Physics • Year 11 • Module 2: Dynamics • Lesson 3
Inclined Planes
Build HSC Band 5–6 extended-response technique: synthesise force analysis, experimental design, and real-world evaluation on inclined plane problems.
1. Multi-step calculation — skateboard ramp at Bondi (Band 4–5)
8 marks Band 4–5
Scenario. A skateboarder of mass 65 kg (including board) approaches a ramp inclined at 18° to the horizontal. The coefficient of kinetic friction between the board wheels and the ramp surface is μ = 0.055. Use g = 9.8 m/s².
(a) Draw a fully labelled free body diagram of the skateboarder on the ramp. Label all forces with their names and formulas. Show the positive direction clearly. (2 marks)
Draw FBD here. Show W (= mg) vertically down, N perpendicular to slope, f along slope (specify direction), W∥ and W⊥ components.
(b) Calculate the weight component along the slope (W∥), the normal force (N), and the kinetic friction force (f). Show all working with units. (3 marks)
(c) Calculate the skateboarder’s acceleration down the ramp. State the direction clearly. (1 mark)
(d) The ramp designers want the skateboarder to accelerate at no more than 2.0 m/s² for safety. Determine the maximum slope angle that achieves this, assuming the same μ = 0.055. Show your reasoning and state one assumption you make. (2 marks)
2. Experimental design — measuring the coefficient of friction (Band 5)
7 marks Band 5
Research question: What is the coefficient of static friction between a wooden block and a laminated plywood ramp surface?
Constraints: You have access to a wooden block, a laminated plywood board, a protractor, a ruler, a set of masses (10 g to 500 g), a digital balance, and a retort stand. You have 40 minutes.
Q2. Design a complete experiment to answer the research question. In your response you must:
- State the independent variable, dependent variable, and at least two controlled variables.
- Describe the procedure in enough detail that another student could replicate it, referencing the critical angle method (tan θc = μ).
- Explain how you would improve reliability (include a sample size justification).
- Identify one source of systematic error in this design and suggest how to reduce it.
- Explain why the mass of the block does not affect the result, using the physics of the critical angle.
3. Data + scenario — Galileo’s inclined plane (Band 5–6)
8 marks Band 5–6
Background. Around 1604, Galileo Galilei used a polished bronze ball rolled along a groove in a tilted wooden plank to study motion. He recorded the distance the ball rolled in successive equal time intervals and noted that the distances grew in the ratio 1 : 3 : 5 : 7 : 9 (odd numbers), implying the ball was uniformly accelerating. He tested this at several ramp angles and found the acceleration was always proportional to sin θ — confirming it was caused by a component of gravity. He also tested balls of different masses and found the slope angle alone (not the mass) determined the acceleration. The table below summarises the pattern in his distances-fallen data.
| Time interval | Distance rolled in each interval (arbitrary units) | Total distance from start (arbitrary units) |
|---|---|---|
| 1st | 1 | 1 |
| 2nd | 3 | 4 |
| 3rd | 5 | 9 |
| 4th | 7 | 16 |
| 5th | 9 | 25 |
Adapted from Galileo, Discourses and Mathematical Demonstrations Relating to Two New Sciences (1638). Distances are proportional, not absolute.
Q3. Analyse and evaluate Galileo’s inclined plane results in terms of modern inclined plane physics. In your response you must:
- Explain what the odd-number pattern in intervals tells us about the type of motion (use the kinematic relationship s ∝ t² to justify).
- Show that the total distance values in the table are consistent with uniform acceleration from rest.
- Using the modern formula a = g sin θ for a frictionless slope, explain why the acceleration changes with slope angle but not with mass — with reference to what happens when Newton’s second law is applied and mass is cancelled.
- Evaluate one limitation of Galileo’s experimental design and explain how modern sensors (e.g. a light gate) would address it.
- State whether Galileo’s conclusion that “mass does not affect the rate of acceleration on a slope” is consistent with the modern formula, and assess what assumption is required for this to hold exactly.
Q1 — Skateboard ramp calculation
(a) FBD marking criteria (2 marks): 1 mark for showing W = mg vertically downward AND N perpendicular to slope, directed away from surface. 1 mark for showing both W∥ = mg sin 18° (along slope, downhill) and f = μmg cos 18° (along slope, uphill, opposing motion), with a clearly labelled positive direction.
(b) Calculations (3 marks — 1 per quantity):
W∥ = mg sin 18° = 65×9.8×sin 18° = 65×9.8×0.3090 ≈ 196.8 N.
N = mg cos 18° = 65×9.8×0.9511 ≈ 605.9 N.
f = μN = 0.055×605.9 ≈ 33.3 N (up the slope).
