3. Moment of Inertia

Moment of Inertia

Moment of inertia describes how difficult it is to change an object's rotational motion.

In linear motion, mass measures an object's resistance to linear acceleration. In rotational motion, moment of inertia plays a similar role: it measures an object's resistance to angular acceleration.

Moment of inertia is represented by:

I

Its SI unit is:

kg·m²

An object's moment of inertia depends on two important things:

• how much mass the object has
• how that mass is distributed relative to the axis of rotation

This second factor is especially important. Two objects with the same mass can have very different moments of inertia.

---

Mass and Rotational Inertia

In linear motion:

F = ma

A larger mass requires a larger force to produce the same acceleration.

The rotational equivalent is:

τ = Iα

where:

• τ = net torque
• I = moment of inertia
• α = angular acceleration

Therefore:

α = τ/I

For the same torque:

• smaller I → greater angular acceleration
• larger I → smaller angular acceleration

Moment of inertia is sometimes called rotational inertia because it describes resistance to changes in rotational motion.

---

Mass Distribution Matters

Moment of inertia does not depend only on total mass.

It also depends strongly on where the mass is located relative to the axis of rotation.

Mass farther from the axis contributes more to the moment of inertia than mass close to the axis.

For a single point mass:

I = mr²

where:

• m = mass
• r = perpendicular distance from the axis

Because r is squared, distance has a very large effect.

---

Worked Example: Point Mass

A:

3.0 kg

mass is located:

0.40 m

from an axis of rotation.

I = mr²

I = (3.0)(0.40²)

I = (3.0)(0.16)

I = 0.48 kg·m²

---

Moving the Mass Farther Out

Now move the same 3.0 kg mass to:

0.80 m

from the axis.

I = mr²

I = (3.0)(0.80²)

I = (3.0)(0.64)

I = 1.92 kg·m²

The radius doubled:

0.40 m → 0.80 m

but the moment of inertia increased by a factor of:

4

because:

I ∝ r²

---

Why Radius Has Such a Large Effect

For a point mass:

I = mr²

If the distance from the axis doubles:

I_new = m(2r)²

I_new = 4mr²

Therefore:

I_new = 4I

If the radius triples:

I_new = m(3r)²

I_new = 9I

If the radius is halved:

I_new = m(r/2)²

I_new = I/4

Mass distribution can therefore change rotational behaviour dramatically.

---

Several Point Masses

If an object consists of several point masses, their individual moments of inertia can be added.

I_total = Σmr²

or:

I_total = m₁r₁² + m₂r₂² + m₃r₃² + ...

Each mass is considered separately.

---

Worked Example: Two Masses

Two 2.0 kg masses are each located:

0.50 m

from an axis.

For each mass:

I = mr²

I = (2.0)(0.50²)

I = 0.50 kg·m²

There are two masses:

I_total = 0.50 + 0.50

I_total = 1.0 kg·m²

---

Worked Example: Masses at Different Distances

Three point masses rotate around the same axis:

• 2 kg at 1 m
• 3 kg at 2 m
• 1 kg at 3 m

Calculate the total moment of inertia.

I = Σmr²

I = (2)(1²) + (3)(2²) + (1)(3²)

I = 2 + 12 + 9

I = 23 kg·m²

Notice that the 1 kg mass contributes:

9 kg·m²

because it is farthest from the axis.

Despite having the smallest mass, it makes a large contribution to I.

---

Moment of Inertia Depends on the Axis

The same object can have different moments of inertia depending on the axis around which it rotates.

Consider a long rod.

It could rotate:

• around its centre
• around one end
• around another axis

The mass of the rod has not changed, but its distribution relative to the axis has changed.

Therefore:

moment of inertia must always be considered relative to a particular axis.

---

Common Moments of Inertia

For objects with continuous mass distributions, formulas have been derived for common shapes.

Here:

• M = total mass
• R = radius
• L = length

These equations apply to idealized uniform objects.

---

Comparing a Ring and a Solid Disk

Consider a thin ring and a solid disk with the same:

• mass M
• radius R

For the ring:

I_ring = MR²

For the disk:

I_disk = 1/2 MR²

Therefore:

I_ring = 2I_disk

The ring has twice the moment of inertia.

Why?

In the ring, essentially all the mass is located at radius R.

In the disk, much of the mass is closer to the axis.

