurement.
Q: Can I measure kinetic friction with this setup?
R: Yes. Let the phone slide down the plane and record the acceleration. The kinetic friction coefficient is μk = tan θ - a/(g cos θ), where a is the measured acceleration during sliding.