(c) Acceleration (1 mark): Fnet = W∥ − f = 196.8 − 33.3 = 163.5 N down the slope. a = Fnet/m = 163.5/65 ≈ 2.52 m/s² down the slope. (Alternatively: a = 9.8(sin 18° − 0.055×cos 18°) = 9.8(0.3090 − 0.0523) = 9.8×0.2567 ≈ 2.52 m/s².)
(d) Maximum angle (2 marks): Set a = g(sin θ − 0.055 cos θ) = 2.0. 2.0/9.8 = sin θ − 0.055 cos θ = 0.2041. By trial: θ = 15°: sin 15° − 0.055×cos 15° = 0.2588 − 0.0531 = 0.2057 ≈ 0.204. So θ ≈ 15° (accept 14.9°–15.1°). Assumption: kinetic friction coefficient remains 0.055 at all speeds and angles. (1 mark correct answer with working; 1 mark stated assumption.)
Q2 — Experimental design marking criteria (7 marks)
1 mark: Correctly identifies independent variable (slope angle θ), dependent variable (angle at which block just starts to slide), and at least two controlled variables (surface condition — same block face and same board section used each trial; same environmental conditions such as temperature and humidity; initial placement of block; measurement technique).
2 marks: Describes a clear, replicable procedure: place block on ramp at low angle; slowly raise ramp using protractor to measure angle; record angle at which block just begins to slide; repeat for 5–10 trials; calculate mean θc; compute μ = tan θc. (1 mark for reference to the critical angle method; 1 mark for clear procedural steps.)
1 mark: Explains reliability improvement with sample size justification: repeat 8–10 trials to reduce the effect of random error in identifying the exact onset of sliding; use mean critical angle to calculate μ; discard outliers beyond 2 standard deviations.
1 mark: Identifies systematic error (e.g. friction of the measuring setup/protractor reading; difficulty identifying the exact moment of first sliding; ramp surface not uniform across its length) and a valid improvement (e.g. use a video camera in slow motion; sand the surface to ensure uniformity; use a digital clinometer instead of a protractor).
1 mark: Explains correctly that mass does not affect the result because at the critical angle mg sin θc = μmg cos θc; mass m cancels from both sides, leaving tan θc = μ. Therefore the critical angle — and hence μ — is the same for any mass of block on the same surface.
1 mark (overall quality): Response is logically structured with clear variable identification, step-by-step method, and error analysis integrated cohesively. Accept responses that also use the set of masses to vary total mass and show the critical angle is unchanged.
Q3 — Galileo’s inclined plane marking criteria (8 marks)
1 mark: Correctly explains that the odd-number pattern (1, 3, 5, 7, 9) in successive equal intervals means the total distance grows as 1, 4, 9, 16, 25 — proportional to t² (the perfect squares). Since s ∝ t² describes uniformly accelerated motion from rest (kinematics: s = ½at²), the ball undergoes constant acceleration.
1 mark: Demonstrates consistency by noting that the cumulative distances (1, 4, 9, 16, 25) are 1², 2², 3², 4², 5² — exactly the pattern predicted by s = ½at² for equal time steps, confirming uniform acceleration.
2 marks: Explains using Newton’s second law: Fnet = ma; for a frictionless slope, Fnet = mg sin θ; therefore mg sin θ = ma, and mass cancels to give a = g sin θ. As θ increases, sin θ increases, so acceleration increases. Since a depends only on g and θ (not m), all masses accelerate equally on the same frictionless slope. (1 mark for mass cancellation with formula; 1 mark for link to sin θ explaining angle dependence.)
2 marks: Identifies a valid limitation of Galileo’s design (e.g. relied on water-clock for timing, which is imprecise compared to modern electronic sensors; could not easily verify uniform acceleration at steep angles; rolling friction in the groove was not accounted for; used arbitrary distance units making absolute acceleration difficult to determine). Explains how a light gate would address it: records exact time at two or more positions automatically, allowing precise calculation of velocity and hence acceleration, eliminating human reaction-time error. (1 mark limitation + 1 mark improvement with light gate.)
1 mark: States that Galileo’s conclusion is consistent with the modern formula a = g sin θ, which does not contain mass. The required assumption is that friction is negligible (frictionless slope), because with friction, a = g sin θ − μg cos θ still does not contain mass — accept this reasoning as also valid (mass still cancels with friction). The deeper assumption is that the object does not slip or roll in a way that introduces a rotational inertia correction.
1 mark (synthesis): Response draws a coherent conclusion that links the historical experiment to modern physics: Galileo’s odd-number result is a direct empirical confirmation of constant acceleration; the modern formula explains why (via Newton’s law and the component of gravity), and the mass-independence is not a coincidence but a consequence of mass appearing identically in both Fnet and the inertia term.