---

Numerical Example

Suppose both objects have:

M = 4.0 kg

R = 0.50 m

For the ring:

I = MR²

I = (4.0)(0.50²)

I_ring = 1.0 kg·m²

For the disk:

I = 1/2 MR²

I = 1/2(4.0)(0.50²)

I_disk = 0.50 kg·m²

The ring is more difficult to angularly accelerate.

---

Comparing a Solid Sphere and Spherical Shell

For a solid sphere:

I = 2/5 MR²

For a thin spherical shell:

I = 2/3 MR²

For equal mass and radius:

2/3 > 2/5

Therefore:

I_shell > I_solid sphere

Again, the object with more mass concentrated farther from the axis has the larger moment of inertia.

---

Comparing Different Shapes

For objects with the same mass and outer radius:

more mass near the axis → smaller I

more mass far from the axis → larger I

For example:

The exact ranking depends on the objects and axes being compared, but the general principle remains:

moving mass outward increases moment of inertia.

---

Moment of Inertia and Angular Acceleration

Newton's Second Law for rotation is:

τ_net = Iα

Therefore:

α = τ_net/I

Suppose the same torque is applied to two objects.

Object A:

I = 2 kg·m²

Object B:

I = 8 kg·m²

The angular accelerations are:

α_A = τ/2

α_B = τ/8

Therefore:

Object A has four times the angular acceleration of Object B.

The object with the smaller moment of inertia responds more easily to torque.

---

Worked Example

A net torque of:

12 N·m

acts on an object with:

I = 3.0 kg·m²

Then:

α = τ/I

α = 12/3.0

α = 4.0 rad/s²

---

Larger Moment of Inertia

Suppose the same 12 N·m torque acts on an object with:

I = 6.0 kg·m²

Then:

α = 12/6

α = 2.0 rad/s²

Doubling I halved the angular acceleration.

For constant torque:

α ∝ 1/I

---

Linear and Rotational Inertia

There is an important parallel between linear and rotational motion.

Mass measures resistance to linear acceleration.

Moment of inertia measures resistance to angular acceleration.

However, there is an important difference:

mass depends only on how much matter an object contains.

moment of inertia also depends on how that matter is distributed relative to the axis.

---

Changing Mass Distribution

Consider a rotating person holding weights.

If the person holds the weights close to their body, the masses are relatively close to the axis.

This gives a smaller moment of inertia.

If the person extends their arms, the masses move farther from the axis.

This gives a larger moment of inertia.

If there is little external torque, changing I also changes angular velocity through conservation of angular momentum, which will be explored in more detail later.

The important idea here is:

arms outward → larger I

arms inward → smaller I

---

Figure Skaters

A spinning figure skater provides a familiar example.

With arms extended:

• more mass is farther from the rotation axis
• moment of inertia is larger

With arms pulled inward:

• more mass is closer to the axis
• moment of inertia is smaller

The skater's mass remains essentially unchanged.

What changes is the distribution of mass.

---

Divers and Gymnasts

Divers and gymnasts use the same principle.

When they tuck their bodies:

• mass moves closer to the rotation axis
• moment of inertia decreases
• their rotational motion can change more rapidly

When they straighten their bodies:

• mass spreads farther from the axis
• moment of inertia increases

This allows athletes to control their rotation during jumps and flips.

---

A Rotating Chair Experiment

Imagine sitting on a low-friction rotating chair while holding two heavy objects.

With your arms extended, the masses are relatively far from your rotation axis.

Your moment of inertia is relatively large.

Pull the objects toward your body.

The masses move closer to the axis and your moment of inertia decreases.

This simple demonstration illustrates that I can change even though:

the total mass remains constant.

---

Why Heavy Rims Matter

Consider two wheels with the same total mass and radius.

Wheel A has much of its mass near the hub.

Wheel B has much of its mass concentrated around the rim.

Wheel B has the larger moment of inertia.

Therefore, the same torque produces less angular acceleration in Wheel B.

This is important when designing:

• vehicle wheels
• flywheels
• turbines
• motors
• machinery

---

Flywheels

A flywheel is designed to have a relatively large moment of inertia.

Much of its mass may be distributed away from the axis.

A large moment of inertia makes its rotational speed more resistant to rapid changes.

Flywheels can therefore help smooth variations in rotational motion and store rotational kinetic energy.

