Physics — Rotational Motion (Torque, Angular Momentum)

Physics — Rotational Motion (Torque, Angular Momentum). Practice questions to deepen understanding of rotational motion in physics — torque and angular momentum. Online physics practice with full solutions and step-by-step explanations.

Rotational motion physics practice — 50 questions: torque τ = Iα, moment of inertia, angular momentum L = Iω, conservation of L, rolling, rotational energy.

50 questions

Question 1

2.00 pts

⭕ Radian:

What is a radian?

An angle for which arc length = radius: θ = s/r

360 degrees

180 degrees

A constant angle

Explanation:

💡 Detailed explanation:

Radian! ⭕

Radian:

θ = s/r

or:

1 rad = the angle for which s = r

🔍 Definition:

A radian = the "natural" unit for angle

When the arc length = the radius
→ the angle = 1 radian

💡 Conversions:

• Full circumference: 2πr
• Full angle: θ = 2πr/r = 2π rad

360° = 2π rad

So:

180° = π rad
90° = π/2 rad
45° = π/4 rad
1° = π/180 rad ≈ 0.01745 rad
1 rad = 180°/π ≈ 57.3°

📊 Examples:

Circle of radius r=2m:

• Arc s=2m → θ=1 rad ≈ 57.3°
• Arc s=4m → θ=2 rad ≈ 114.6°
• Arc s=2πr=12.56m → θ=2π rad = 360°

⚡ Why radians?

• Cleanest formulas
• v = ωr (in rad/s)
• Without conversion factors
• The natural unit in physics

Question 2

2.00 pts

🔄 Angular velocity:

What is ω (omega)?

ω = Δθ/Δt, the rate of change of the angle (rad/s)

Ordinary velocity

The force

The energy

Explanation:

💡 Detailed explanation:

Angular velocity! 🔄

Angular velocity (ω):

ω = Δθ/Δt

or:

ω = dθ/dt

🔍 Components:

• ω: angular velocity (rad/s)
• θ: angle (rad)
• t: time (s)
• Vector! Direction: right-hand rule

Meaning:

How many radians per second
the body rotates

Example:
ω = 2π rad/s
→ one full revolution per second
→ frequency f = 1 Hz

📊 Common units:

• rad/s (the basic one)
• RPM (revolutions per minute)
• deg/s (degrees per second)

Conversions:
1 RPM = 2π/60 rad/s ≈ 0.105 rad/s
1 Hz = 2π rad/s

💡 Connection to linear velocity:

v = ωr

The point at the perimeter
moves at speed v

Far from center → larger v

Conservation of T:

Period: T = 2π/ω
Frequency: f = 1/T = ω/(2π)

Question 3

2.00 pts

⚡ Angular acceleration:

What is α (alpha)?

α = Δω/Δt, the rate of change of angular velocity (rad/s²)

The velocity

The angle

The force

Explanation:

💡 Detailed explanation:

Angular acceleration! ⚡

Angular acceleration (α):

α = Δω/Δt

or:

α = dω/dt = d²θ/dt²

🔍 Components:

• α: angular acceleration (rad/s²)
• ω: angular velocity (rad/s)
• t: time (s)

Meaning:
How much the angular velocity changes
per second

Example:

A wheel accelerated from 0 to 20 rad/s
in 4s

α = (20-0)/4 = 5 rad/s²

Each second ω increases by 5 rad/s

💡 Connection to ordinary acceleration:

Tangential acceleration:
a_t = αr

Centripetal acceleration:
a_c = ω²r

Total acceleration:
a = √(a_t² + a_c²)

Question 4

2.00 pts

📐 Rotational kinematics:

What are the formulas (for constant α)?

ω = ω₀ + αt, θ = θ₀ + ω₀t + ½αt², ω² = ω₀² + 2αθ

Only one

No formulas

Completely different

Explanation:

💡 Detailed explanation:

Rotational kinematics formulas! 📐

🔄 3 central formulas:

1️⃣ Velocity:
ω = ω₀ + αt

2️⃣ Angle:
θ = θ₀ + ω₀t + ½αt²

3️⃣ Without time:
ω² = ω₀² + 2α(θ-θ₀)

📊 Comparison with linear kinematics:

Linear	Rotational
x	θ
v	ω
a	α
v = v₀ + at	ω = ω₀ + αt
x = x₀ + v₀t + ½at²	θ = θ₀ + ω₀t + ½αt²
v² = v₀² + 2a·Δx	ω² = ω₀² + 2α·Δθ

💡 Insight:

The formulas are identical in form!

Just replace:
x → θ
v → ω
a → α

This is true for all of mechanics!

Question 5

2.00 pts

🧮 Exercise:

A wheel accelerates from ω₀=0 with α=2 rad/s²

What is the angular velocity after t=5s?

\(\omega = 10\) rad/s

\(\omega = 2\) rad/s

\(\omega = 5\) rad/s

\(\omega = 0\)

Explanation:

💡 Detailed explanation:

Basic calculation! 🧮

📐 Solution:

Given:
ω₀ = 0 (starts from rest)
α = 2 rad/s²
t = 5 s

Formula:

ω = ω₀ + αt

ω = 0 + 2×5

ω = 10 rad/s

💡 Meaning:

The wheel rotates at 10 rad/s

How many revolutions per second?
f = ω/(2π) = 10/(2π) ≈ 1.59 Hz

Almost 1.6 revolutions per second!

Angle traversed:
θ = ½αt² = ½×2×25 = 25 rad
≈ 4 full revolutions

Question 6

2.00 pts

🔗 Connection between v and ω:

What is the relation between linear and angular velocity?

v = ωr, the velocity depends on the radius!

v = ω

No relation

v = ω/r

Explanation:

💡 Detailed explanation:

v and ω relation! 🔗

Central relation:

v = ωr

linear velocity
= angular velocity × radius

🔍 Derivation:

Distance along the perimeter: s = rθ

Differentiation with respect to time:
ds/dt = r(dθ/dt)

v = rω ✓

💡 Meaning:

Same ω, different radius
→ different velocity!

Example:
A disc rotating ω=10 rad/s

• Point at r=0.1m: v=1 m/s
• Point at r=0.5m: v=5 m/s
• Point at r=1m: v=10 m/s

Farther → faster!

🎡 Example: a carousel

A child at the center vs a child at the edge

Same ω for both
but:
• At the center: small v
• At the edge: large v

Why is it scarier at the edge? Larger v!

Question 7

2.00 pts

⏱️ Frequency and period:

What is the relation between ω, f, T?

ω = 2πf = 2π/T, all are related!

