3.3.5 Coefficient of Friction - Harder Problems

infoNote

In harder coefficient of friction problems, multiple forces and varying conditions come into play. Here's a summary of the approach:

Identify all forces: Draw a free body diagram, showing forces like weight, normal reaction, friction, applied forces, and tension (if present).
Resolve forces into components: On inclined planes, break forces into parallel and perpendicular components. Use $mg \sin(\theta)$ and $mg \cos(\theta)$ for weight components.
Apply friction formulas:

Static friction: $F_{\text{friction}} \leq \mu_s F_{\text{normal}}$ (when preventing motion).
Kinetic friction: $F_{\text{friction}} = \mu_k F_{\text{normal}}$ (when the object is moving).

Use $F = ma$ : Apply Newton's second law to solve for unknowns like acceleration, tension, or friction. Consider equilibrium conditions if the object is stationary. These problems require careful force resolution, consistent sign convention, and correct use of friction formulas.

infoNote

Problem Statement:

A box of mass 10 kg rests in limiting equilibrium on a rough plane inclined at 20° above the horizontal.
A horizontal force of magnitude P is applied to the box.
Given that the box remains in equilibrium, find:

F_{\text{max}} = 10g \sin 20° = \mu R

R = 10g \cos 20°

\mu = \frac{10g \sin 20°}{10g \cos 20°} = \frac{33.518}{92.0899} \approx 0.3640

Answer (a): The coefficient of friction $\mu \approx 0.3640$ .

The force P is horizontal and can be resolved into components parallel and perpendicular to the slope:
Parallel: $P \cos 20°$
Perpendicular: $P \sin 20°$

-F_{\text{max}} + P \cos 20° - 10g \sin 20° = 0

R = P \sin 20° + 10g \cos 20°

\mu (P \sin 20° + 10g \cos 20°) + P \cos 20° - 10g \sin 20° = 0

0.3640(P \sin 20° + 92.0899) + P \cos 20° = 33.518

0.8159P = 67.0387

P \approx 82.24 \, \text{N}

Answer (b): The maximum possible value of P is approximately 82.24 N.

:::

Tips:

infoNote

Resolve forces correctly: Always resolve forces into horizontal and vertical components (or parallel/perpendicular to an inclined plane). Pay special attention to the normal force, as it may not always equal $mg$ if other forces are acting at angles.
Check motion status: Determine whether static or kinetic friction applies. If the object is on the verge of moving, use the maximum static friction $F_{\text{friction}} = \mu_s F_{\text{normal}}$ . If it's already moving, apply kinetic friction.
Solve systematically: Use $F = ma$ in both horizontal and vertical directions to set up equations for unknowns. Make sure to account for all forces, including tension, applied forces, and weight components, for precise solutions.