🥋 Newton’s Second Law with Friction — Problem Breakdown (White to Black Belt Mastery)

Quote:
Train your mind like a fighter. One clean step at a time.

White Belt (Easy) — Core Problem

Problem:
A 5.00 kg box is pulled across a horizontal floor by a constant 12.0 N force. The kinetic friction coefficient is μₖ = 0.200. The box starts from rest.
Find:

1. The acceleration.
2. The time to reach 3.00 m/s.

Given:
m = 5.00 kg, Fₐₚₚ = 12.0 N, μₖ = 0.200, v₀ = 0, v = 3.00 m/s, g = 9.80 m/s².

Plan:
We’ll follow Newton’s second law (see Newton’s Laws Study Guide):

1. Compute friction: Fₖ = μₖ N, and N = m g on level ground.
2. Net force: Fₙₑₜ = Fₐₚₚ − Fₖ.
3. Use Fₙₑₜ = m a to get a.
4. Use v = v₀ + a t to find t.

Solve:
N = m g = 5.00 × 9.80 = 49.0 N.
Fₖ = μₖ N = 0.200 × 49.0 = 9.80 N.
Fₙₑₜ = 12.0 − 9.80 = 2.20 N.
a = Fₙₑₜ / m = 2.20 / 5.00 = 0.440 m/s².
t = (v − v₀) / a = 3.00 / 0.440 = 6.82 s.

Answer (3SF, with units):
a = 0.440 m/s²
t = 6.82 s

Common Mistakes and Pitfalls:

• Forgetting friction: use Fₖ = μₖ m g.
• Mixing up static and kinetic friction.
• Dropping units.
• Rounding too early.
• Assuming Fₙₑₜ = Fₐₚₚ without subtracting Fₖ.

Sensei’s Shortcuts:

• On level ground, N = m g (review Vectors Study Guide for components).
• If pull ≈ friction, expect small acceleration.
• To hit a target speed, just use t ≈ v/a.

Practice Drill (Easy):

1. m = 3.00 kg, Fₐₚₚ = 10.0 N, μₖ = 0.300 → a = 0.393 m/s².
2. m = 8.00 kg, Fₐₚₚ = 25.0 N, μₖ = 0.150 → t = 2.42 s.
3. m = 4.00 kg, μₖ = 0.250. Find Fₐₚₚ for a ≥ 0.500 m/s² → 11.8 N.

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Yellow Belt (Medium) — Extension Problem

Variation:
A 5.00 kg box slides down a 10.0° incline with μₖ = 0.100. It starts from rest.
Find the acceleration and speed after 4.00 s.

Outline:
Along slope: a = g (sin θ − μₖ cos θ).
Numbers: sin 10° = 0.173, cos 10° = 0.985.
a = 9.80 (0.173 − 0.100 × 0.985) = 0.737 m/s².
v = a t = 0.737 × 4.00 = 2.95 m/s.

👉 This shows how inclines plus friction work — a great follow-up to the Kinematics in 1D Study Guide.
For a visual, try the PhET Forces & Motion Simulation.

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Black Belt (Hard) — Exam Challenge

Challenge:
Two masses are connected: m₁ = 6.00 kg on a rough table (μₖ = 0.250) and m₂ = 4.00 kg hanging off the edge. Find acceleration and tension.

Strategy:
Use free-body diagrams (see Vectors Study Guide) and Newton’s laws. Combine both equations.

Solution:
Fₖ = μₖ m₁ g = 14.7 N.
Equation for m₁: T − Fₖ = m₁ a.
Equation for m₂: m₂ g − T = m₂ a.
Add: 39.2 − 14.7 = 10.0 a → a = 2.45 m/s².
T = 39.2 − 4.00 × 2.45 = 29.4 N.

👉 For more practice, see OpenStax University Physics — Newton’s Laws.

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Sensei’s Final Words

Every rep counts. Small steps today, mastery tomorrow.

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