🧱 Static vs. Kinetic Friction

Part 1 of 7 — Friction

Friction is the force that resists relative motion (or attempted motion) between surfaces in contact. It's what lets you walk, drive a car, and hold objects. Without friction, life would be... slippery.

Understanding the two types of friction — static and kinetic — is essential for AP Physics 1 dynamics problems.

What Is Friction?

Friction is a contact force that acts parallel to the contact surface and opposes relative motion (or the tendency of relative motion) between two surfaces.

Two Types of Friction

Key Differences

Static Friction in Detail

Static friction keeps an object from starting to move. It's a self-adjusting force:

• Push a box with 5 N → $f_s = 5$ N (box stays still)
• Push with 10 N → $f_s = 10$ N (box stays still)
• Push with 15 N → $f_s = 15$ N (box stays still)
• Push with 20 N → $f_{s,\max}$ is exceeded → box starts moving!

The Inequality

$f_s \leq \mu_s N$

• $f_s$ matches the applied force up to a maximum value
• The maximum is $f_{s,\max} = \mu_s N$
• Once the applied force exceeds $f_{s,\max}$, the object starts to move

Static Friction Can Point in Any Direction

Common misconception: friction always opposes motion. Static friction opposes the tendency of motion:

• A box on a truck accelerating forward: static friction points forward (prevents the box from sliding backward relative to the truck)
• Walking: static friction points forward (your foot pushes back, friction pushes you forward)

Kinetic Friction in Detail

Once an object starts sliding, kinetic friction takes over.

$f_k = \mu_k N$

Key Properties

• $f_k$ is constant (doesn't depend on speed in the AP model)
• $f_k$ opposes the direction of sliding (velocity relative to the surface)
• $\mu_k < \mu_s$ for the same surfaces → it's easier to keep something moving than to start it moving

Why $\mu_k < \mu_s$?

When surfaces are stationary relative to each other, microscopic bonds form between surface irregularities. Once sliding begins, these bonds are continuously broken before they can fully form, reducing the friction force.

Typical Coefficients

Static vs. Kinetic Friction Concepts 🎯

Friction Calculations 🧮

A 20 kg box sits on a horizontal surface. $\mu_s = 0.50$, $\mu_k = 0.40$, $g = 10$ m/s².

1. What is the maximum static friction force (in N)?

2. If the box is sliding, what is the kinetic friction force (in N)?

3. You push with 60 N horizontally. Does the box move? What is the friction force (in N)?

Classify the Friction 🔍

Exit Quiz — Static vs. Kinetic Friction ✅

📐 Friction Equations

Part 2 of 7 — Friction

Now that we understand the two types of friction, let's master the mathematical relationships. These equations are used in nearly every dynamics problem on the AP exam.

$f_s \leq \mu_s N \qquad f_k = \mu_k N$

The Static Friction Inequality

$f_s \leq \mu_s N$

What This Means

• $f_s$ = actual static friction force (what it is right now)
• $\mu_s$ = coefficient of static friction (dimensionless, depends on surfaces)
• $N$ = normal force (perpendicular contact force)
• $\mu_s N$ = maximum static friction force

The Inequality Is Important!

The $\leq$ sign means static friction can be anywhere from zero up to $\mu_s N$:

$0 \leq f_s \leq \mu_s N$

When to Use $f_s = \mu_s N$

Only when the problem says:

• "On the verge of sliding"
• "About to move"
• "Maximum static friction"
• "What is the largest force before the object moves?"