No relation

Only ω matters

Completely different

Explanation:

💡 Detailed explanation:

Frequency and period! ⏱️

🔄 3 ways to describe rotation:

1️⃣ Angular velocity (ω):
ω (rad/s)

2️⃣ Frequency (f):
f (Hz) = revolutions/second

3️⃣ Period (T):
T (s) = time for one revolution

🔗 The relations:

Formulas:

f = 1/T
(frequency = inverse of period)

ω = 2πf
(one revolution = 2π radians)

ω = 2π/T

or:

T = 2π/ω
f = ω/(2π)

💡 Examples:

A wheel ω = 100 rad/s:
• f = 100/(2π) ≈ 15.9 Hz
• T ≈ 0.063 s

Earth (rotation):
T = 24 hours = 86,400 s
ω = 2π/86,400 ≈ 7.27×10⁻⁵ rad/s

💡 RPM:

An engine 3000 RPM:
= 3000/60 = 50 Hz
= 50×2π = 314 rad/s

Question 8

2.00 pts

🎯 Centripetal acceleration:

What is a_c?

a_c = v²/r = ω²r, directed toward the center!

No acceleration

Only in a straight line

Directed outward

Explanation:

💡 Detailed explanation:

Centripetal acceleration! 🎯

Centripetal acceleration:

a_c = v²/r

or:

a_c = ω²r

Always directed toward the center! →

🔍 Why is there acceleration?

In circular motion:
• Magnitude of velocity is constant (v)
• But direction changes!
• Direction change = acceleration

Derivation:

v⃗ changes direction
Δv⃗ points toward the center

|Δv| = v·Δθ
Δt = Δθ/ω

a = Δv/Δt = (v·Δθ)/(Δθ/ω)
a = vω = v·(v/r)

a_c = v²/r ✓

💡 Examples:

1. Car on a curve:
v = 20 m/s, r = 50m
a_c = 400/50 = 8 m/s²

Friction needed:
f = ma_c = 8000 N (for a 1000kg car)

2. Satellite:
v = 7800 m/s, r = 6.67×10⁶ m
a_c ≈ 9.1 m/s² (~g!)

Gravity supplies it!

⚠️ Important:

a_c is not a "new" acceleration
but a consequence of changing direction

For a real force: a_c requires a real force
(F_c = ma_c)

Question 9

2.00 pts

💪 Centripetal force:

What is F_c?

F_c = ma_c = mv²/r = mω²r, a name for the force that causes rotation

A new force

No such force

Only in space

Explanation:

💡 Detailed explanation:

Centripetal force! 💪

Centripetal force:

F_c = mv²/r

or:

F_c = mω²r

🔍 What is F_c?

It's not a new force!

F_c = a label for the force
that causes circular motion

Who supplies F_c?

• Car on a curve: friction
• Ball on a string: tension
• Satellite: gravity
• Electron in an atom: electric force
• Washing machine: normal force

💡 Examples:

1. Car on a curve:

m = 1000 kg
v = 15 m/s
r = 30 m

F_c = 1000×15²/30 = 7500 N

Static friction must supply this force.
If μ_s·mg < F_c → the car skids out!

2. Ball on a string:

m = 0.5 kg
v = 4 m/s
r = 0.5 m

T = F_c = 0.5×16/0.5 = 16 N

If the string can't bear it → snaps!

⚠️ Common mistake:

"Centrifugal force" - doesn't exist!
It's a fictitious force in a rotating frame

The real force is centripetal (inward)

Question 10

2.00 pts

🛰️ Exercise:

Satellite in orbit r=7×10⁶ m
mass M_earth=6×10²⁴ kg
G=6.67×10⁻¹¹

What is the velocity?

v ≈ 7550 m/s

v = 1000 m/s

v = 100,000 m/s

v = 0

Explanation:

💡 Detailed explanation:

Satellite! 🛰️

📐 Solution:

Given:
r = 7×10⁶ m
M = 6×10²⁴ kg
G = 6.67×10⁻¹¹ N·m²/kg²

Orbital condition:

Gravitational force = centripetal force

GMm/r² = mv²/r

Cancel m:

GM/r² = v²/r

v² = GM/r

v = √(GM/r)

Calculation:

v = √(6.67×10⁻¹¹ × 6×10²⁴ / 7×10⁶)

v = √(4×10¹⁴ / 7×10⁶)

v = √(5.71×10⁷)

v ≈ 7550 m/s

About 7.5 km/s!

💡 Insight:

Velocity is independent of the satellite's mass!

Only on r (orbit radius)

Closer → faster
Farther → slower

Period:
T = 2πr/v ≈ 5830 s ≈ 97 minutes

Question 11

2.00 pts

🎢 Vertical loop:

What is the minimum velocity at the top?

v_min = √(gr), so that gravity is enough for F_c

v = 0

v = √(2gr)

Not possible

Explanation:

💡 Detailed explanation:

Vertical loop! 🎢

🎯 Analysis at the top of the loop:

Forces on the body:
• Gravity mg ↓ (toward center)
• Normal N ↓ (toward center)

Condition:

Sum of forces = F_c

N + mg = mv²/r

Minimum velocity:
When N = 0 (just doesn't fall)

mg = mv²/r

g = v²/r

v_min = √(gr)

💡 Insight:

Right at this threshold:
gravity alone is enough
to provide F_c

• v < √(gr) → falls ✗
• v = √(gr) → just barely makes it ✓
• v > √(gr) → safe ✓✓

Example:

Loop r = 5m
g = 10 m/s²

v_min = √(10×5) = √50 ≈ 7.07 m/s

At a lower velocity → falls!

Starting height:

From energy conservation:
mgh = ½mv_min² + mg(2r)

gh = ½gr + 2gr

h = 2.5r

You need to start from a height of 2.5r!

Question 12

2.00 pts

📚 Summary:

What are the 4 central variables?

θ (angle), ω (angular velocity), α (acceleration), r (radius)

Only ω

No variables

Too many

Explanation:

💡 Detailed explanation:

Kinematics summary! 📚

🔄 Rotational kinematics:

The variables:

• θ - angle (rad)
• ω - angular velocity (rad/s)
• α - angular acceleration (rad/s²)
• r - radius (m)

📊 Central formulas:

Formula	Description
θ = s/r	Angle from arc
ω = Δθ/Δt	Angular velocity
α = Δω/Δt	Angular acceleration
v = ωr	Linear velocity
a_c = v²/r = ω²r	Centripetal accel
F_c = mv²/r	Centripetal force

✅ What we learned:

• Radians
• Angular velocity ω
• Angular acceleration α
• Connection v = ωr
• Frequency and period
• Centripetal acceleration
• Centripetal force
• Satellite orbits
• Vertical loops

Question 13

2.00 pts

🔧 Torque:

What is torque (τ)?

