Cylinder rolling down a ramp: v² against x sim
Measure sin θ with a half-metre rule and set square, time a rolling cylinder (or a trolley) over eight distances, fill the table, plot v² against x and find G ÷ (2 sin θ).
Kinematics: v² against x on a ramp · covers the O-Level, IGCSE, GCSE topic
What the sim shows
The ramp's top rests on a rod in a boss and clamp; point A is s = 0.800 m along the ramp and h = 0.139 m above the bench, so sin θ = h ÷ s = 0.174. Released from rest the object speeds up steadily, so v² = 2ax: a graph of v² against x is a straight line through the origin with gradient G = 2a, and G ÷ (2 sin θ) = a ÷ sin θ. For a solid cylinder of slotted masses a = g sin θ ÷ (1 + I/mr²) with I/mr² = 0.511 (annular discs, 6 mm hole in 40 mm), less a little rolling resistance (μ = 0.0015), so G ÷ (2 sin θ) ≈ 6.44 m/s²; for a low-friction trolley (light wheels, μ = 0.0037) a ≈ 0.96 g sin θ and G ÷ (2 sin θ) ≈ 9.42 m/s². g = 9.81 m/s². Time runs only while a run is in progress, on the real clock from the moment you release the object (click it, press the big Release button under the drawing, or press Space); stop the stopwatch as the object's back edge passes the marker by clicking the stopwatch, the same button (now Stop) or Space again. Slow motion plays the roll at ¼ speed, but the stopwatch still counts real experiment seconds. A press within 0.2 s of the true passing time counts; the stopwatch reads to 0.01 s and each recorded time carries seeded human start–stop scatter of about 0.015 s, so five trials spread as they do in a real lab. The mean, v = 2x ÷ t_ave and v² are given to 2 d.p., as the answer key does (1.036 → 1.04, 1.737 → 1.74, 3.02), with v worked from the unrounded t_ave; a wrong typed answer shows the unrounded value, the correct rounding and whether it looks like a rounding slip or a method slip. Each row's point is plotted when its v² is checked correct, and G ÷ (2 sin θ) is checked only once part (a) is right. Model limits: rolling without slipping on a rigid ramp, no air resistance, a thin ramp (sin θ = h ÷ s ignores the board's thickness); v = 2x ÷ t assumes uniform acceleration from rest. The exam paper's own data fit a near-frictionless trolley (9.42 m/s², close to g), not a rolling cylinder.
- Finish marker (tape) on the near metre rule: x runs from the object's back edge at the start line to the marker
- Rolling object: five 100 g slotted masses taped into a cylinder, or a red low-friction trolley
- Your best-fit line on the graph (dashed), with a gradient triangle
Use it in your lessons
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