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7.5MechanicsStretch

Dynamics and Inclined Planes

When a particle accelerates on a slope, Newton’s second law replaces equilibrium along the plane. Resolve perpendicular for the reaction (and friction), then apply F = ma down the slope.

26 min Applications of Forces
Edexcel A Level Maths: 7.5 Dynamics and Inclined PlanesCurated by Kepler, not produced by us. Watch it before the notes below.
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What you'll be able to do

  • Apply F = ma on an inclined plane
  • Include friction in the dynamics
  • Find acceleration on a slope
  • Combine with suvat for motion
1

Newton’s second law on a slope

Perpendicular to the slope there is no acceleration, so R=mgcos⁡θR = mg\cos\theta. Along the slope, the net force gives mama: e.g. sliding down a rough slope, mgsin⁡θ−μR=mamg\sin\theta - \mu R = ma.

mgsin⁡θ−μmgcos⁡θ=mamg\sin\theta - \mu mg\cos\theta = ma
Newton’s second law down a rough slope.
Example 1A 2 kg block slides down a rough 30° slope, μ = 0.2 (g = 9.8). Find a.
1mgsin⁡θ=9.8mg\sin\theta = 9.8 N; μmgcos⁡θ=0.2(2)(9.8)cos⁡30°≈3.39\mu mg\cos\theta = 0.2(2)(9.8)\cos30° \approx 3.39 N.
2a=(9.8−3.39)/2≈3.2a = (9.8 - 3.39)/2 \approx 3.2 m/s².
Answer
a≈3.2a \approx 3.2 m/s²

Tip — Friction acts UP the slope when the particle slides down — opposing the motion.

Formula recap

R=mgcos⁡θR = mg\cos\theta
Normal reaction.
mgsin⁡θ−μR=mamg\sin\theta - \mu R = ma
Sliding down a rough slope.

Common mistakes to avoid

Putting friction down the slope when the block slides down.
Friction opposes motion, so it acts up the slope.
Forgetting the slope is rough (omitting μR).
Include friction unless told the slope is smooth.

Key takeaways

  • Perpendicular: R = mg cos θ (no acceleration).
  • Along slope: net force = ma.
  • Friction opposes the direction of motion.

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

  1. Perpendicular to a slope, the acceleration is:

    • zero
    • gg
    • gsin⁡θg\sin\theta
    • maximum
    Show answer and explanation

    Answer: zero

    No motion into the slope.

  2. When a block slides down, friction acts:

    • up the slope
    • down the slope
    • into the slope
    • horizontally
    Show answer and explanation

    Answer: up the slope

    Opposes motion.

  3. Newton’s law down a rough slope is:

    • mgsin⁡θ−μR=mamg\sin\theta - \mu R = ma
    • mgcos⁡θ=mamg\cos\theta = ma
    • mgsin⁡θ+μR=mamg\sin\theta + \mu R = ma
    • R=maR = ma
    Show answer and explanation

    Answer: mgsin⁡θ−μR=mamg\sin\theta - \mu R = ma

    Net force = ma.

  4. For a 2 kg block, 30°, μ = 0.2, a ≈ (g=9.8):

    • 3.23.2 m/s²
    • 4.94.9 m/s²
    • 1.71.7 m/s²
    • 9.89.8 m/s²
    Show answer and explanation

    Answer: 3.23.2 m/s²

    (9.8 − 3.39)/2.

  5. After finding a, motion is analysed with:

    • suvat
    • moments
    • the binomial theorem
    • integration by parts
    Show answer and explanation

    Answer: suvat

    Constant acceleration.

Next step

Check your understanding

Work through questions on Dynamics and Inclined Planes. Use the feedback to decide what needs another look.

3 past-paper questions · 7 marks

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