τ = r × F = rF·sin(θ), the ability of a force to cause rotation

The force

The mass

The velocity

Explanation:

💡 Detailed explanation:

Torque! 🔧

Torque - τ:

τ = r × F

or in scalars:

τ = rF·sin(θ)

or:

τ = r_⊥·F

🔍 Components:

• τ: torque (N·m)
• r: lever arm (m) - distance from axis
• F: force (N)
• θ: angle between r and F
• Vector! Direction: right-hand rule

💡 Meaning:

Torque = "rotational force"

Measures the ability of a force
to cause rotation around an axis

Depends on:
1. Magnitude of the force (F)
2. Distance from the axis (r)
3. Angle between them (θ)

Maximum: when θ=90° (perpendicular)

💡 Example - a door:

Where do you push?

• Far from the hinge: small force is enough
• Near the hinge: need a large force
• Push parallel to the door: nothing happens!

That's why the handle is far from the hinge!

📊 Examples of values:

• Engine: 200-400 N·m
• Bike crank: 50-100 N·m
• Door handle: 5-10 N·m

Question 14

2.00 pts

⚖️ Zero torque:

When does τ = 0?

When F=0 or r=0 or θ=0°/180° (F passes through the axis)

Always

Never

Only in space

Explanation:

💡 Detailed explanation:

Zero torque! ⚖️

When does τ = 0?

From the formula: τ = rF·sin(θ)

🔍 3 cases:

1️⃣ F = 0

No force → no torque

Trivial

2️⃣ r = 0

The force acts on the axis

No lever arm → no torque

Example:
A door - you press on the hinge
→ the door doesn't move!

3️⃣ θ = 0° or 180°

The force is through the axis
(directed straight at it or away from it)

sin(0°) = 0
sin(180°) = 0

→ τ = 0

Example:
A bicycle pedal
when it's exactly up/down
pushing on it doesn't help!

💡 Maximum:

Conversely:
τ_max = rF
when θ = 90° (perpendicular)

Then sin(90°) = 1
→ Maximum effect from the force

Question 15

2.00 pts

➕ Multiple forces:

How is τ_net calculated?

τ_net = Στ_i, algebraic sum (with ± signs)

Only the largest

Average

Product

Explanation:

💡 Detailed explanation:

Total torque! ➕

Net torque:

τ_net = Στ_i

Algebraic sum!

🔍 How:

1️⃣ Determine sign:
• Counterclockwise: + (⟲)
• Clockwise: - (⟳)

2️⃣ Calculate each torque:
τ_i = r_i·F_i·sin(θ_i)

3️⃣ Add with signs:
τ_net = τ_1 + τ_2 + τ_3 + ...

💡 Example:

Rod about a pivot at the middle:

• Force F1=10N at r1=2m to the left
→ τ1 = +20 N·m (⟲)

• Force F2=15N at r2=1m to the right
→ τ2 = -15 N·m (⟳)

Net:
τ_net = 20 + (-15) = +5 N·m

→ rotates counterclockwise ⟲

⚖️ Equilibrium:

If τ_net = 0
→ no net rotation
→ rotational equilibrium!

Not rotating
or rotating at constant velocity

Question 16

2.00 pts

⚙️ Second law for rotation:

What is it?

τ_net = Iα, torque causes angular acceleration!

τ = ma

τ = F

No such law

Explanation:

💡 Detailed explanation:

Newton's second law for rotation! ⚙️

Newton's second law for rotation:

τ = Iα

Torque = moment of inertia × angular acceleration

🔍 Meaning:

Just like F = ma!

Only for rotation:

Linear	Rotational
F (force)	τ (torque)
m (mass)	I (moment of inertia)
a (acceleration)	α (angular acceleration)
F = ma	τ = Iα

💡 Insight:

Same equation, different variables!

This is one of the deepest analogies
in physics

The same idea applies wherever there's rotation
(rigid bodies, planets, gyroscopes, stars)

📊 Example:

τ = 10 N·m on a body with I = 2 kg·m²

α = τ/I = 10/2 = 5 rad/s²

Each second ω increases by 5 rad/s

Question 17

2.00 pts

🎯 Moment of inertia:

What is I?

I = Σ(m_i·r_i²), "rotational mass" - resistance to changes in ω

The mass

The force

The velocity

Explanation:

💡 Detailed explanation:

Moment of inertia! 🎯

Moment of inertia (I):

I = Σ(m_i·r_i²)

or continuous:

I = ∫ r² dm

🔍 Meaning:

I = "rotational mass"

Measures: how hard it is to change ω

Properties:

• Units: kg·m²
• Always non-negative ≥ 0
• Depends on the axis!
• Depends on the shape
• Grows with r² (quadratic!)

📊 Common formulas:

Body	I (about center)
Particle	mr²
Hoop (radius R)	MR²
Solid disc	½MR²
Solid sphere	⅖MR²
Thin rod (length L)	ML²/12

💡 Why does it depend on the axis?

Mass farther from the axis
→ contributes more to I

(because of r²)

Same body, different axis
→ different I!

Question 18

2.00 pts

📐 Parallel axis theorem:

What is it?

I = I_cm + Md², transition between parallel axes

No such theorem

Only for discs

I = mr²

Explanation:

💡 Detailed explanation:

Parallel axis theorem! 📐

Parallel axis theorem:

I = I_cm + Md²

🔍 Meaning:

Connects moments of inertia
about different axes

Components:

• I: moment of inertia about a new axis
• I_cm: moment of inertia about the center of mass
• M: total mass
• d: distance between the axes

Condition: parallel axes!

💡 Example:

Uniform rod M, L

About the center:
I_cm = ML²/12

About the end:
d = L/2

I = ML²/12 + M(L/2)²
I = ML²/12 + ML²/4
I = ML²/12 + 3ML²/12
I = 4ML²/12

I = ML²/3 ✓

Identical to the formula!

⚡ Useful application:

Saves calculations!

Just need:
1. I_cm (often known/in a table)
2. The distance to the new axis

Done!

Question 19

2.00 pts

🧮 Exercise:

Disc M=2kg, R=0.3m
Torque τ=5 N·m

What is the angular acceleration?

α = 55.6 rad/s²

α = 2.5 rad/s²

α = 5 rad/s²

α = 10 rad/s²

Explanation:

💡 Detailed explanation:

Rotating disc! 🧮

📐 Solution:

Given:
M = 2 kg
R = 0.3 m
τ = 5 N·m

Step 1: Moment of inertia

Solid disc:

I = ½MR²

I = ½×2×(0.3)²
I = 1×0.09
I = 0.09 kg·m²

Step 2: Second law

τ = Iα

α = τ/I

α = 5/0.09

α ≈ 55.6 rad/s²

💡 Meaning:

High angular acceleration!