---

Rotational Kinetic Energy

A rotating object has rotational kinetic energy:

E_rot = 1/2 Iω²

This resembles linear kinetic energy:

E_k = 1/2 mv²

The comparison is:

Moment of inertia therefore affects not only angular acceleration but also the energy stored in rotation.

---

Worked Example: Rotational Energy

A flywheel has:

I = 5.0 kg·m²

and:

ω = 20 rad/s

Then:

E_rot = 1/2 Iω²

E_rot = 1/2(5.0)(20²)

E_rot = 2.5(400)

E_rot = 1000 J

---

Comparing Rotational Energy

Suppose two objects rotate at the same angular velocity.

Object A:

I = 2 kg·m²

Object B:

I = 6 kg·m²

Since:

E_rot = 1/2 Iω²

Object B has:

three times as much rotational kinetic energy

at the same angular velocity.

---

Moment of Inertia of a Rod

For a uniform thin rod rotating around its centre:

I = 1/12 ML²

For the same rod rotating around one end:

I = 1/3 ML²

Compare them:

(1/3)/(1/12) = 4

Therefore:

I_end = 4I_centre

The rod is much harder to angularly accelerate about its end because, on average, its mass is farther from the axis.

---

Worked Rod Example

A uniform rod has:

M = 6.0 kg

L = 2.0 m

About Its Centre

I = 1/12 ML²

I = 1/12(6)(2²)

I = 2.0 kg·m²

About One End

I = 1/3 ML²

I = 1/3(6)(2²)

I = 8.0 kg·m²

Therefore, rotating the rod around one end gives four times the moment of inertia.

---

Moment of Inertia of a Disk

For a solid disk rotating about its central axis:

I = 1/2 MR²

Suppose:

M = 8 kg

R = 0.40 m

Then:

I = 1/2(8)(0.40²)

I = 4(0.16)

I = 0.64 kg·m²

---

Moment of Inertia of a Ring

For a thin ring:

I = MR²

Suppose the ring has the same mass and radius:

M = 8 kg

R = 0.40 m

Then:

I = (8)(0.40²)

I = 1.28 kg·m²

The ring has twice the moment of inertia of the disk.

---

Predicting the Effect of Changing Mass

For objects whose shape and size remain the same:

I ∝ M

If mass doubles:

I doubles

If mass triples:

I triples

For example, a solid disk has:

I = 1/2 MR²

If M changes from M to 2M:

I_new = 1/2(2M)R²

I_new = 2I

---

Predicting the Effect of Changing Size

For many similar objects:

I ∝ R²

If radius doubles while mass remains constant:

I becomes four times larger.

For example, for a disk:

I = 1/2 MR²

If R becomes 2R:

I_new = 1/2 M(2R)²

I_new = 4(1/2 MR²)

I_new = 4I

---

Mass and Radius Changing Together

Suppose a thin ring changes from:

M → 2M

and:

R → 3R

Since:

I = MR²

the new moment of inertia is:

I_new = (2M)(3R)²

I_new = 18MR²

Therefore:

I_new = 18I

The radius has a particularly strong effect because it is squared.

---

Rotating Systems with Added Masses

Moment of inertia can also be changed by attaching masses to an existing rotating object.

Suppose a disk initially has:

I_disk = 2.0 kg·m²

Two 1.0 kg masses are attached:

0.50 m

from the axis.

Each contributes:

I = mr²

I = (1.0)(0.50²)

= 0.25 kg·m²

Two masses contribute:

0.50 kg·m²

Therefore:

I_total = 2.0 + 0.50

I_total = 2.50 kg·m²

---

Moving the Added Masses Outward

Now move both masses to:

1.0 m

from the axis.

Each contributes:

I = (1.0)(1.0²)

= 1.0 kg·m²

Together:

2.0 kg·m²

Therefore:

I_total = 2.0 + 2.0

I_total = 4.0 kg·m²

Moving the same masses outward significantly increased the rotational inertia.

---

Effect on Angular Acceleration

Suppose the net torque remains:

τ = 8 N·m

With:

I = 2.50 kg·m²

α = τ/I

α = 8/2.50

α = 3.2 rad/s²

After moving the masses outward:

I = 4.0 kg·m²

α = 8/4.0

α = 2.0 rad/s²

Moving mass outward reduced the angular acceleration produced by the same torque.

---

Rolling Objects

Moment of inertia also affects how objects roll.

