Chapter Review

Rotational Motion and Gravitation

Angular Kinematics · Rotational Dynamics · Gravitation and Satellites

Angular Position and Velocity

Angular displacement (Δθ) measures rotation in radians, linking to arc length via s = rθ. Angular velocity ω = Δθ/Δt measures spin rate, with the right-hand rule giving its vector direction along the rotation axis.

Key Points

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1 revolution = 2π rad = 360°; always convert to radians before using formulas
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Linear (tangential) velocity relates to angular velocity by v = rω — outer points move faster
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ω = 2πf = 2π/T connects angular velocity to frequency and period
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RPM to rad/s: multiply by 2π/60
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Positive ω = counterclockwise, negative ω = clockwise by convention

Formula

$$v = r\omega$$

Angular Acceleration and Components

Angular acceleration α = Δω/Δt describes how quickly rotation speeds up or slows down. A rotating point experiences tangential acceleration (speed change) and centripetal acceleration (direction change) simultaneously.

Key Points

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Tangential acceleration: $a_t = r\alpha$ — along the tangent, changes speed
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Centripetal acceleration: $a_c = \omega^2 r = v^2/r$ — toward center, changes direction
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Total acceleration: $a = \sqrt{a_t^2 + a_c^2}$ since the two components are perpendicular
•
Uniform circular motion: α = 0, constant ω, but centripetal acceleration still exists
•
Speeding up: ω and α same sign; slowing down: opposite signs

Formula

$$a_t = r\alpha, \quad a_c = \omega^2 r$$

Rotational Kinematic Equations

Under constant angular acceleration, four kinematic equations mirror their linear counterparts exactly, with the mapping x→θ, v→ω, a→α.

Key Points

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$\omega_f = \omega_i + \alpha t$
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$\Delta\theta = \omega_i t + \frac{1}{2}\alpha t^2$
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$\Delta\theta = \frac{\omega_i + \omega_f}{2} t$
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$\omega_f^2 = \omega_i^2 + 2\alpha\Delta\theta$
•
Strategy: list knowns, identify the unknown, pick the equation containing all knowns plus the unknown

Formula

$$\omega_f^2 = \omega_i^2 + 2\alpha\Delta\theta$$

Centripetal Force

Centripetal force is not a new force type — it is the net radial component of real forces (tension, friction, gravity, normal) that maintains circular motion.

Key Points

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$F_c = mv^2/r = mr\omega^2$ — net inward force for circular path
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Vertical circle top: $T + mg = mv^2/r$ (both forces point inward)
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Vertical circle bottom: $T - mg = mv^2/r$ (tension opposes weight)
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Minimum speed at top of vertical circle: $v_{min} = \sqrt{gr}$ (when T = 0)
•
Never draw centripetal force as a separate arrow in free-body diagrams

Formula

$$F_c = \frac{mv^2}{r}$$

Moment of Inertia

Moment of inertia I measures resistance to angular acceleration, depending on both total mass and how that mass is distributed relative to the rotation axis.

Key Points

•
$I = \sum m_i r_i^2$ — mass farther from axis contributes quadratically more
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Thin ring/hoop: $I = mr^2$ (all mass at maximum radius)
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Solid disk/cylinder: $I = \frac{1}{2}mr^2$
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Solid sphere: $I = \frac{2}{5}mr^2$
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Same object has different I about different axes
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Ranking for same m and r: $I_{hoop} > I_{disk} > I_{sphere}$

Formula

$$I = \sum m_i r_i^2$$

Torque and Rotational Newton's Law

Torque τ is the rotational analogue of force, determined by force magnitude, distance from pivot, and the angle of application. Net torque causes angular acceleration via τ = Iα.

Key Points

•
$\tau = rF\sin\theta$ — maximum when force is perpendicular to lever arm
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$\tau_{net} = I\alpha$ is the rotational form of Newton's second law
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Zero torque when force acts through the pivot (r = 0) or along the lever arm (θ = 0°)
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Counterclockwise torque is conventionally positive
•
Static equilibrium requires $\sum \tau = 0$

Formula

$$\tau = rF\sin\theta = I\alpha$$

Rotational Energy and Rolling

A spinning body stores rotational kinetic energy ½Iω². Rolling objects have both translational and rotational KE, with the energy partition determined by shape.