After 1 second:
ω = αt = 55.6 rad/s

About 9 revolutions per second

Edge speed:
v = ωR = 55.6×0.3 ≈ 16.7 m/s
(after one second)

Question 20

2.00 pts

🪀 Yo-yo:

What accelerates the yo-yo?

Torque from the tension: τ = TR, causes α

Gravity alone

No torque

Only the string

Explanation:

💡 Detailed explanation:

Yo-yo! 🪀

🔍 Yo-yo analysis:

Forces on the yo-yo:
• Gravity: mg ↓
• Tension: T ↑ (in the string)

Linear motion:

mg - T = ma

(moves down with acceleration a)

Rotational motion:

The tension creates a torque!

τ = T·R (R = axle radius)

τ = Iα

T·R = Iα

Connection:

The string doesn't slip:
a = αR

Two equations, two unknowns (T, a)

Solution:

a = g/(1 + I/(mR²))

If I = ½mR² (disc):

a = g/(1 + ½) = ⅔g

Less than free fall!

💡 Why slower?

Some of the falling energy
goes into rotation

E_p → E_k (translation) + E_k (rotation)

Question 21

2.00 pts

⚖️ Equilibrium:

When is a body in rotational equilibrium?

When Στ = 0, the sum of torques = zero

Always

Never

Only at rest

Explanation:

💡 Detailed explanation:

Rotational equilibrium! ⚖️

Equilibrium condition:

Στ = 0

Sum of torques = 0

→ α = 0
→ ω constant (or zero)

📊 Full equilibrium:

Requires two conditions:

1️⃣ Translational equilibrium:

ΣF = 0

No linear acceleration
v constant (or zero)

2️⃣ Rotational equilibrium:

Στ = 0

No angular acceleration
ω constant (or zero)

💡 Example: ladder against a wall

Ladder m, L leans against the wall

Forces:
• Weight mg
• Friction with the floor
• Normal from the floor
• Normal from the wall

Two conditions:
• ΣF = 0
• Στ = 0

Together they determine the minimum friction
needed to prevent slipping

Question 22

2.00 pts

⚖️ Seesaw:

Two children: m₁=30kg at r₁=2m, m₂=? at r₂=3m

Balance - what is m₂?

\(m_2 = 20\) kg

\(m_2 = 30\) kg

\(m_2 = 45\) kg

\(m_2 = 60\) kg

Explanation:

💡 Detailed explanation:

Seesaw! ⚖️

📐 Solution:

Given:
m₁ = 30 kg, r₁ = 2 m (left)
m₂ = ?, r₂ = 3 m (right)

Balance condition:

Στ = 0

Torques about the pivot:

τ₁ (counterclockwise) = m₁gr₁
τ₂ (clockwise) = m₂gr₂

Balance:
m₁gr₁ = m₂gr₂

Cancel g:
m₁r₁ = m₂r₂

30×2 = m₂×3
60 = 3m₂

m₂ = 20 kg

💡 Insight:

Lighter child
but farther away!

Product: m×r
30×2 = 60
20×3 = 60 ✓

Equal!

Principle:

Farther from the pivot
→ less mass needed

Closer
→ more mass needed

Question 23

2.00 pts

⚙️ Pulley:

What is the mechanical advantage?

Changes the direction of force, sometimes increases it (pulley system)

Only changes direction

Always increases

No advantage

Explanation:

💡 Detailed explanation:

Pulley! ⚙️

🔍 Types of pulleys:

1️⃣ Simple (fixed) pulley:

Hangs from the ceiling

• Changes the direction of force
• Doesn't increase! F_in = F_out
• Mechanical advantage = 1

Why useful?
→ It's easier to pull down
than to lift up

2️⃣ Movable pulley:

Attached to the load

• The string supports it from both sides
• F_in = F_out/2
• Mechanical advantage = 2!

Need half the force
but twice the distance

3️⃣ Pulley system:

Combination of fixed and movable

• n movable pulleys
• Mechanical advantage = 2n

Example:
4 movable pulleys
→ advantage = 8
→ F = W/8

But: must pull 8× the distance!

⚠️ Principle:

No free lunches!

Reduce force → increase distance
Energy is conserved

W = F·d (constant)

Question 24

2.00 pts

🎡 Wheel and axle:

What is the advantage?

Advantage = R/r, small force at large radius → large force at small radius

No advantage

Just decoration

Always 1

Explanation:

💡 Detailed explanation:

Wheel and axle! 🎡

Wheel and axle:

Two cylinders of different diameters
rotating together

🔍 Principle:

• Force F_in at radius R (large)
• Force F_out at radius r (small)

Torque analysis:

The torques are equal:
(same body, same α)

τ_in = τ_out

F_in × R = F_out × r

F_out/F_in = R/r

Mechanical advantage = R/r

💡 Example:

R = 0.5m (wheel)
r = 0.05m (axle)

Advantage = 0.5/0.05 = 10

Push with 10N
→ pull with 100N!

Applications:
• Door handle
• Wheels in machines
• Vehicle steering
• Water wheel

⚠️ Cost:

Wheel motion ×10
→ axle motion ÷10

Same work!

Question 25

2.00 pts

📚 Torque summary:

What are the 3 central points?

τ = r×F (torque), τ = Iα (second law), Στ=0 (equilibrium)

Only τ = F

No formulas

Too many

Explanation:

💡 Detailed explanation:

Torque summary! 📚

🔧 Torque summary:

1️⃣ Definition:
τ = r × F = rF·sin(θ)

2️⃣ Second law:
τ = Iα

3️⃣ Equilibrium:
Στ = 0

📊 Central formulas:

Formula	Description
τ = rF·sin(θ)	Torque
τ = Iα	Newton's 2nd
I = Σm_i·r_i²	Moment of inertia
I = I_cm + Md²	Steiner's theorem
Στ = 0	Equilibrium

✅ What we learned:

• Torque τ = r × F
• When τ = 0 (3 cases)
• Net torque (algebraic sum)
• Newton's 2nd law: τ = Iα
• Moment of inertia I
• Steiner's theorem
• Disc/wheel exercises
• Yo-yo
• Equilibrium ΣF=0 and Στ=0
• Seesaw
• Pulleys
• Wheel and axle

Question 26

2.00 pts

💫 Angular momentum:

What is L?