The Kinetic Friction Equation

$f_k = \mu_k N$

What This Means

• $f_k$ = kinetic friction force (constant while sliding)
• $\mu_k$ = coefficient of kinetic friction
• $N$ = normal force

Key Properties

1. $f_k$ is constant — it doesn't depend on speed (AP model)
2. $f_k$ is always less than $f_{s,\max}$ — since $\mu_k < \mu_s$
3. $N$ is NOT always $mg$ — it depends on other vertical forces

Normal Force Affects Friction

Since $f = \mu N$, anything that changes the normal force changes friction:

Using Friction in $F = ma$ Problems

Horizontal Surface — Constant Velocity

If an object slides at constant velocity ($a = 0$):

$F_{\text{applied}} - f_k = 0$ $F_{\text{applied}} = \mu_k N = \mu_k mg$

Horizontal Surface — Accelerating

If an object slides with acceleration $a$:

$F_{\text{applied}} - f_k = ma$ $F_{\text{applied}} - \mu_k mg = ma$ $a = \frac{F_{\text{applied}} - \mu_k mg}{m}$

Example

A 10 kg box is pushed with 60 N on a surface where $\mu_k = 0.30$. ($g = 10$ m/s²)

$f_k = \mu_k mg = 0.30 \times 10 \times 10 = 30 \text{ N}$ $a = \frac{60 - 30}{10} = \frac{30}{10} = 3 \text{ m/s}^2$

Friction Equation Concepts 🎯

Friction Equation Practice 🧮

Use $g = 10$ m/s² for all problems.

1. A 15 kg block on a surface ($\mu_k = 0.20$) is pushed with 50 N horizontally. What is the acceleration (in m/s²)?

2. A 10 kg block slides on a surface ($\mu_k = 0.50$). What horizontal force is needed for the block to slide at constant velocity (in N)?

3. A 25 kg box sits on a surface with $\mu_s = 0.60$. What is the maximum horizontal push before it starts moving (in N)?

Round all answers to 3 significant figures.

Equation Selection 🔍

Exit Quiz — Friction Equations ✅

📋 Free Body Diagrams with Friction

Part 3 of 7 — Friction

Adding friction to FBDs requires care. You must determine the direction of friction (it depends on the situation!) and correctly write Newton's Second Law equations for both axes.

Drawing Friction on FBDs

Step 1: Determine the Type

• Is the object sliding? → Kinetic friction ($f_k = \mu_k N$)
• Is the object stationary? → Static friction ($f_s \leq \mu_s N$)

Step 2: Determine the Direction

• Kinetic friction: Opposes the direction of sliding (velocity relative to surface)
• Static friction: Opposes the tendency to slide

Step 3: Draw the Arrow

• Friction is parallel to the surface
• At the contact point, pointing in the direction determined in Step 2

Complete FBD for a Block Pushed Across a Rough Floor

Forces:

• $\vec{W} = mg$ downward
• $\vec{N}$ upward (perpendicular to surface)
• $\vec{F}_{\text{applied}}$ in the direction of push
• $\vec{f}_k$ opposite to the direction of motion (along the surface)

FBD Examples with Friction

Example 1: Block Pushed Right, Sliding Right

• Weight: down
• Normal: up
• Applied force: right
• Kinetic friction: left (opposes rightward sliding)

Example 2: Block Sliding Right with NO Applied Force (Slowing Down)

• Weight: down
• Normal: up
• Kinetic friction: left (opposes rightward motion)
• No applied force!
• Net force is to the left → block decelerates

Example 3: Box on a Truck Accelerating Forward (Box Not Sliding)

• Weight: down
• Normal: up
• Static friction: forward (prevents box from sliding backward relative to the truck)

Key insight: Static friction can point in ANY direction along the surface. It points in whatever direction is needed to prevent relative motion.

Example 4: Block on a Ramp (Not Sliding)

• Weight: straight down
• Normal: perpendicular to ramp surface (tilted)
• Static friction: up the ramp (prevents sliding down)

Writing Newton's Second Law with Friction

Horizontal Surface, Object Sliding Right

x-direction (horizontal): $\sum F_x = F_{\text{app}} - f_k = ma_x$

y-direction (vertical): $\sum F_y = N - mg = 0 \quad \Rightarrow \quad N = mg$

Combining:

$F_{\text{app}} - \mu_k mg = ma$ $a = \frac{F_{\text{app}} - \mu_k mg}{m}$

What If No Applied Force? (Object Sliding and Slowing)

$\sum F_x = -f_k = ma$ $-\mu_k mg = ma$ $a = -\mu_k g$

This is powerful: A sliding object on a horizontal surface decelerates at $\mu_k g$, regardless of mass!

