The 42nd Period | STEM Education Hub

At the exact moment an object is about to slide, gravity's pull down the slope exactly equals the maximum friction holding it in place. Today you'll predict that exact tipping point — then find out how close your math gets to reality.

🎯 Goals

Decompose weight into components parallel and perpendicular to an incline.
Predict the critical angle where an object begins to slide, using tanθ = μs.
Test the predicted critical angle on a real ramp and compare to your measurement.
Calculate the acceleration of an object sliding down the ramp above the critical angle.

🧰 Materials

📏An adjustable ramp (board propped up with books or blocks)

📐Protractor

📦An object with a known-ish μs (reuse your block from Activity 2-2-1 if possible)

⏱️Stopwatch

📓Physics notebook

Section 1 of 4

Predict the Critical Angle

Recall or measure your object's μs.

If you measured μs for this block in Activity 2-2-1, use that value. Otherwise, your teacher will provide one.

Set mg sinθ equal to maximum static friction and solve for θ.

mg sinθ = μs mg cosθ. The mg on both sides cancels, and sinθ/cosθ = tanθ, leaving tanθ = μs. Solve for your predicted critical angle using θ = tan⁻¹(μs).

⚠️ This shortcut only works at the exact tipping point — don't use tanθ=μs for angles where the object is already sliding or clearly not sliding.

Section 2 of 4

Test It

Place your object on the ramp at a low angle and slowly raise one end.

Raise it gradually and evenly — a jerky lift can make the object slide before it actually would have on its own.

Use the protractor to measure the exact angle the instant the object starts to slide.

Repeat at least twice more and average your measured angles.

Trial	Measured critical angle (°)

📓 Physics Notebook

How close was your predicted critical angle to your measured one? If they didn't match closely, what about the real ramp or block might explain the gap?

Section 3 of 4

Measure Acceleration Down the Ramp

Set the ramp at an angle noticeably above your critical angle.

You want the object to accelerate down the ramp clearly, not just barely creep.

Release the object from rest and time how long it takes to slide a measured distance.

Use Δx = ½at² to calculate the measured acceleration from your distance and time.

Calculate the predicted acceleration using F = ma with mg sinθ and friction.

a_predicted = g sinθ − μk g cosθ (the mass cancels out of this version, just like it did in the critical-angle formula).

Angle (°)	Distance (m)	Time (s)	a measured (m/s²)	a predicted (m/s²)

Section 4 of 4

Conclusion

Question 1

Derive tanθ = μs starting from mg sinθ = μs mg cosθ, showing each algebra step.

Question 2

Explain why the critical-angle formula and the acceleration formula both end up mass-independent, the same way the falling-objects puzzle from Lesson 2-2-2 did.

Question 3

If your measured acceleration was smaller than your predicted acceleration, suggest one realistic reason why — something about the real ramp, not just 'measurement error.'

📤 Turn In

Predicted critical angle calculation and completed measurement table
Predicted vs. measured acceleration table with work shown
Written answers to the three Conclusion questions

← Back to Activity 2-2-3 Overview Review the Deep Dive →Next up: Project 2-2-4, Friction Ramp Lab.