Friction

Types of Forces in Mechanics

In classical mechanics, forces are categorized based on how they arise and how they interact with objects.

1. Contact Forces: Forces that arise due to physical contact.

• Normal Force ($\vec{N}$): Perpendicular reaction force by a surface.
• Friction ($\vec{f}$): Opposes relative motion between surfaces.
• Tension ($\vec{T}$): Force exerted by strings or ropes.
• Spring Force: Follows Hooke's Law: $F = -kx$
• Applied Force ($\vec{F}_a$): Force exerted manually or mechanically.

2. Non-contact (Action-at-a-Distance) Forces: Forces that act without direct contact.

• Gravitational Force ($\vec{F}_g$): $F = G\frac{m_1 m_2}{r^2}$
• Electrostatic Force: $F = k\frac{q_1 q_2}{r^2}$
• Magnetic Force: Depends on moving charges and magnetic fields

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Friction

Friction is the force that opposes the relative motion or tendency of such motion of two surfaces in contact. It arises due to microscopic irregularities and electromagnetic forces between the surfaces.

• Static Friction ($f_s$): This force acts when there is no relative motion. It is self-adjusting, meaning it matches the applied force up to a certain maximum limit, beyond which the object starts to move. This limit is called the limiting friction. $$f_s \le f_{s,max} = \mu_s N$$ where $\mu_s$ is the coefficient of static friction and N is the normal force.

• Kinetic Friction ($f_k$): This force acts when there is relative motion between the surfaces. Its magnitude is approximately constant and is given by: $$f_k = \mu_k N$$ where $\mu_k$ is the coefficient of kinetic friction. Generally, for the same pair of surfaces, $\mu_k < \mu_s$.

• Rolling Friction: Occurs when one surface rolls over another. Much smaller than kinetic friction.

Laws of Friction:

1. Friction is proportional to normal force: $f = \mu N$
2. Friction is independent of contact area.
3. Static friction > Kinetic friction

Coefficient of Friction ($\mu$):

• $\mu_s$ for static, $\mu_k$ for kinetic
• Depends on the nature of surfaces

Angle of Friction ($\theta$): $$\tan\theta = \mu$$

Angle of Repose ($\theta_r$):

• The angle of incline at which an object just starts to slide
• $\tan\theta_r = \mu_s$

Applications:

• Walking, rolling tires, gripping objects
• Machines: minimized with lubrication
• Brakes, clutches use friction intentionally

Example:

Question: A 2 kg block is placed on a rough horizontal surface. Coefficient of static friction is 0.4. Find the minimum force required to move it.

Solution: $f_{s,max} = \mu_s N = 0.4 \times (2 \times 9.8) = 0.4 \times 19.6 = 7.84\, N$ Minimum force required = 7.84 N