Consider objects with the same mass and radius rolling down a slope.

An object must gain both:

• translational kinetic energy
• rotational kinetic energy

An object with a larger moment of inertia requires a larger share of its energy for rotation.

As a result, its centre of mass generally accelerates less rapidly down the slope.

For ideal rolling without slipping, objects such as a solid sphere generally reach the bottom sooner than a ring of the same mass and radius.

The reason is their different mass distributions and moments of inertia.

---

Comparing Rolling Objects

For equal M and R:

Thin ring:

I = MR²

Solid disk:

I = 1/2 MR²

Solid sphere:

I = 2/5 MR²

The solid sphere has the smallest rotational inertia relative to MR².

The ring has the largest.

This affects how quickly they accelerate while rolling.

---

Moment of Inertia and Engineering

Engineers deliberately control moment of inertia.

A small I may be useful when:

• rapid acceleration is needed
• rapid braking is needed
• machinery must respond quickly
• energy use during repeated acceleration should be reduced

A large I may be useful when:

• steady rotation is desirable
• rotational speed should resist sudden changes
• significant rotational energy needs to be stored

---

Vehicle Wheels

Reducing wheel mass can improve how quickly a vehicle's wheels change rotational speed.

Removing mass near the rim is especially effective because:

I depends on r²

One kilogram removed near the outer edge can affect moment of inertia much more than one kilogram removed close to the axle.

This is one reason mass distribution matters in wheel design.

---

Rotating Machinery

Moment of inertia affects how quickly motors, turbines, fans, and other rotating machines respond.

A motor must produce torque to change angular velocity:

τ = Iα

A machine with a large I requires greater torque to achieve the same angular acceleration.

It may also require more time to start or stop.

---

Moment of Inertia Is Not Torque

Moment of inertia and torque describe different things.

Moment of inertia describes resistance to angular acceleration.

Torque causes angular acceleration.

Their relationship is:

τ_net = Iα

This parallels:

F_net = ma

Moment of inertia is not itself a force or turning effect.

---

Moment of Inertia Is Not Angular Momentum

Moment of inertia is also different from angular momentum.

Moment of inertia:

I

describes the mass distribution.

Angular momentum:

L

describes rotational motion.

For a rigid object rotating about a fixed axis:

L = Iω

Changing I can therefore affect rotational behaviour, especially when angular momentum is conserved.

---

A Complete Example

A solid disk has:

M = 10 kg

R = 0.50 m

A net torque of:

τ = 15 N·m

acts on it.

Moment of Inertia

I = 1/2 MR²

I = 1/2(10)(0.50²)

I = 5(0.25)

I = 1.25 kg·m²

Angular Acceleration

τ = Iα

α = τ/I

α = 15/1.25

α = 12 rad/s²

If the disk starts from rest and the torque acts for 3.0 s:

ω_f = ω_i + αt

ω_f = 0 + (12)(3)

ω_f = 36 rad/s

This shows how moment of inertia connects torque to rotational motion.

---

Predicting Changes Without Full Calculations

Many moment-of-inertia problems can be solved using proportional reasoning.

If mass doubles:

I × 2

If radius doubles:

I × 4

If radius triples:

I × 9

If radius halves:

I × 1/4

If the same torque acts and I doubles:

α × 1/2

If the same torque acts and I becomes four times larger:

α × 1/4

Recognizing these patterns can make problems much faster to solve.

---

A Strategy for Moment of Inertia Problems

First identify the axis of rotation.

Then determine whether the object can be treated as:

• a point mass
• several point masses
• a ring
• a disk
• a cylinder
• a sphere
• a rod

Choose the appropriate expression for I.

Check that all distances are measured relative to the correct axis.

Calculate I.

If torque is involved, use:

τ_net = Iα

If rotational energy is involved, use:

E_rot = 1/2 Iω²

Finally, check whether the answer makes physical sense based on the mass distribution.

---

Common Mistakes

Thinking moment of inertia depends only on mass

Mass distribution is also essential.

Ignoring the axis of rotation

The same object can have different moments of inertia about different axes.

Forgetting to square the radius

For a point mass:

I = mr²

not mr.

Assuming objects with equal mass and radius have equal I

Their internal mass distributions may be different.

Thinking mass farther from the axis makes rotation easier

It increases I and makes angular acceleration harder to produce for the same torque.

Using the wrong shape formula

A ring and a solid disk do not have the same moment of inertia.

