3.3.3 Coefficient of Friction - F = ma

In mechanics, the coefficient of friction plays a crucial role in determining the force of friction that opposes the motion of objects. When combined with Newton's second law, $F = ma$, we can analyse the motion of objects on surfaces where friction is present.

1. Understanding Friction

• Friction: Friction is a resistive force that acts against the relative motion of two surfaces in contact.
• Coefficient of Friction ( $\mu$ ): This is a dimensionless quantity that represents the ratio between the force of friction $F_{\text{friction}}$ and the normal force $N$ acting perpendicular to the surfaces.

$\mu = \frac{F_{\text{friction}}}{N}$

• Types of Friction:
  • Static Friction ( $\mu_s$ ): The frictional force that must be overcome to start the motion.
  • Kinetic Friction ( $\mu_k$ ): The frictional force acting on an object that is already in motion.

2. Force of Friction

The frictional force $F_{\text{friction}}$ is given by:

$F_{\text{friction}} = \mu N$

Where:

• $\mu$ is the coefficient of friction (static or kinetic).
• $N$ is the normal force, which is often $N = mg$ when the object is on a horizontal surface, with $m$ being the mass and $g$ the acceleration due to gravity.

3. Newton's Second Law: $F = ma$ with Friction

When friction is involved, Newton's second law is applied by accounting for the frictional force as part of the net force acting on the object.

• Scenario 1: Object on a Horizontal Surface (with Friction)
  • Suppose a horizontal force $F$ is applied to an object of mass $m$ , causing it to accelerate.
  • The net force $F_{\text{net}}$ acting on the object is the applied force $F$ minus the frictional force $F_{\text{friction}}$ :

$F_{\text{net}} = F - F_{\text{friction}} = ma$

• Substituting F_{\text{friction}} = $$\mu N and $N = mg$ :

$F - \mu mg = ma$

• Solving for the acceleration $a$ :

$a = \frac{F - \mu mg}{m}$

• Scenario 2: Object on an Inclined Plane (with Friction)
  • Consider an object of mass $m$ on an inclined plane with angle $\theta$ to the horizontal.
  • The forces acting on the object are:
  • Gravity: Acts downward with a force $mg$.
  • Normal Force: Acts perpendicular to the plane, $N = mg \cos \theta$ .
  • Frictional Force: Opposes motion, $F_{\text{friction}} = \mu N = \mu mg \cos \theta$ .
  • Component of Gravity along the Plane: $mg \sin \theta$ , which drives the object down the incline.
  • The net force along the plane is:

$F_{\text{net}} = mg \sin \theta - F_{\text{friction}}$

$F_{\text{net}} = mg \sin \theta - \mu mg \cos \theta$

• Applying $F_{\text{net}} = ma$:

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

• Simplifying for acceleration $a$ :

$a = g (\sin \theta - \mu \cos \theta)$

4. Example Problems

5. Summary

When friction is involved, Newton's second law $F = ma$ requires adjusting for the frictional force, which is determined by the coefficient of friction $\mu$ and the normal force $N$ . Whether on a horizontal surface or an inclined plane, understanding how to incorporate friction into your calculations is crucial for accurately analysing the motion of objects.

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