L = Iω, rotational momentum (like p = mv)

The force

The torque

The energy

Explanation:

💡 Detailed explanation:

Angular momentum! 💫

Angular momentum (L):

L = Iω

or for a particle:

L = r × p = mvr

🔍 Meaning:

Angular momentum = "amount of rotation"

Comparison:

Linear	Angular
p = mv	L = Iω
F = dp/dt	τ = dL/dt
p is conserved	L is conserved

📊 Properties:

• Units: kg·m²/s
• A vector! Direction: right-hand rule
• Conserved when Στ_ext = 0
• Plays a fundamental role in quantum mechanics too

💡 Examples:

• Bicycle wheel: L is large
→ harder to tip over
(gyroscopic stability)

• Earth: L huge
→ axis stable for billions of years

• Galaxies: L gigantic
→ preserve their rotation

Question 27

2.00 pts

⚖️ Conservation of L:

When is L conserved?

When Στ_ext = 0, no external torques

Always

Never

Only in space

Explanation:

💡 Detailed explanation:

Conservation of angular momentum! ⚖️

Conservation of angular momentum:

L = constant

if:

Στ_ext = 0

🔍 Derivation:

From the second law:
τ = dL/dt

If Στ_ext = 0:
dL/dt = 0
→ L = constant ✓

Condition:

No external torques
(or their sum = 0)

Internal forces
do not affect L_total!

Just like linear momentum

💡 Meaning:

If I changes
→ ω changes!

L = Iω = constant

I₁ω₁ = I₂ω₂

• I small → ω large
• I large → ω small

⭐ Examples:

• Ice skater pulling arms in
• Diver curling up
• Star collapse
• Planet formation
• Spinning tops
• Helicopters (tail rotor)

It's one of the deepest laws
in physics, valid even in quantum mechanics!

Question 28

2.00 pts

⛸️ Spinning ice skater:

Pulls arms in - what happens to ω?

ω increases! Smaller I → larger ω (L is conserved)

ω decreases

ω is constant

Stops

Explanation:

💡 Detailed explanation:

Spinning ice skater! ⛸️

🔍 Analysis:

State 1: arms outstretched

• Large I (mass far from the axis)
• Small ω
• L = I₁ω₁

State 2: arms pulled in

• Small I (mass near the axis)
• Large ω!
• L = I₂ω₂

Conservation of angular momentum:

No external torques
(zero friction on the ice)

L₁ = L₂

I₁ω₁ = I₂ω₂

ω₂/ω₁ = I₁/I₂

Numerical example:
I₁ = 4 kg·m², ω₁ = 1 rad/s
I₂ = 0.4 kg·m² (10 times smaller)

L = 4 kg·m²/s (conserved)

ω₂ = L/I₂ = 4/0.4

ω₂ = 10 rad/s

10 times faster!

💡 Why does this happen?

The internal forces
do work to pull the arms in

This work goes into
kinetic energy of rotation

L is conserved
but kinetic energy increases!
(at the expense of muscle work)

Question 29

2.00 pts

🤸 Diver:

How do they spin faster in the air?

Curls into a ball - smaller I → larger ω

Stretches out

Cannot

Only in water

Explanation:

💡 Detailed explanation:

Diver! 🤸

🔍 The diving process:

1️⃣ Take-off:

Pushes off the board
Jumps with a small initial ω

L = I_stretched · ω₀

2️⃣ In the air - curls up:

Folds the body
→ I becomes very small

L is conserved (no external torque)

L = I_curled · ω

I_curled << I_stretched
→ ω >> ω₀

Spins very fast!
Can do several rotations

3️⃣ Before entering the water:

Stretches out
→ I increases
→ ω decreases

Enters straight into the water!

L is still conserved

💡 Control:

The diver controls I
→ controls ω
→ controls the number of rotations

It's an art!

Numbers:
I_stretched ≈ 15 kg·m²
I_curled ≈ 3 kg·m²

5× difference!
→ ω is 5× larger when curled

⚡ The same principle:

Astronauts use the same idea
to rotate in space
by changing their body shape!

Question 30

2.00 pts

⭐ Collapsing star:

What happens to ω?

ω increases enormously! Smaller R → smaller I → huge ω

ω decreases

ω is constant

The star disappears

Explanation:

💡 Detailed explanation:

Collapsing star! ⭐

Star collapse process:

A massive star runs out of fuel
→ collapses inward
→ very small radius

🔍 Conservation of L:

Before collapse:

R ≈ 7×10⁸ m (like the Sun)
T ≈ 25 days = 2.16×10⁶ s
ω₁ = 2π/T ≈ 2.9×10⁻⁶ rad/s

Very slow

After collapse (neutron star):

R ≈ 10 km = 10⁴ m

I ∝ MR²

I₂/I₁ = (R₂/R₁)²
= (10⁴/7×10⁸)²
≈ 2×10⁻¹⁰

I 5 billion times smaller!

Conservation of L:
ω₂ = ω₁(I₁/I₂)
ω₂ ≈ 2.9×10⁻⁶ × 5×10⁹

ω₂ ≈ 14,500 rad/s

About 2300 revolutions per second!

💫 Pulsar:

A spinning neutron star
emits radio waves

We see it pulse
at every revolution

The fastest known pulsar:
716 Hz (716 revolutions/sec)

Each pulse is incredibly precise
used as cosmic clocks

Question 31

2.00 pts

🔗 Connection τ and L:

What is the relation?

τ = dL/dt, torque changes angular momentum

No relation

τ = L

τ = Iω

Explanation:

💡 Detailed explanation:

Connection τ and L! 🔗

Central relation:

τ = dL/dt

Torque = rate of change of angular momentum

🔍 Derivation:

L = Iω

dL/dt = d(Iω)/dt

If I is constant:
dL/dt = I(dω/dt)
dL/dt = Iα

τ = Iα = dL/dt

Identical to the second law!

📊 Comparison:

Linear	Angular
F = dp/dt	τ = dL/dt
p = mv	L = Iω
F = ma	τ = Iα

💡 Insight:

Same physics, different variables!

The same equations apply
to both linear and rotational motion

One of the deepest analogies in physics

Question 32

2.00 pts

🎯 Gyroscope:

Why doesn't it fall?

L is large and preserves direction; forces create precession, not falling

Magic

Magnetism

Very light

Explanation:

💡 Detailed explanation:

Gyroscope! 🎯

🔍 What is a gyroscope?

A wheel spinning very fast
on an axle

Why doesn't it fall?

1️⃣ Large angular momentum:

L = Iω (very large ω)
→ L is large!

L preserves its direction
(robust to disturbances)

2️⃣ Precession:

Gravity creates torque τ

τ = dL/dt

L doesn't decrease in magnitude
but changes direction!

→ The axle rotates horizontally
(precession)

Instead of falling!

💡 Precession rate:

Ω = τ/L = mgr/(Iω)

The larger ω
→ smaller Ω
→ slower precession

Applications:

• Aircraft navigation
• Spacecraft
• Smartphones
• Electric scooters
• Robots

Question 33

2.00 pts

🎯 Particle:

What is L of a particle in a circular orbit?

