Coefficient of Friction - Inclined Planes (AQA A-Level Mathematics): Revision Notes

The frictional force $F$ opposes the motion of the block sliding down the slope. The maximum value of this frictional force is given by:

$F_{\text{max}} = \mu R$

where $\mu$ is the coefficient of friction, and $R$ is the normal force.

Since the normal force is balancing the perpendicular component of the gravitational force:

$R = mg \cos \theta$

The block will start to slide down the plane if the parallel component of the gravitational force $(mg \sin \theta)$ exceeds the maximum frictional force $(\mu R)$. So, the condition for the block to just start moving is:

$mg \sin \theta = \mu R$

Substituting $R = mg \cos \theta$, we get:

$mg \sin \theta = \mu mg \cos \theta$

Cancelling $mg$ from both sides:

$\mu = \tan \theta$

• Coefficient of Friction $(\mu)$: This determines how much frictional force is available to oppose motion. On an inclined plane, $\mu = \tan \theta$ if the object is just about to slide.
• Key Forces: The weight of the object is split into components parallel and perpendicular to the slope, influencing the friction and normal force.
• Critical Angle: The angle $\theta$ where the object starts to slide is directly related to $\mu$. If $\theta$ increases beyond a certain point, the object will start to slide down the plane.

Example Problem

Problem: A block of mass $10$ kg is resting on an inclined plane that makes an angle of $30$° with the horizontal. If the coefficient of friction between the block and the plane is $0.5$,

Question : will the block slide down the plane?

Solution:

1. Calculate the forces:

• $mg = 10 \times 9.8 = 98 \, \text{N}$
• Parallel force: $mg \sin 30° = 98 \times 0.5 = 49 \, \text{N}$
• Normal force: $mg \cos 30° = 98 \times \frac{\sqrt{3}}{2} \approx 84.9 \, \text{N}$

2. Calculate the maximum frictional force:

• $F_{\text{max}} = \mu R = 0.5 \times 84.9 \approx 42.45 \, \text{N}$

3. Compare the forces:

• The parallel force ($49 N$) is greater than the maximum frictional force ($42.45 N$), so the block will slide down the plane. This example shows how the balance between gravitational and frictional forces determines whether an object will slide down an inclined plane.

Example: Force on a Slope with Friction