Coefficient of Friction & F = ma

For problems on a horizontal surface with friction:

Step 1: Resolve any angled forces into vertical and horizontal components.

Step 2: Calculate the normal reaction force (R). With a horizontal surface, (R) acts vertically upwards. The magnitude of (R) makes the total vertical force zero. Be careful—vertical forces other than gravity affect (R).

Step 3: Calculate (F_{MAX} = \mu R) and find the resultant of all horizontal forces. If the resultant of other horizontal forces is (\leq F_{MAX}), friction balances them and the object stays stationary.

Step 4: Use (F = ma) to determine acceleration (only when the object is moving).

When an object is stationary:

• If resultant horizontal force (\leq F_{MAX}): object remains at rest; friction equals the resultant (opposite direction)
• If resultant = (F_{MAX}): limiting equilibrium (on the point of moving)
• If resultant > (F_{MAX}): object accelerates; friction = (F_{MAX}) opposing motion

When moving: friction always equals (F_{MAX} = \mu R) in the direction opposite to motion.

A wooden block of mass 2 kg rests on a rough horizontal floor ((\mu = 0.6)). A force of 15 N is applied at 30° to the horizontal. Find the acceleration.

Vertical: (R + 15\sin 30° = 2g) → (R = 20 - 7.5 = 12.5,\text{N})

Horizontal: (15\cos 30° - F = 2a). (F_{MAX} = 0.6 \times 12.5 = 7.5,\text{N}). Resultant = (15\cos 30° \approx 13,\text{N} > 7.5) → block moves.

(13 - 7.5 = 2a) → (a = 2.75,\text{m/s}^2)

When a force acts at an angle, resolve into components:

• Horizontal: (F_h = F\cos\theta)
• Vertical: (F_v = F\sin\theta)

The parallel component causes the effect; the perpendicular component may affect the normal reaction. Use trigonometry and draw a vector triangle.