L = mvr, linear momentum × radius

L = 0

L = mv

L = Iω

Explanation:

💡 Detailed explanation:

Particle in orbit! 🎯

Angular momentum of a particle:

L = r × p

or in scalars (circular orbit):

L = mvr

🔍 Connection to Iω:

For a particle: I = mr²

v = ωr

L = Iω = mr² · (v/r)

L = mvr ✓

Identical!

💡 Example:

Satellite:
m = 1000 kg
v = 7500 m/s
r = 7×10⁶ m

L = mvr
L = 1000×7500×7×10⁶
L = 5.25×10¹³ kg·m²/s

Enormous!

⚠️ Important:

Even for an elliptical orbit
L is conserved!

Close to Sun: large v
Far: small v

But mvr = constant
(vector argument: only the perpendicular component
contributes to L when r and v are perpendicular)

Question 34

2.00 pts

🪐 Kepler's Second Law:

What does it state?

Equal areas in equal times - a consequence of conservation of L!

Circular orbit

No relation

Only for Mars

Explanation:

💡 Detailed explanation:

Kepler's Second Law! 🪐

Kepler's Second Law:

"The line connecting a planet to the Sun
sweeps out equal areas
in equal time intervals"

🔍 The physics behind it:

This is a consequence of conservation of L!

Derivation:

L = mvr·sin(θ) = constant

If the planet is at distance r
and velocity v (perpendicular)

Area swept in time dt:
dA = ½r²dθ

dA/dt = ½r²(dθ/dt)
dA/dt = ½r²ω
dA/dt = ½rv

But L = mvr = constant

→ dA/dt = L/(2m) = constant!

Area per time = constant ✓

💡 Meaning:

Close to Sun (perihelion):
• Small r
• Large v (to preserve L)

Far from Sun (aphelion):
• Large r
• Small v

But area/time = constant!

⭐ Note:

Summer/winter is not because of distance
but because of the axis tilt!

(Earth is closest to Sun in January
during the Northern Hemisphere winter)

Question 35

2.00 pts

💥 Off-center collision:

What is conserved?

Both p and L (if no external forces/torques)

Only p

Only L

Nothing

Explanation:

💡 Detailed explanation:

Off-center collision! 💥

Two conservation laws!

1️⃣ Linear momentum:
Σp_i = Σp_f

2️⃣ Angular momentum:
ΣL_i = ΣL_f

🔍 Why both?

Collision forces:
• Internal (between the bodies)
• Don't affect p_total
• Don't affect L_total

Example:

A ball strikes the end of a rod

Before:
• Ball: p, L=mvr
• Rod: at rest

After:
• Both move and rotate
• p_total conserved
• L_total conserved

2 equations → 2 unknowns
(v_cm and ω)

⚠️ Energy:

E_k is not necessarily conserved
(depends if elastic)

But p and L are always conserved!

Question 36

2.00 pts

🧮 Exercise:

Ball m at v strikes the end of a rod M, L (at rest)
Sticks - what is ω?

ω = mvL / (I_rod + mL²), conservation of L

ω = 0

ω = v/L

ω = mv/(ML)

Explanation:

💡 Detailed explanation:

Rod and ball! 🧮

📐 Solution:

Given:
Ball: mass m, velocity v
Rod: mass M, length L
Pivot: at the other end of the rod

Before:

L_ball = mvL (about the pivot)
L_rod = 0 (at rest)

L_total = mvL

After (stuck together):

Rotate together with ω

Need total I:

I_rod = ML²/3 (about end)
I_ball = mL² (particle at radius L)

I_total = ML²/3 + mL²

Conservation of L:

mvL = I_total · ω

mvL = (ML²/3 + mL²)ω

ω = mvL/(ML²/3 + mL²)

or:

ω = 3mv/((M+3m)L)

💡 Sanity check:

If M >> m:
ω ≈ 3mv/(ML)

If m >> M:
ω ≈ v/L

Question 37

2.00 pts

📚 L summary:

What are the 3 central points?

L = Iω, τ = dL/dt, L is conserved (if Στ_ext=0)

Only L = Iω

No formulas

Too many

Explanation:

💡 Detailed explanation:

Angular momentum summary! 📚

💫 Angular momentum:

1️⃣ Definition:
L = Iω = r × p

2️⃣ Connection to torque:
τ = dL/dt

3️⃣ Conservation:
L = constant (if Στ_ext=0)

✅ What we learned:

• L = "amount of rotation"
• Units: kg·m²/s
• Vector (direction: axis)
• Conservation: I↓ → ω↑
• Examples: skater, diver, star
• Connection to Kepler
• Collisions: p and L conserved

Question 38

2.00 pts

⚡ Rotational energy:

What is the E_k of a rotating body?

E_k = ½Iω², analogous to ½mv²

E_k = Iω

E_k = ω²

E_k = 0

Explanation:

💡 Detailed explanation:

Rotational energy! ⚡

Rotational kinetic energy:

E_k = ½Iω²

🔍 Derivation:

Body = sum of particles

For each particle i:
E_k,i = ½m_i·v_i²

But v_i = ωr_i

E_k,i = ½m_i·(ωr_i)²
E_k,i = ½m_i·r_i²·ω²

Sum:
E_k = Σ(½m_i·r_i²·ω²)
E_k = ½(Σm_i·r_i²)·ω²

E_k = ½Iω²

Just like ½mv²!

📊 Comparison:

Linear	Rotational
E_k = ½mv²	E_k = ½Iω²
m	I
v	ω

💡 Example:

Disc M = 2 kg, R = 0.5 m, ω = 10 rad/s

I = ½MR² = ½×2×0.25 = 0.25 kg·m²

E_k = ½Iω² = ½×0.25×100
E_k = 12.5 J

(at the same kinetic energy
m moving at v = √(2E_k/m) = 3.5 m/s)

Question 39

2.00 pts

⚡ Rolling:

What is the total energy?

E = ½mv² + ½Iω², translation + rotation

Only ½mv²

Only ½Iω²

Zero

Explanation:

💡 Detailed explanation:

Rolling energy! ⚡

Total energy in rolling:

E = ½mv² + ½Iω²

Translation of center of mass + rotation about it

🔍 Two components:

1️⃣ Translational energy:

E_trans = ½mv²

v = velocity of the center of mass

As if all the mass were at the center

2️⃣ Rotational energy:

E_rot = ½Iω²

I = moment of inertia about center of mass
ω = angular velocity

💡 Rolling without slipping:

Condition: v = ωR

Then:
E = ½mv² + ½I(v/R)²
E = ½v²(m + I/R²)

Example - solid disc:
I = ½mR²

E = ½v²(m + ½m)
E = ¾mv²

50% more than just translation!

