7.3MechanicsStretch

Friction and Static Particles

Combining friction with equilibrium lets you analyse a particle on the point of sliding — often on a rough inclined plane. The limiting friction F = μR closes the system of equations.

26 min Video by Zeeshan Zamurred Applications of Forces

Edexcel A Level Maths: 7.3 Friction and Static ParticlesWatch the full walkthrough before the notes below.

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What you'll be able to do

Combine friction with equilibrium
Analyse a particle on a rough slope
Use F = μR at the point of slipping
Find the limiting angle or force

On a rough slope

For a particle in limiting equilibrium on a rough slope: resolve perpendicular to find $R = m g cos θ$ , then up the slope friction balances $m g sin θ$ , with $F = μ R$ at the point of slipping.

m g sin θ = μ m g cos θ \Rightarrow μ = tan θ

At the point of sliding on a slope.

1Perpendicular:

R = m g cos θ

2Along:

m g sin θ = μ R = μ m g cos θ

3Divide:

μ = tan θ

Answer

μ = tan θ

Tip — On the point of sliding down a slope, μ = tan θ where θ is the slope angle.

Formula recap

R = m g cos θ

Normal reaction on a slope.

μ = tan θ

At the point of sliding on a slope.

Common mistakes to avoid

Using F = μR before the particle is at the point of slipping.

F = μR only in limiting equilibrium; otherwise F ≤ μR.

Using mg for the normal reaction on a slope.

R = mg cos θ on an inclined plane.

Key takeaways

Combine friction (F = μR) with equilibrium.
On a rough slope R = mg cos θ.
At the point of sliding, μ = tan θ.

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