Key Points

•
Total KE of rolling object: $K = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2$
•
Rolling without slipping constraint: $v = r\omega$
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Speed at bottom of incline: $v = \sqrt{\frac{2gh}{1+k}}$ where $I = kmr^2$
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Lower k means more translational KE → sphere beats disk beats hoop in rolling races
•
Rotational KE fraction: $k/(1+k)$ of total

Formula

$$K_{total} = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2$$

Angular Momentum Conservation

When no net external torque acts on a system, total angular momentum L = Iω is conserved — decreasing I increases ω and vice versa.

Key Points

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$L = I\omega$ for a rigid body about a fixed axis
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$I_i\omega_i = I_f\omega_f$ — skater pulls arms in, I decreases, ω increases
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KE changes even when L is conserved: $K = L^2/(2I)$
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Direction of $\vec{L}$ also stays fixed (gyroscopic stability)
•
Internal work by muscles accounts for the KE change

Formula

$$I_i\omega_i = I_f\omega_f$$

Newton's Law of Gravitation and Field Strength

Every mass attracts every other mass with a force proportional to their product and inversely proportional to the square of their separation. Gravitational field strength g = GM/r² gives acceleration due to gravity at distance r.

Key Points

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$F = G\frac{m_1 m_2}{r^2}$ with $G = 6.674 \times 10^{-11}$ Nm²/kg²
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Inverse square law: double the distance → force drops to 1/4
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Surface gravity: $g = GM/R^2$ where R is planet radius
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Planet comparison: $g_1/g_2 = (M_1/M_2)(R_2/R_1)^2$
•
Gravity is always attractive and acts along the line joining centers

Formula

$$F = G\frac{m_1 m_2}{r^2}$$

Orbital Motion and Kepler's Third Law

A satellite orbits when gravity provides the centripetal force. Kepler's third law states T² ∝ r³, connecting orbital period to radius for all objects orbiting the same central mass.

Key Points

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Orbital speed: $v = \sqrt{GM/r}$ — independent of satellite mass
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Kepler's third law: $T^2 = \frac{4\pi^2}{GM}r^3$
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Ratio form: $T_1/T_2 = (r_1/r_2)^{3/2}$ — no need for G or M
•
Orbital radius r = R_planet + altitude — always measure from center
•
Higher orbit → slower speed but longer period

Formula

$$v = \sqrt{\frac{GM}{r}}$$

Weightlessness and Apparent Weight

Apparent weight depends on acceleration: a lift accelerating upward increases it, downward decreases it. In orbit, continuous free fall makes apparent weight zero — everything falls together.

Key Points

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Apparent weight: $T = m(g + a)$ upward, $T = m(g - a)$ downward
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Free fall ($a = g$): $T = 0$ — weightlessness
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Orbiting is continuous free fall sideways around the planet
•
Gravity still acts in orbit — weightlessness is due to shared acceleration, not zero gravity
•
Geostationary orbit: r ≈ 42,300 km from center (≈ 36,000 km altitude), T = 24 hours

Formula

$$T = m(g \pm a)$$

Formulas

Arc-Angle Relationship

Links linear distance along a curve to angular displacement

Formula

$s = r\theta$

Linear-Angular Velocity

Tangential speed from spin rate and radius

Formula

$v = r\omega$

Centripetal Force

Net inward force for circular motion

Formula

$F_c = \frac{mv^2}{r} = mr\omega^2$

Moment of Inertia

Rotational resistance from mass distribution

Formula

$I = \sum m_i r_i^2$

Rotational Newton's Second Law

Net torque causes angular acceleration

Formula

$\tau_{net} = I\alpha$

Conservation of Angular Momentum

When no external torque acts, Iω stays constant

Formula

$I_i\omega_i = I_f\omega_f$

Universal Gravitation

Attractive force between any two masses

Formula

$F = G\frac{m_1 m_2}{r^2}$

Orbital Velocity

Speed for stable circular orbit

Formula

$v = \sqrt{\frac{GM}{r}}$

Kepler's Third Law

Period-radius relationship for orbits

Formula

$T^2 = \frac{4\pi^2}{GM}r^3$

Apparent Weight

Scale reading in an accelerating system

Formula

$T = m(g \pm a)$