⚡ Race down an incline:

Hoop (I = mR²): E = mv²
Solid disc (I = ½mR²): E = ¾mv²
Solid sphere (I = ⅖mR²): E = 0.7mv²

For the same potential energy:
The sphere rolls fastest!
(less goes to rotation)

Question 40

2.00 pts

⛷️ Rolling on an incline:

What is the formula for v at the bottom?

v = √(2gh/(1+I/(mR²))), depends on the body shape!

v = √(2gh)

v = 0

v = gh

Explanation:

💡 Detailed explanation:

Rolling on an incline! ⛷️

📐 Derivation:

Conservation of energy:

E_p = E_k,trans + E_k,rot

mgh = ½mv² + ½Iω²

Rolling without slipping: v = ωR

mgh = ½mv² + ½I(v/R)²

gh = ½v²(1 + I/(mR²))

v² = 2gh/(1 + I/(mR²))

v = √(2gh/(1 + I/(mR²)))

💡 For different bodies:

Body	I/(mR²)	v
Solid sphere	2/5	√(10gh/7)
Solid disc	1/2	√(4gh/3)
Hollow sphere	2/3	√(6gh/5)
Hoop	1	√(gh)

⚡ Race winners:

The solid sphere wins!
(smaller I/(mR²) → less rotational energy → more translational energy)

Order: sphere → disc → hollow sphere → hoop

Question 41

2.00 pts

🎯 Rolling condition:

What is it?

v = ωR, no slipping

v = ω

v = 0

No condition

Explanation:

💡 Detailed explanation:

Rolling condition! 🎯

Rolling without slipping:

v = ωR

Center-of-mass velocity
= angular velocity × radius

🔍 Meaning:

The point in contact with the ground
has velocity = 0 (doesn't slip!)

Proof:

Center of mass: v →
Rotation: ωR ← (at the bottom)

Net velocity at the contact point:
v_contact = v - ωR

Rolling: v_contact = 0
→ v = ωR ✓

💡 Consequences:

• Distance: s = θR
• Velocity: v = ωR
• Acceleration: a = αR

⚠️ Slipping:

If v ≠ ωR
→ slipping
→ energy turns into heat
→ kinetic friction

Rolling:
v = ωR
→ no slipping
→ static friction
→ no energy loss!

Question 42

2.00 pts

🪀 Yo-yo:

What is v at the bottom (height h)?

v = √(2gh/(1+I/(mR²))), like rolling!

v = √(2gh)

v = 0

v = gh

Explanation:

💡 Detailed explanation:

Yo-yo and energy! 🪀

🔍 Energy analysis:

Start (top):

E_p = mgh
E_k = 0

E_total = mgh

End (bottom):

E_p = 0
E_k = ½mv² + ½Iω²

String doesn't slip: v = ωR

E_k = ½mv² + ½I(v/R)²

Conservation of energy:

mgh = ½mv² + ½I(v/R)²

gh = ½v²(1 + I/(mR²))

v = √(2gh/(1 + I/(mR²)))

Exactly like rolling on an incline!

💡 Example:

Yo-yo: I = ½mR²

v = √(2gh/(1 + ½))
v = √(4gh/3)

Less than free fall (√(2gh))

Why? Some of the energy
goes into rotation!

Question 43

2.00 pts

⛷️ Acceleration on an incline:

What is the a of a rolling body?

a = g·sin(θ)/(1+I/(mR²)), depends on the shape!

a = g·sin(θ)

a = 0

a = g

Explanation:

💡 Detailed explanation:

Acceleration on an incline! ⛷️

📐 Derivation:

Forces on a rolling body:
• Gravity: mg
• Normal: N
• Friction: f (up the incline)

Newton's 2nd (translation):

mg·sin(θ) - f = ma

Newton's 2nd (rotation):

τ = Iα
f·R = Iα

Rolling condition: a = αR

f·R = I(a/R)
f = Ia/R²

Solution:

Substitute f into the first equation:

mg·sin(θ) - Ia/R² = ma

mg·sin(θ) = a(m + I/R²)

a = g·sin(θ)/(1 + I/(mR²))

💡 Comparison:

• Sliding: a = g·sin(θ)
• Sphere: a = 5g·sin(θ)/7
• Disc: a = 2g·sin(θ)/3
• Hoop: a = g·sin(θ)/2

Sphere = fastest down the slope!

Question 44

2.00 pts

🔧 Friction:

What role does friction play in rolling?

Causes rotation (τ=fR), but does no work!

Slows it down

No friction

Does negative work

Explanation:

💡 Detailed explanation:

Friction in rolling! 🔧

Role of friction:

Static friction
causes rotation
but does no work!

🔍 How is that possible?

1️⃣ Creates torque:

f acts at radius R

τ = f·R

Causes the body to rotate ✓

2️⃣ Does no work:

W = F·d

But at the contact point:
v_contact = 0!

No displacement → W = 0

Static friction does no work ✓

💡 Why is this important?

• Without friction → only sliding
• With friction → rolling

Friction is essential for rolling
but energy is conserved!

⚠️ Kinetic friction (slipping):

Different! Does negative work
→ energy turns into heat
→ wastes energy

That's why we want rolling, not slipping

Question 45

2.00 pts

🧮 Comprehensive exercise:

Solid sphere m=2kg, R=0.1m
rolls down from h=5m

What are v, ω, E_k at the bottom? (g=10)

v=8.37 m/s, ω=83.7 rad/s, E_k=100J

v=10 m/s

v=0

v=5 m/s

Explanation:

💡 Detailed explanation:

Comprehensive exercise! 🧮

📐 Full solution:

Given:
m = 2 kg
R = 0.1 m
h = 5 m
g = 10 m/s²
Solid sphere: I = ⅖mR²

1️⃣ Velocity:

v = √(2gh/(1 + I/(mR²)))

I/(mR²) = ⅖mR²/(mR²) = ⅖

v = √(2×10×5/(1 + ⅖))
v = √(100/1.4)
v = √(71.43)

v ≈ 8.45 m/s

2️⃣ Angular velocity:

v = ωR

ω = v/R = 8.45/0.1

ω ≈ 84.5 rad/s

3️⃣ Kinetic energy:

Check:
E_k = E_p = mgh
E_k = 2×10×5

E_k = 100 J

(conservation of energy ✓)

Verification:
E_trans = ½mv² = ½×2×71.43 ≈ 71.4 J
E_rot = ½Iω² = ⅖mv² × ½ = ⅕×2×71.43 ≈ 28.6 J
Total: 100 J ✓

💡 Insights:

• 71% translational, 29% rotational
• Lower than free fall √(100) = 10 m/s
• Some energy went into rotation

Question 46

2.00 pts

🏗️ Engineering:

Where are rotational principles used?

