On the Edge of Glory: How Many Sheets ‘Til It Tips?

Hey there, Physics Friend!

Week, what, 1? 2? Or maybe even week 3 for some of you of the school year already?! Please, take me back to vacation! No, wait, we love juggling lesson plans, lab setups and those ever‑present “When will we ever use this?” questions. This week, one of our readers sent us a brilliant puzzle that stopped us in our tracks:

It sounds simple… but it’s a clever geometry challenge. Before you scroll down, take a wild guess. Remove a handful of sheets? Half the roll? Let’s review it together.

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The Tipping Paper Towel Roll

The question: How many sheets of paper would you have to peel off a roll standing upright on a 30° incline before it tips?

Why the roll might tip:
Picture a paper‑towel roll as a short cylinder standing on its flat end. It will stay upright as long as its centre of mass (its “balance point”) sits above its base. When you put it on a slope, the whole roll leans downhill. At some point, that balance point moves outside the base and gravity pulls it over. The steeper the slope, the more the balance point shifts, and the wider the base must be to keep it stable.

How to tell when it will tip:
A simple way to estimate the tipping point is to compare the roll’s height to its width. For a slope of angle θ, the roll needs a minimum half‑width of about

half‑width ≈ (roll height ÷ 2) × tan(θ).

(In trig terms, tan θ measures how steep the slope is.) For a 30° hill, tan 30° is about 0.577. A typical roll is roughly 28 cm tall, so the half‑width needed to stay upright is

(28 cm ÷ 2) × 0.577 ≈ 8 cm.

That means the full diameter of the roll would have to be about 16 cm to avoid tipping on a 30° slope.

How the roll shrinks as you unroll it:
As you pull off sheets, the roll gets thinner. Each sheet adds a tiny bit of thickness; a fresh roll might be about 6 cm in radius (12 cm wide) and it gets smaller as you use it up. There’s a formula that can relate the number of sheets left to the roll’s radius if you know the length and thickness of each sheet, but you don’t really need it here.

So how many sheets must you remove?
Because the critical half‑width for a 30° slope is around 8 cm (16 cm total width) and a full roll is only about 6 cm in radius (12 cm total width), the roll is already too narrow to stand stably on such a steep incline. In other words, on a 30° slope a paper‑towel roll is already beyond the tipping point—you wouldn’t need to remove any sheets at all. The only reason it might not fall right away is friction; it’s more likely to slide down before it tips.

Key takeaway: a paper‑towel roll’s base simply isn’t wide enough to keep it upright on a steep 30° slope, no matter how many sheets are left on it. In other words, it’s quite literally on the edge of glory. (Lady Gaga fans, you know the song!)

If you do want to try the math yourself, use the simple “half‑width = (height/2) × tan θ” rule above and compare it to the roll’s actual width.

🔬 Experiment: The Paper‑Towel Tipping Puzzle

Time needed: ~10 minutes and a few basic supplies (a paper‑towel roll, a board or stack of books to make a ramp, and a ruler).

Hook the class. Tell students: “I need your predictions! How many sheets do I have to tear off this roll on a 30° ramp before it falls over?” Let them guess; don’t reveal the answer yet.

Set up the demo. Prop one end of a board on a book so it makes about a 30° incline. Stand the roll upright at the top and slowly increase the angle (or gently release the roll) to see whether it slides or tips. Ask: did removing sheets make a difference?

Explain the concept: Stability comes down to the center of mass and the base of support. On a slope, gravity shifts the center of mass downhill. If that “balance point” moves outside the roll’s base, gravity produces a turning moment and it topples.

Introduce the simple formula: Geometry shows there’s a tipping point angle related to the roll’s height and half‑width. A cylinder will fall when its tilt_angle exceeds

tilt_angle_tip = arctan( height / (2 * radius) )

Rearranging for a given slope (slope_angle) gives the minimum radius needed to stay upright:

radius_needed = ( height / 2 ) * tan( slope_angle )

For a typical 28 cm‑tall roll and a 30° slope, this calculates to about 8 cm of radius (16 cm diameter).

Compare to the real roll. Have students measure the actual radius of your roll (a fresh roll is about 6 cm). Since 6 cm < 8 cm, the roll is already too narrow to avoid tipping on such a steep hill. That’s why the demo shows it toppling even with all the sheets on.

Wrap up & extend. Emphasize that the shape’s stability depends on both its height and width: tall, narrow objects tip more easily. Challenge students to bring in different cylinders (cans, water bottles) and predict their tipping angles using the same formula. It’s a quick way to connect trigonometry, center of mass, and real‑world stability.

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😂 This Week’s Physics Laugh

Joke Explanation: This joke plays on the word 'sulfur' sounding like 'suffering' and the idea of a detective solving a case. The periodic table is a key concept in chemistry, and sulfur is an essential element in the universe, playing crucial roles in various chemical reactions and life processes. This humor connects the serious pursuit of scientific discovery with the lightheartedness of a detective story.

Stay Curious,

The Phantastic Physics Team

P.S. Forward this to your physics teacher friend and see if they can figure out the answer to the paper towel roll question without peeking!