FC - PlanetaryGeophysics_I_MoI

A set of 50 question-and-answer flashcards covering key definitions, formulas, and examples related to the Moment of Inertia as discussed in the lecture notes.

What does the Moment of Inertia (MoI) measure for a rotating body?

Its resistance to changes in rotational motion (angular acceleration).

Write the equation for rotational kinetic energy in terms of MoI and angular velocity.

E_rot = ½ Θ ω²

Which equation links angular momentum to MoI?

L = Θ ω

What are the SI units of the Moment of Inertia?

Kilogram metre squared (kg · m²)

On which property of a body does MoI primarily depend: total mass or mass distribution?

The distribution of mass relative to the rotation axis.

How does increasing the distance of mass elements from the rotation axis affect MoI?

MoI increases proportionally to the square of that distance (r²).

When the rotation axis of a body is changed, what happens to its MoI?

The MoI value changes because it is axis-dependent.

What name is given to the three perpendicular axes along which MoI takes extreme values?

Principal axes.

Around which of the principal axes does a free rigid body tend to spin?

The axis with the maximum MoI.

Provide the Cartesian integral definition of MoI for a continuous body.

Θ = ∫(x² + y²) ρ(x,y,z) dV

State the formula for the MoI of a homogeneous solid sphere.

Θ = (2/5) m R²

What is the numerical value of the MoI coefficient for a homogeneous sphere?

0.4 (2/5).

Give the general formula expressing MoI with the MoI coefficient I.

Θ = I m R²

How does concentrating mass toward the center of a sphere influence its MoI coefficient?

It lowers the coefficient below 0.4.

Which type of sphere can have an MoI coefficient greater than 0.4?

A hollow (shell-like) sphere with mass concentrated away from the center.

What is Earth’s MoI coefficient?

0.3307

What is Mars’s MoI coefficient?

0.3644

What is Jupiter’s MoI coefficient?

0.2756

What is the Moon’s MoI coefficient?

0.3929

What is the Sun’s MoI coefficient?

0.07

Give the MoI formula for a thin hollow sphere of radius r.

Θ = (2/3) m r²

Provide the MoI formula for a thick-walled hollow sphere with outer radius R and inner radius r.

Θ = (2/5) m (R⁵ − r⁵)/(R³ − r³)

How can the MoI of any sphere be expressed in terms of density ρ, volume V, and radius R?

Θ = (2/5) ρ V R²

In spherical coordinates, what is the expression for x?

x = r sin θ cos φ

In spherical coordinates, what is the expression for y?

y = r sin θ sin φ

What is the value of the integral ∫₀^π sin³θ dθ used in sphere MoI derivation?

2/3

Give the equatorial MoI (A) of an oblate spheroid with equatorial radius a and polar radius c.

A = (1/5)(a² + c²) m

Give the polar MoI (C) of an oblate spheroid about its rotation (c) axis.

C = (2/5) a² m

For an oblate spheroid where a > c, which MoI (A or C) is larger?

C is larger.

Give the principal MoI A for a triaxial ellipsoid with semi-axes a, b, c.

A = (1/5)(b² + c²) m

Give the principal MoI B for a triaxial ellipsoid with semi-axes a, b, c.

B = (1/5)(a² + c²) m

Give the principal MoI C for a triaxial ellipsoid with semi-axes a, b, c.

C = (1/5)(a² + b²) m

For a triaxial ellipsoid with a > b > c, which principal MoI is the largest?

C (about the shortest axis).

Provide the MoI of a thin-walled hollow cylinder of mass m and radius r about its symmetry axis.

Θ = m r²

Provide the MoI of a solid cylinder about its symmetry axis.

Θ = ½ m r²

Provide the MoI of a thick-walled cylinder with inner radius r₁ and outer radius r₂ about its axis.

Θ = ½ m (r₁² + r₂²)

Provide the MoI of a solid cylinder about a diameter through its center.

Θ = (1/12) m (3r² + h²)

In the Explorer I satellite, which axis has the smallest MoI?

The z-axis (along the satellite’s length).

Around which axes will Explorer I tend to spin when disturbed?

Around its x or y axes, where the MoI is greatest.

What significant discovery was made by Explorer I?

The Van Allen radiation belts.

Why is a planet’s MoI coefficient important to planetary scientists?

It constrains the planet’s internal density structure and composition.

What symbol is used for angular velocity in the MoI equations?

ω (omega).

How is angular velocity ω related to rotation period T?

ω = 2π / T

Write the expression for linear momentum.

p = m v

Write the expression for translational kinetic energy.

E_kin = ½ m v²

What mathematical object collects the MoI values about three axes?

The inertia tensor (MoI tensor).

Why does a hollow sphere have a greater MoI than a solid sphere of equal mass and radius?

Because its mass is concentrated farther from the rotation axis, increasing r² contributions.

For a homogeneous sphere of constant density, how does MoI scale with radius?

It is proportional to R⁵ (since m ∝ R³ and Θ ∝ m R²).

How does the presence of a large dense core affect a planet’s MoI coefficient?

It lowers the coefficient below the homogeneous value of 0.4.

In these notes, what physical quantity is denoted by the Greek letter Θ?

The Moment of Inertia.