Everywhere: engines, gears, turbines, spin

Only in the lab

No applications

Only in space

Explanation:

💡 Detailed explanation:

Engineering applications! 🏗️

🌍 Areas of application:

⚙️ Engines:

• Torque → rotation
• Gears: mechanical advantage
• Large I → flywheel (energy storage)
• ω → power

💨 Turbines:

• Energy → rotation
• Large L → stability
• E_k = ½Iω²
• Wind/water harvesting

🚗 Vehicles:

• Wheels: v=ωR
• Gearbox: τ vs ω
• ABS: prevents slipping
• Brake discs: E_k→heat

🚁 Aviation:

• Helicopter rotor: L
• Gyroscope: navigation
• Satellite spin: stability
• Propeller: τ→thrust

🏃 Sports:

• Diver: I changes
• Baseball: spin and Magnus effect
• Bicycle: gyroscopic stability
• Golf: spin for control

⭐ Bottom line:

Rotational physics is everywhere!
From watches to galaxies

Question 47

2.00 pts

⚠️ Common error:

Which statement is wrong?

"Friction in rolling does negative work" (no! W=0)

τ = r×F

L = Iω

E_k = ½Iω²

Explanation:

💡 Detailed explanation:

Common errors! ⚠️

❌ The error:

"Friction in rolling
does negative work"

Wrong!

The truth:

In rolling without slipping:

v_contact = 0

W = f·d = f·0 = 0

No work!

⚠️ Other errors:

❌ "I depends only on mass"
✓ depends also on shape and axis!

❌ "In rolling E_k = ½mv²"
✓ E_k = ½mv² + ½Iω²

❌ "τ = F (like Newton's 2nd)"
✓ τ = r×F (need a lever arm!)

❌ "L is always conserved"
✓ Only if Στ_ext = 0

❌ "All bodies roll at the same speed"
✓ Depends on I!

❌ "Torque = force"
✓ Torque = force × lever arm

❌ "ω points in the direction of rotation"
✓ ω is perpendicular to the plane (right-hand rule)

Question 48

2.00 pts

📚 Central formulas:

What are the 10 important formulas?

θ,ω,α,τ,I,L,E_k + relations (v=ωR, τ=Iα, L=Iω, E=½Iω²)

Only one

No formulas

Too many

Explanation:

💡 Detailed explanation:

Full formula table! 📚

🔄 Rotational motion - all formulas:

📊 Basic formulas:

Topic	Formula	Units
Angle	θ = s/r	rad
Angular velocity	ω = Δθ/Δt = 2πf	rad/s
Angular acceleration	α = Δω/Δt	rad/s²
Linear velocity	v = ωr	m/s

⚙️ Dynamics:

Topic	Formula
Torque	τ = rF·sin(θ)
Newton's 2nd	τ = Iα
Moment of inertia	I = Σm_i·r_i²
Equilibrium	Στ = 0

💫 Angular momentum:

• L = Iω
• L = mvr (particle)
• τ = dL/dt
• L = constant (if Στ_ext=0)

⚡ Energy:

• E_k = ½Iω²
• Rolling: E = ½mv² + ½Iω²
• Conservation: mgh = ½mv² + ½Iω²
• Rolling condition: v = ωR

Question 49

2.00 pts

🎨 Concept map:

What are the central concepts?

Kinematics (θ,ω,α), dynamics (τ,I), angular momentum (L), energy (E_k), rolling

Only ω

No concepts

Too many

Explanation:

💡 Detailed explanation:

Concept map! 🎨

🗺️ The full map:

Rotational motion

📦 5 central domains:

1️⃣ Rotational kinematics

• θ (angle) - rad
• ω (angular velocity) - rad/s
• α (angular acceleration) - rad/s²
• Relations: v=ωr, a=αr
• Formulas: ω=ω₀+αt, θ=½αt²
• Frequency: f, period: T
• a_c = ω²r (centripetal)

2️⃣ Torque and dynamics

• τ (torque) - N·m
• I (moment of inertia) - kg·m²
• Newton's 2nd: τ = Iα
• Equilibrium: Στ = 0
• Parallel axis theorem: I=I_cm+Md²
• Applications: seesaw, pulleys

3️⃣ Angular momentum

• L = Iω - kg·m²/s
• L = mvr (particle)
• τ = dL/dt
• Conservation: I↓ → ω↑
• Examples: ice skater, diver, star

4️⃣ Rotational energy

• E_k = ½Iω²
• Rolling: E = ½mv² + ½Iω²
• Conservation of energy
• Rolling on incline

5️⃣ Rolling

• Condition: v = ωR
• Static friction (no work!)
• Yo-yo
• Sphere/disc/hoop on incline

Question 50

2.00 pts

🎓 Summary of Exam 166:

What is the central takeaway?

Rotation = a complete analogy to linear motion! Every concept has a counterpart

Only formulas

Only calculations

No takeaway

Explanation:

💡 Detailed explanation:

Final summary of Exam 166! 🎓

🎉 Exam 166 completed! 🎉

Rotational motion

50 questions | full comprehensive coverage

📚 What we learned:

🔄 Part A: Kinematics (1-12)

• Radian: θ = s/r
• Angular velocity: ω = Δθ/Δt
• Angular acceleration: α = Δω/Δt
• Kinematics formulas (identical to linear!)
• Relations: v=ωr, a=αr
• Frequency and period: ω=2πf=2π/T
• Centripetal acceleration: a_c=ω²r
• Centripetal force: F_c=mω²r
• Satellite, vertical loop

Insight: rotation = motion in a circle

🔧 Part B: Torque (13-25)

• Torque: τ = r×F
• Moment of inertia: I = Σ(mr²)
• Newton's 2nd: τ = Iα
• Parallel axis theorem: I = I_cm + Md²
• Equilibrium: Στ = 0
• Yo-yo, seesaw, pulleys
• Wheel and axle

Insight: rotation has its own mechanics

💫 Part C: Angular momentum (26-37)

• L = Iω, L = mvr
• τ = dL/dt
• Conservation: I↓ → ω↑
• Ice skater, diver, collapsing star
• Connection to Kepler
• Gyroscope, off-center collisions

Insight: a fundamental conservation law

⚡ Part D: Energy and rolling (38-50)

• E_k = ½Iω²
• Rolling: E = ½mv² + ½Iω²
• Rolling condition: v = ωR
• Sphere/disc/hoop on incline
• Yo-yo, friction
• Engineering applications

Insight: rotation is everywhere

💡 The central takeaway:

Linear ↔ Rotational analogy!

m → I, v → ω, a → α
F → τ, p → L
F=ma → τ=Iα
p=mv → L=Iω
½mv² → ½Iω²

Same physics, different variables!