Example

A box slides across a surface with $\mu_k = 0.30$ at initial speed 12 m/s. How far does it slide?

$a = -\mu_k g = -0.30 \times 9.8 = -2.94 \text{ m/s}^2$ $v^2 = v_0^2 + 2a\Delta x$ $0 = 144 + 2(-2.94)\Delta x$ $\Delta x = 144/5.88 = 24.5 \text{ m}$

FBD with Friction 🎯

FBD with Friction — Calculations 🧮

1. A 5 kg block slides to the right on a surface with $\mu_k = 0.40$. No other horizontal force acts. What is the block's acceleration (magnitude, in m/s²)? Use $g = 10$ m/s².

2. The block from #1 has an initial speed of 10 m/s. How far does it slide before stopping (in m)?

3. A 12 kg box is pushed with 80 N across a rough floor ($\mu_k = 0.30$). What is the acceleration (in m/s²)?

Round all answers to 3 significant figures.

Friction Direction Practice 🔍

Exit Quiz — FBDs with Friction ✅

📐 Friction with Angled Forces

Part 4 of 7 — Friction

When you pull or push an object at an angle, the vertical component of your force changes the normal force, which in turn changes the friction force. This is a very common AP Physics 1 scenario.

Pulling at an Angle Above Horizontal

A force $F$ is applied at angle $\theta$ above horizontal to a block on a rough surface.

Force Components

• $F_x = F\cos\theta$ (horizontal, in direction of motion)
• $F_y = F\sin\theta$ (vertical, upward)

y-direction (no vertical acceleration):

$N + F\sin\theta - mg = 0$ $N = mg - F\sin\theta$

Friction Force

$f_k = \mu_k N = \mu_k(mg - F\sin\theta)$

Pulling up reduces the normal force, which reduces friction. This is why it's easier to pull a suitcase at an angle than to push it!

x-direction:

$F\cos\theta - f_k = ma$ $F\cos\theta - \mu_k(mg - F\sin\theta) = ma$

Example

Pull a 20 kg box with 80 N at 30° ($\mu_k = 0.25$, $g = 10$ m/s²):

$N = 200 - 80\sin 30° = 200 - 40 = 160$ N

$f_k = 0.25 \times 160 = 40$ N

$F_x = 80\cos 30° = 69.3$ N

$a = (69.3 - 40)/20 = 29.3/20 = 1.47$ m/s²

Pushing at an Angle Below Horizontal

A force $F$ is applied at angle $\theta$ below horizontal (pushing downward and forward).

Force Components

• $F_x = F\cos\theta$ (horizontal, forward)
• $F_y = -F\sin\theta$ (vertical, downward)

y-direction:

$N - mg - F\sin\theta = 0$ $N = mg + F\sin\theta$

Friction Force

$f_k = \mu_k N = \mu_k(mg + F\sin\theta)$

Pushing down increases the normal force, which increases friction. More effort wasted fighting friction!

Comparing Pull vs. Push

For the same force magnitude and angle, pulling upward is always more efficient:

The Optimal Pulling Angle

There's actually an optimal angle that maximizes acceleration (or minimizes the force needed to move at constant velocity).

For Constant Velocity ($a = 0$):

$F\cos\theta = \mu_k(mg - F\sin\theta)$

Solving for $F$: $F = \frac{\mu_k mg}{\cos\theta + \mu_k \sin\theta}$

To minimize $F$, take $dF/d\theta = 0$:

$\theta_{\text{optimal}} = \tan^{-1}(\mu_k)$

Example

For $\mu_k = 0.40$: $\theta_{\text{opt}} = \tan^{-1}(0.40) = 21.8°$

This means pulling at about 22° above horizontal requires the least force to keep the object moving at constant velocity.

This explains why suitcase handles are angled — it's not just for comfort; it's physics!

Angled Force with Friction 🎯

Angled Force Calculations 🧮

A 10 kg block on a rough surface ($\mu_k = 0.40$) is pulled with 50 N. Use $g = 10$ m/s².

