Friction

• Static friction: Force that operates between bodies not slipping on each other

  • Self-adjustable in magnitude (adjusts to maintain relative rest)
  • Limited by maximum value: $f_s \leq \mu_s N$
  • Direction: configured to prevent relative motion

• Kinetic friction: Force between bodies slipping over each other

  • Constant magnitude: $f_k = \mu_k N$ (for given normal force)
  • Direction: always opposes relative motion
  • Generally $\mu_k < \mu_s$ for same surfaces

• Transition occurs when applied force exceeds maximum static friction

The kinetic friction on the block acts forward (in the direction of the cart's motion).

Key principle: Kinetic friction on a body A slipping against body B is opposite to the velocity of A with respect to B.

Since the block is sliding backward relative to the cart:

• The block's velocity relative to the cart is backward
• Therefore, friction from the cart acts on the block in the forward direction
• This counterintuitive case demonstrates that friction doesn't always oppose absolute motion, but rather opposes relative motion between contacting surfaces

Increasing the angle θ creates multiple competing effects:

Direct effects:

1. Horizontal component decreases: $F_h = F\cos\theta$ decreases as θ increases

   • Reduces tendency to move horizontally
   • Decreases required friction to maintain equilibrium

2. Vertical component increases: $F_v = F\sin\theta$ increases as θ increases

   • Reduces normal force: $N = mg - F\sin\theta$
   • Decreases maximum possible friction: $f_{max} = \mu_s N = \mu_s(mg - F\sin\theta)$

Combined effect on static case:

• Two opposing influences on friction required for equilibrium
• At small angles: friction decreases as θ increases (horizontal component effect dominates)
• At large angles: maximum available friction decreases more rapidly (vertical component effect dominates)
• There exists an optimal angle that minimizes the required force magnitude

For kinetic friction:

• As θ increases, normal force decreases
• Results in lower kinetic friction: $f_k = \mu_k(mg - F\sin\theta)$
• Makes it easier to maintain motion once started

To verify friction's independence from contact area:

1. Measure the coefficient of friction with the block resting on its largest face
2. Repeat the measurement with the block positioned on smaller faces
3. Keep the normal force constant by:
   • Either using the same block weight in all trials
   • Or adding weights to maintain identical normal force when necessary
4. Compare the resulting friction coefficients (μ = W₂/W₁)

If friction is truly independent of area, the measured coefficients will be the same regardless of which face contacts the surface, provided the normal force remains constant and the surface materials are identical.

• Sliding friction: Energy is lost through breaking and reforming molecular bonds between surfaces, causing heat generation and material deformation
• Rolling friction: Energy losses occur primarily through:
  1. Minor deformation at the contact point (elastic hysteresis)
  2. Micro-slippage at the contact surface
  3. Surface adhesion effects
• In sliding, the entire contact area experiences relative motion
• In rolling, only a small portion of the wheel makes momentary contact with the surface
• This fundamental difference explains why moving heavy objects with wheels conserves significantly more energy than dragging them

The coefficient of static friction (μs) can be determined by:

1. Placing a block on an adjustable inclined plane
2. Gradually increasing the angle of inclination until the block just begins to slide
3. Measuring the angle θ at which sliding begins
4. Calculating: μs = tan θ

This works because at the critical angle:

• The component of weight parallel to the plane (mg·sin θ) equals the maximum static friction force (μs·mg·cos θ)
• At this point: μs = tan θ

The method provides a direct measurement without needing pulleys or additional weights.

Ball bearings reduce friction by substituting rolling motion for sliding motion:

• Steel balls roll between fixed and rotating parts, minimizing surface contact
• Rolling friction is significantly lower than sliding friction (often 10-20 times less)
• The balls maintain a small point of contact rather than broad surface contact
• As parts move relative to each other, the balls simply roll rather than slide
• This mechanism transforms potential energy loss into rotational motion of the balls

Example: In industrial machinery, ball bearings often reduce energy consumption by 15-25% compared to simple bushings or sliding surfaces.

Before reaching the critical angle for slipping:

The static friction force (fs) has:

• Direction: Up the inclined plane (opposite to potential motion)
• Magnitude: $f_s = mg\sin θ$

Key characteristics:

• The force is self-adjusting, automatically taking the exact value needed to prevent motion
• It varies with the angle: as θ increases, fs increases
• It is always less than or equal to the maximum possible static friction: $f_s ≤ μ_s N = μ_s mg\cos θ$

At any angle below the critical angle:

1. The normal force equals $N = mg\cos θ$
2. The component of weight down the plane equals $mg\sin θ$
3. For equilibrium, the static friction matches this component: $f_s = mg\sin θ$

Example: If a 2 kg block is on a 15° incline, the static friction force is $f_s = 2 \text{ kg} \times 9.8 \text{ m/s}^2 \times \sin(15°) = 5.08 \text{ N}$ up the plane.

Note: Unlike kinetic friction, static friction is not constant but adjusts to maintain equilibrium until it reaches its maximum value.

Newton's First Law application to this equilibrium situation:

• For an object at rest to remain at rest, the net force must equal zero
• In the horizontal direction: applied force (F) = static friction force (f)
• In the vertical direction: normal force (N) = weight (mg)
• The static friction force automatically adjusts its magnitude to maintain zero net force
• This exemplifies a force-balancing mechanism that creates equilibrium

Key insight: Newton's First Law explains why static friction is self-adjusting rather than constant; it must create the precise opposing force needed to maintain zero acceleration, regardless of the magnitude of the applied force (up to f_max).

The weight (mg) of an object on an inclined plane can be resolved into:

• Component parallel to incline: mg·sin(θ)

  • This component causes the object to slide down
  • Must be counteracted by friction or an external force to prevent sliding

• Component perpendicular to incline: mg·cos(θ)

  • Equals the normal force (N) in static situations
  • Determines the magnitude of the maximum static friction force

These components explain why objects tend to slide down inclined surfaces, and why steeper angles (larger θ) make sliding more likely.