Chapter 6: Circular Motion, Orbits, and Gravity – Study Notes

Chapter 6: Circular Motion, Orbits, and Gravity

Overview

This chapter covers the fundamental concepts of circular motion, the dynamics involved, apparent forces, orbits, weightlessness, and Newton’s law of gravity. These topics are essential for understanding motion in physics, especially in systems involving rotation and gravitational interactions.

Uniform Circular Motion

Definition and Properties

Uniform circular motion refers to the motion of an object traveling at a constant speed along a circular path. Although the speed remains constant, the direction of the velocity changes continuously, resulting in acceleration.

• Velocity is always tangent to the circle.

• Acceleration is always directed toward the center of the circle (centripetal acceleration).

• The magnitude of centripetal acceleration is given by: $a = \frac{v^2}{r}$

Period, Frequency, and Speed

The period (T) is the time required for one complete revolution. The frequency (f) is the number of revolutions per second.

• Frequency: $f = \frac{1}{T}$

• Speed in terms of period and frequency: $v = 2\pi r f = \frac{2\pi r}{T}$

• Centripetal acceleration in terms of frequency and period: $a = (2\pi f)^2 r = \left(\frac{2\pi}{T}\right)^2 r$

Example: Car Rounding a Corner

When a car turns a corner at constant speed, its velocity is tangent to the curve, but its acceleration is directed toward the center of the curve (centripetal direction).

Example: Spinning Table Saw Blade

• Given: Blade diameter = 25 cm, spins at 3600 rpm.

• Radius: $r = 0.125$ m

• Frequency: $f = 60$ s-1 (after unit conversion)

• Speed: $v = 2\pi f r = 47$ m/s

• Acceleration: $a = (2\pi f)^2 r = 1.8 \times 10^4$ m/s2

Dynamics of Uniform Circular Motion

Forces in Circular Motion

According to Newton’s second law, a net force must act on an object to produce centripetal acceleration in circular motion.

• Net force toward the center: $F_{\text{net}} = m a = m \frac{v^2}{r}$

• This force can be provided by tension, friction, gravity, or other familiar forces.

Free-Body Diagrams

• In horizontal circular motion, the net force points toward the center of the circle.

• In vertical circular motion, forces such as gravity and normal force must be considered.

Maximum Speed on a Curve

• For a car of mass $m$ on a curve of radius $r$ with friction coefficient $\mu$:

• Maximum speed: $v_{\text{max}} = \sqrt{\mu g r}$

Apparent Forces in Circular Motion

Centrifugal Force (Fictitious Force)

When observed from a rotating frame, an apparent outward force (centrifugal force) seems to act on objects. However, this is not a real force and does not appear in Newton’s laws or free-body diagrams.

• What is felt is the tendency of the body to move in a straight line, while the actual force (e.g., car door) pushes it toward the center.

Circular Orbits and Weightlessness

Orbits

Objects in orbit around Earth (or any planet) are in continuous free fall toward the center, but their tangential velocity keeps them moving around the planet.

• For a circular orbit: $F_{\text{gravity}} = F_{\text{centripetal}}$ $\frac{GMm}{r^2} = m \frac{v^2}{r}$ $v = \sqrt{\frac{GM}{r}}$

Weightlessness

• Astronauts in orbit experience weightlessness because both they and their spacecraft are in free fall.

Newton’s Law of Gravity

Universal Law of Gravitation

Newton’s law states that every mass attracts every other mass with a force proportional to the product of their masses and inversely proportional to the square of the distance between them.

• Law of gravity: $F = G \frac{m_1 m_2}{r^2}$

• Where $G$ is the gravitational constant: $G = 6.674 \times 10^{-11}$ N·m2/kg2

Weight on Other Worlds

• Mass remains constant, but weight depends on the planet’s mass and radius.

• Weight on a planet: $W = m g_{\text{planet}}$ $g_{\text{planet}} = \frac{GM_{\text{planet}}}{R_{\text{planet}}^2}$

Gravity and Orbits

Satellites and Orbital Speed

• For a satellite of mass $m$ in a circular orbit of radius $r$ around a planet of mass $M$:

• Orbital speed: $v = \sqrt{\frac{GM}{r}}$

• Orbital period: $T = 2\pi \sqrt{\frac{r^3}{GM}}$

Galactic Orbits

• Gravity holds galaxies together; stars orbit the galactic center with periods depending on their distance from the center.

Summary Table: Key Equations in Circular Motion and Gravity

Concept	Equation	Description
Centripetal Acceleration	$a = \frac{v^2}{r}$	Acceleration toward center in circular motion
Frequency	$f = \frac{1}{T}$	Revolutions per second
Speed in Circular Motion	$v = 2\pi r f$	Speed along circular path
Newton’s Law of Gravity	$F = G \frac{m_1 m_2}{r^2}$	Gravitational force between two masses
Orbital Speed	$v = \sqrt{\frac{GM}{r}}$	Speed for stable circular orbit
Orbital Period	$T = 2\pi \sqrt{\frac{r^3}{GM}}$	Time for one complete orbit

Additional info: These notes expand on the provided slides and text, adding definitions, formulas, and academic context for clarity and completeness. The summary table is inferred for exam preparation.