1. If pulled horizontally, what is the friction force (in N)?

2. If pulled at 37° above horizontal ($\sin 37° = 0.60$, $\cos 37° = 0.80$), what is the normal force (in N)?

3. For the 37° pull, what is the acceleration (in m/s²)?

Round all answers to 3 significant figures.

Pull vs. Push Comparison 🔍

Exit Quiz — Friction with Angled Forces ✅

🔬 Finding $\mu$ Experimentally

Part 5 of 7 — Friction

How do physicists and engineers actually measure the coefficient of friction? This part covers the classic experimental methods — which are also common AP lab questions.

Method 1: Horizontal Pull

Setup

Place a block on a horizontal surface. Attach a spring scale and pull horizontally, gradually increasing the force.

Finding $\mu_s$

Record the force when the block just starts to move. That's $f_{s,\max}$.

$\mu_s = \frac{f_{s,\max}}{N} = \frac{f_{s,\max}}{mg}$

Finding $\mu_k$

Pull the block at constant velocity. The applied force equals kinetic friction.

$\mu_k = \frac{f_k}{N} = \frac{F_{\text{constant velocity}}}{mg}$

Example

A 2 kg block requires 9.8 N to just start moving and 7.8 N to slide at constant speed.

$\mu_s = \frac{9.8}{2(9.8)} = \frac{9.8}{19.6} = 0.50$

$\mu_k = \frac{7.8}{19.6} = 0.40$

Method 2: The Incline Method

Finding $\mu_s$ with a Ramp

Place a block on a ramp. Slowly increase the angle until the block just begins to slide. Record this critical angle $\theta_c$.

At the critical angle, the block is on the verge of sliding:

$f_{s,\max} = mg\sin\theta_c$ $N = mg\cos\theta_c$

$\mu_s = \frac{f_{s,\max}}{N} = \frac{mg\sin\theta_c}{mg\cos\theta_c} = \tan\theta_c$

$\boxed{\mu_s = \tan\theta_c}$

Beautiful result: $\mu_s$ equals the tangent of the critical angle. No need to know the mass!

Finding $\mu_k$ with a Ramp

Give the block a push and find the angle where it slides at constant velocity:

$\mu_k = \tan\theta_{\text{constant velocity}}$

Why This Method Is Elegant

• Mass cancels out — you don't need to measure it
• Simple equipment: just a ramp and a protractor
• Highly reproducible

Method 3: Graphical Analysis

Using an $f$ vs. $N$ Graph

Vary the normal force (by stacking masses) and measure friction at each value. Plot $f$ vs. $N$.

$f = \mu N$

This is a linear relationship passing through the origin:

• Slope = $\mu$
• No y-intercept (friction is zero when $N = 0$)

Using an Acceleration Method

Push a block with known force $F$ and measure acceleration $a$:

$F - \mu_k mg = ma$ $a = \frac{F}{m} - \mu_k g$

Plot $a$ vs. $F/m$:

• Slope = 1
• y-intercept = $-\mu_k g$
• So $\mu_k = -\text{(y-intercept)}/g$

Sources of Error

Experimental Methods Quiz 🎯

Experimental Calculations 🧮

1. A 3 kg block on a horizontal surface requires 12 N to just start sliding. What is $\mu_s$? ($g = 10$ m/s²)

2. A block slides at constant velocity down a ramp inclined at 20°. What is $\mu_k$? ($\tan 20° = 0.364$, round to 3 significant figures)

3. In an experiment, doubling the mass of a block doubles the measured friction force. This confirms that friction is proportional to what variable?

Lab Reasoning 🔍

Exit Quiz — Measuring Friction ✅

🛠️ Problem-Solving Workshop

Part 6 of 7 — Friction

This workshop brings together all friction concepts: static vs. kinetic, FBDs with friction, angled forces, and experimental methods. Work through these problems systematically using the FBD → equations → solve approach.

