When a body is stationary on an inclined plane, can we assume F = μR?

No. When stationary, F ≤ μR. Only at limiting equilibrium (on the point of moving) is F = μR.

What is the least value of μ to keep a block at rest on a slope of angle α?

μ = tan α. At limiting equilibrium, mg sin α = μ mg cos α, so μ = tan α.

How do you determine which way connected particles on slopes will move?

Compare mg sin α for each particle. The particle with the larger component down its slope will move down.

Two particles, same mass, on slopes 30° and 60°. Which moves down?

The one on 60° slope. mg sin 60° > mg sin 30°. The heavier component down slope wins.

Where does friction act in connected particle problems?

Only on particles resting on a rough surface. It acts opposite to the direction of motion or potential motion.

A block is in limiting equilibrium on a 40° slope. What is μ?

μ = tan 40° ≈ 0.839. At limiting equilibrium, μ = tan α.

When can we use F = μR?

When the object is moving, or when it is in limiting equilibrium (on the point of moving). Not when stationary in general equilibrium.

Particle A (2 kg) on 20° slope, particle B (2 kg) on 50° slope. Which moves down?

B. 2g sin 50° > 2g sin 20°. The steeper slope has the larger component down.

In connected particle equilibrium, F = ma gives:

F = 0 for each particle (or a = 0). Write equations for each particle in the direction of motion.

Coefficient of Friction (Harder Problems) | Pearson Edexcel International AS Maths

Dynamics & Statics of a Particle

Coefficient of Friction (Harder Problems)

Summary

Harder friction problems involve equilibrium vs limiting equilibrium, finding minimum μ, and connected particles on rough inclined planes. When stationary, F ≤ μR (cannot assume F = μR). When moving, F = μR. For connected particles, compare weight components down slopes to determine direction of motion; friction acts only on particles on rough surfaces.

1. Friction in Equilibrium

When a body is in static equilibrium under several forces including friction:

The value of friction is (F \leq \mu R)
When (F = \mu R), the body is in limiting equilibrium (on the point of moving)
Cannot assume (F = \mu R) when the body is stationary!
A stationary body is either in equilibrium ((F < \mu R)) or limiting equilibrium ((F = \mu R))

2. Friction When Moving

When a body is moving under forces including friction:

Friction always equals (F = \mu R)
Direction is always opposite to the direction of motion

3. Finding Minimum μ (Least Possible Value)

4. Connected Particles with Friction

When two particles are connected by a light inextensible string over a smooth pulley:

Friction acts only on particles resting on a rough surface
On an inclined slope, friction acts opposite to the direction the particle is moving or about to move
Direction of motion depends on: - Mass of each particle
- Angle of each slope

Key: Compare the components of weight in the direction of motion for each particle. The particle with the heavier component will move down its slope.

5. Determining Direction of Motion

6. Worked Example – Least μ

7. Worked Example – Connected Particles

Setup: Particles A and B, same mass, connected by light inextensible string over smooth pulley. A on rough plane at 30°, B on rough plane at 60°. μ_A = 0.2. Limiting equilibrium. Show A is on point of moving down; find μ_B.

Direction: Compare (mg\sin 30° = 0.5mg) vs (mg\sin 60° = 0.866mg). B is heavier down slope → B tends to move down, A up. So A is on point of moving down (B pulling it).

A (down positive): (mg\sin 30° + F_A - T = 0) with (F_A = \mu_A R_A = 0.2 \times mg\cos 30°)

B (down positive): (mg\sin 60° - F_B - T = 0) with (F_B = \mu_B R_B = \mu_B \times mg\cos 60°)

Solve simultaneously for (T) and (\mu_B).

8. Examiner Tips

Coefficient of Friction (Harder Problems)

Summary

1. Friction in Equilibrium

When a body is in static equilibrium under several forces including friction:

The value of friction is (F \leq \mu R)
When (F = \mu R), the body is in limiting equilibrium (on the point of moving)
Cannot assume (F = \mu R) when the body is stationary!
A stationary body is either in equilibrium ((F < \mu R)) or limiting equilibrium ((F = \mu R))

2. Friction When Moving

When a body is moving under forces including friction:

Friction always equals (F = \mu R)
Direction is always opposite to the direction of motion

3. Finding Minimum μ (Least Possible Value)

Problem type: Block of mass (m) stationary on plane inclined at (\alpha). Find the least value of (\mu) to keep it at rest.

At limiting equilibrium (on the point of sliding down): $mg\sin\alpha = \mu R = \mu \cdot mg\cos\alpha$

So: $\mu = \tan\alpha$

The least value of (\mu) is (\tan\alpha). For the block to stay at rest, (\mu \geq \tan\alpha).

4. Connected Particles with Friction

When two particles are connected by a light inextensible string over a smooth pulley:

Friction acts only on particles resting on a rough surface
On an inclined slope, friction acts opposite to the direction the particle is moving or about to move
Direction of motion depends on: - Mass of each particle
- Angle of each slope

Key: Compare the components of weight in the direction of motion for each particle. The particle with the heavier component will move down its slope.

5. Determining Direction of Motion

Do not assume the system moves in the direction of the heavier particle! Consider the angle too.

Particle A on slope at (\alpha): component down slope = (m_A g \sin\alpha)
Particle B on slope at (\beta): component down slope = (m_B g \sin\beta)

If (m_A \sin\alpha > m_B \sin\beta) and they're connected, A tends to move down.

If (m_B \sin\beta > m_A \sin\alpha), B tends to move down.

Two particles on slopes. Compare (m_A g \sin 30°) vs (m_B g \sin 60°) to determine direction.