Rational Inertia

Definition of Moment of Inertia

The moment of inertia is a measure of how hard it is to change an object's rotational motion around an axis.

It depends on the object's mass and how far that mass is distributed from the axis of rotation

mr² is called

Answer: Moment of inertia

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Explanation: mr² (mass × radius squared) is the moment of inertia for a single particle.

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Example: A ball (mass m) on a string (length r) spinning has moment of inertia I=mr²

What are the units for 'moment of inertia'?

Answer: kg·m²

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Explanation: Mass (kg) multiplied by radius squared (m²).

Example: A disk with m=2 kg and r = 3m has 18 kg*m²

The moment of inertia of an object depends on the

Answer - Shape, mass, and rotation axis

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Explanation: A thin rod rotated about its end vs. center has different I

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Example: A book spun on its corner vs. its center has different inertia.

The mass elements are not all located at the same — from the axis.

Answer: Distance

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Explanation: Calculating I requires summing mass at varying distances.

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Example: A dumbbell (masses at ends) vs. a stick (mass spread out) have different I

hoop AND write its moment of inertia equation

Answer - I = mr²

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Explanation: All mass is at radius r

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Example: A bicycle wheel (hoop shape)

Disk and Moment of Inertia

Answer = 1/2mr²

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Explanation: Mass is spread evenly across the disk.

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Example: A CD

Which has a bigger moment of inertia, a hoop or a disk?

Bold Answer: Hoop

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Explanation: Hoop’s mass is farther from the axis.

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Example: A hoop and disk with the same m and r: hoop resists spinning more.

the angular acceleration is proportional to the — acting on it.

Bold Answer: Torque

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Explanation: α=τ/I

(angular acceleration depends on torque).

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Example: Pushing harder on a door (more torque) makes it swing faster.

What is the SI units of angular acceleration?

Answer: rad/s²

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Explanation: Rate of change of angular velocity (radians per second squared).

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Example: A wheel speeding up from 0 to 10 rad/s in 2s has α=5 rad/s²α

Rotational version of F=ma

Answer: τ=Iα

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Explanation: Rotational version of F=ma

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Example: Torque on gives α=5 rad/s²

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τ: Torque (vector, SI unit: N·m)

I: Moment of inertia (scalar, SI unit: kg·m²)

α: Angular acceleration (vector, SI unit: rad/s²)

If torque stays the same but I gets smaller, what happens to α?

Answer: α increases

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Explanation: α=τ/I Smaller II = larger α.

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Example: A spinning ice skater pulls arms in (reduces I) to spin faster

What is the answer to Quick Quiz 8.2 pg254?

Answer: The torque increases when the force is applied farther from the pivot.

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Explanation: Torque τ=rFsin⁡θ

Larger r (distance) means more torque.

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Example: A wrench grips a bolt: pushing at the end (longer r) creates more torque than pushing near the bolt.

What is the answer to Quick Quiz 8.3 pg254?

Answer: The hoop has a larger moment of inertia than the disk.

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Explanation: Hoop’s mass is all at radius r, while disk’s mass is spread out.

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Example: A bike wheel (hoop) resists spinning changes more than a DVD (disk).

When an ice skater pulls in their arms, they (increase/decrease) their moment of inertia.

Answer: decrease

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Explanation: Pulling arms in reduces the distance of mass from the axis.

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Example: Skater spins slower with arms out, faster with arms tucked.

If an ice skater's moment of inertia is reduced, their angular speed (increases/decreases).

Answer: increases

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Explanation: Angular momentum L=Iω

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Example: Skater spins faster when pulling arms in.

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L (Angular Momentum):

Definition: Angular momentum is the rotational equivalent of linear momentum, representing the quantity of motion in a rotating system.

its a Vector

When an ice skater extends their arms, they (increase/decrease) their moment of inertia.

Answer: increase

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Explanation: Extending arms increases the distance of mass from the axis.

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Example: Skater slows down when spreading arms out.

If an ice skater's moment of inertia increases, their angular speed (increases/decreases).

Answer: decreases

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L=Iω

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Example: Skater spins slower with arms extended.

If the person in Figure 8.32 draws the weights inward, what happens?

Answer: Angular speed increases

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Explanation: Reducing I increase ω to conserve angular momentum.

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Example: Pulling weights closer makes the person spin faster.

From point A to B, the planet moves more slowly because it’s farther from the Sun.

Answer: True

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Explanation: Angular momentum conservation:
r↑, tangential speed v↓

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Example: Earth moves slower in July (aphelion) than in January (perihelion).

Kepler’s second law: Wedges formed in Figure 7.23 have the same

Answer: area

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Explanation: Equal time = equal area swept by the planet’s orbit.

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Example: A comet near the Sun sweeps a short, wide wedge equal in area to a long, narrow wedge when far away.

Kepler’s second law relates to conservation of _

Answer: angular momentum

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Explanation: No external torque = angular momentum (L) is constant.

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Example: A spinning ice skater (no external torque) speeds up when pulling arms in.