Friction Problem Strategy

Step 1: Determine the Type of Friction

• Is the object sliding? → Kinetic ($f_k = \mu_k N$)
• Is the object stationary? → Static ($f_s \leq \mu_s N$)
• "On the verge"? → Static at maximum ($f_s = \mu_s N$)

Step 2: Find the Normal Force First!

The normal force is NOT always $mg$. Set up the y-direction equation: $N = mg \pm F_y \quad \text{or} \quad N = mg\cos\theta \text{ (on incline)}$

Step 3: Calculate Friction

$f = \mu N$

Step 4: Apply $F = ma$ in the Direction of Motion

$\sum F_{\text{along motion}} = ma$

Worked Example

A 10 kg block is pulled at 30° above horizontal with 60 N across a rough surface ($\mu_k = 0.20$, $g = 10$ m/s²).

Normal force: $N = mg - F\sin 30° = 100 - 30 = 70$ N

Friction: $f_k = 0.20 \times 70 = 14$ N

Horizontal: $F\cos 30° - f_k = ma$ $52 - 14 = 10a \Rightarrow a = 3.8 \text{ m/s}^2$

Worked Example 2: Will It Slide?

A 5 kg block sits on a horizontal surface ($\mu_s = 0.50$, $\mu_k = 0.30$). You push horizontally with 20 N.

Check: $f_{s,\max} = \mu_s mg = 0.50 \times 50 = 25$ N

Since $20 < 25$: The block does NOT move. $f_s = 20$ N.

Now push with 30 N:

Since $30 > 25$: The block starts moving!

Once moving: $f_k = \mu_k mg = 0.30 \times 50 = 15$ N

$a = \frac{30 - 15}{5} = \frac{15}{5} = 3 \text{ m/s}^2$

Worked Example 3: Constant Velocity Pulling

What force is needed to pull a 20 kg box at constant velocity at 25° above horizontal? ($\mu_k = 0.35$, $g = 10$ m/s²)

At constant velocity ($a = 0$): $F\cos 25° = \mu_k(mg - F\sin 25°)$ $F(0.906) = 0.35(200 - 0.423F)$ $0.906F = 70 - 0.148F$ $1.054F = 70$ $F = 66.4 \text{ N}$

Workshop Multiple Choice 🎯

Workshop Calculations 🧮

1. A 8 kg block is pushed with 60 N horizontally across a rough floor ($\mu_k = 0.25$, $g = 10$ m/s²). What is the acceleration (in m/s²)?

2. A block slides to a stop in 4 seconds from an initial speed of 12 m/s on a horizontal surface. What is $\mu_k$? ($g = 10$ m/s²)

3. What angle must a ramp be tilted to for a block to be on the verge of sliding if $\mu_s = 0.577$? (in degrees, given $\tan^{-1}(0.577) = 30°$)

Round all answers to 3 significant figures.

Quick Reasoning Checks 🔍

Exit Quiz — Friction Workshop ✅

🎯 Synthesis & AP Review

Part 7 of 7 — Friction

Congratulations on completing the Friction unit! This final lesson reviews every key idea, connects friction to the broader dynamics framework, and tests you with AP-style questions.

Concept Map: Friction in Dynamics

$\text{Force} \rightarrow \text{FBD} \rightarrow F_{\text{net}} = ma$

Key Friction Equations

Key Principles

• $\mu_s > \mu_k$ always — it's harder to start motion than maintain it
• Friction is independent of contact area and independent of speed
• Friction always opposes relative motion (or tendency of motion)
• The normal force determines friction — anything that changes $N$ changes friction

Conceptual Review 🧠

AP-Style Calculations 📝

1. A 6 kg block slides with $\mu_k = 0.30$ on a horizontal surface. What is the magnitude of its deceleration (m/s²)? ($g = 10$ m/s²)

2. A block on a flat surface requires 24 N to start moving and 18 N to keep it moving at constant velocity. The block weighs 60 N. What is $\mu_s$?

3. A 50 N horizontal force pushes a 10 kg block at constant velocity. What is $\mu_k$? ($g = 10$ m/s²)

Round all answers to 3 significant figures.

AP Reasoning Questions 🎯

Final Exit Quiz — Friction Unit ✅