Using diameter instead of radius

Check which measurement is given.

Forgetting to add contributions from several masses

Use:

I_total = Σmr²

Confusing I with torque

I measures rotational inertia; torque causes changes in rotational motion.

Confusing moment of inertia with angular velocity

Moment of inertia describes mass distribution, not how quickly the object is rotating.

---

Key Terms

Moment of inertia (I) — measure of an object's resistance to angular acceleration about a particular axis.

Rotational inertia — another name for moment of inertia.

Mass distribution — arrangement of an object's mass relative to its rotation axis.

Axis of rotation — line around which an object rotates.

Point mass — idealized object whose mass is treated as concentrated at one position.

Radius (r) — perpendicular distance between a mass and the axis of rotation.

Torque (τ) — turning effect of a force.

Angular acceleration (α) — rate of change of angular velocity.

Rigid body — object whose particles maintain fixed positions relative to each other.

Thin ring — idealized circular object with its mass concentrated near one radius.

Solid disk — circular object whose mass is distributed throughout its area.

Centre of mass — average position of an object's mass distribution.

Rotational kinetic energy — energy associated with rotation, calculated using E_rot = 1/2 Iω².

Flywheel — rotating device designed to store rotational energy and resist rapid changes in angular speed.

Angular momentum — rotational quantity related to moment of inertia and angular velocity.

---

Key Takeaways

• Moment of inertia measures resistance to angular acceleration.
• Its symbol is I and its SI unit is kg·m².
• Moment of inertia is the rotational analogue of mass.
• Unlike ordinary mass, moment of inertia depends on how mass is distributed.
• Mass farther from the axis contributes much more strongly to I.
• For a point mass, I = mr².
• For several point masses, I_total = Σmr².
• Because radius is squared, doubling the distance from the axis increases a point mass's contribution to I by a factor of four.
• The same object can have different moments of inertia about different axes.
• A thin ring has a larger moment of inertia than a solid disk of the same mass and radius.
• Moving mass outward increases moment of inertia.
• Moving mass inward decreases moment of inertia.
• Torque and angular acceleration are related by τ_net = Iα.
• For the same torque, a larger I produces a smaller angular acceleration.
• Rotational kinetic energy is E_rot = 1/2 Iω².
• Figure skaters, divers, gymnasts, flywheels and vehicle wheels demonstrate the importance of mass distribution.
• Moment of inertia affects how quickly rotating machinery can start and stop.
• Understanding I is essential for predicting the behaviour of rotating systems.

---

Check Your Understanding

1. Define moment of inertia and give its SI unit.
2. Explain why moment of inertia is sometimes called rotational inertia.
3. Calculate the moment of inertia of a 4.0 kg point mass located 0.50 m from an axis.
4. The mass in Question 3 is moved to 1.0 m from the axis. Calculate its new moment of inertia and compare it with the original value.
5. Three point masses are located around an axis: 2 kg at 0.50 m, 3 kg at 1.0 m, and 1 kg at 2.0 m. Calculate the total moment of inertia.
6. Explain why a thin ring has a larger moment of inertia than a solid disk with the same mass and radius.
7. A solid disk has mass 6.0 kg and radius 0.40 m. Calculate its moment of inertia.
8. A thin ring has the same mass and radius as the disk in Question 7. Calculate its moment of inertia and compare the two values.
9. A uniform rod has mass 3.0 kg and length 2.0 m. Calculate its moment of inertia when rotating about its centre.
10. Calculate the moment of inertia of the same rod when rotating about one end. Explain why the answer is larger.
11. A net torque of 20 N·m acts on an object with moment of inertia 5.0 kg·m². Calculate its angular acceleration.
12. Two objects experience the same net torque. Object A has I = 2 kg·m², while Object B has I = 8 kg·m². Compare their angular accelerations.
13. A rotating system contains two identical masses. Explain what happens to its moment of inertia if both masses are moved from 0.25 m to 0.50 m from the axis.
14. Explain why pulling the arms inward changes the moment of inertia of a spinning figure skater even though the skater's total mass does not change.
15. A solid disk has mass 8.0 kg, radius 0.50 m, and initially starts from rest. A constant net torque of 10 N·m acts for 4.0 s. Calculate the disk's moment of inertia, angular acceleration, final angular velocity, and rotational kinetic energy after 4.0 s.