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

Chapter 6: Circular Motion, Orbits, and Gravity

Introduction

This chapter explores the physics of objects moving in circles, including uniform circular motion, the forces involved, and the application of these concepts to orbital motion and gravity. Understanding these principles is essential for analyzing everyday phenomena such as cars turning corners, satellites orbiting planets, and the sensation of weightlessness in orbit.

Uniform Circular Motion

Velocity and Acceleration in Circular Motion

In uniform circular motion, an object moves at a constant speed around a circle, but its velocity changes direction continuously. The acceleration responsible for this change is called centripetal acceleration, always directed toward the center of the circle.

• Velocity (\vec{v}): Tangent to the circle at every point.

• Acceleration (\vec{a}): Points toward the center of the circle.

• Centripetal acceleration formula: $a = \frac{v^2}{r}$

Period, Frequency, and Speed

The period (T) is the time for one complete revolution. The frequency (f) is the number of revolutions per second. The speed of an object in circular motion is related to these quantities:

• Period: $T = \frac{1}{f}$

• Speed: $v = \frac{2\pi r}{T} = 2\pi r f$

• Centripetal acceleration: $a = \frac{v^2}{r} = 4\pi^2 r f^2$

Example: Spinning Table Saw Blade

A table saw blade with a diameter of 0.25 m spins at 3600 rpm. To find the period, speed, and acceleration of a tooth at the edge:

• Convert rpm to Hz: $f = \frac{3600}{60} = 60$ Hz

• Period: $T = \frac{1}{f} = 0.017$ s

• Speed: $v = 2\pi r f = 47$ m/s

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

Dynamics of Uniform Circular Motion

Centripetal Force

According to Newton’s second law, a net force is required to keep an object moving in a circle. This force is called centripetal force and is always directed toward the center.

• Centripetal force formula: $F_{\text{net}} = m a = m \frac{v^2}{r}$

• The net force can be provided by tension, friction, or the normal force, depending on the situation.

Forces Producing Centripetal Acceleration

Different forces can act as the centripetal force in various scenarios:

• Tension: For a ball on a string swung in a circle.

• Static friction: For a car turning a corner without skidding.

• Normal force: For a car on a banked curve.

Example: Maximum Speed for a Car to Turn a Corner

The maximum speed a car can turn without sliding is determined by the maximum static friction force:

• Static friction: $f_{s,\text{max}} = \mu_s n = \mu_s mg$

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

Example: Car’s Speed on a Banked Turn

On a banked curve, the normal force provides the centripetal force. The speed at which a car can take the curve without friction is:

• Normal force component: $n \sin \theta$

• Speed: $v = \sqrt{r g \tan \theta}$

Apparent Forces and Weight in Circular Motion

Centrifugal Force (Fictitious Force)

In a rotating reference frame, people often feel a force pushing them outward, called the centrifugal force. However, this is not a real force and does not appear in Newton’s laws or free-body diagrams.

Apparent Weight in Vertical Circular Motion

On roller coasters or loops, the sensation of weight changes due to the normal force supporting you. At the bottom of the loop, apparent weight is greater than true weight; at the top, it can be less.

• Bottom of loop: $n = w + \frac{mv^2}{r}$

• Top of loop: $n = w - \frac{mv^2}{r}$

• Critical speed: $v_c = \sqrt{g r}$ (minimum speed to stay in contact at the top)

Circular Orbits and Weightlessness

Orbital Motion and Free Fall

Satellites and astronauts in orbit are in continuous free fall, experiencing weightlessness. The force of gravity provides the centripetal force for orbital motion.

• Orbital speed: $v_{\text{orbit}} = \sqrt{g r}$

• Period: $T = \frac{2\pi r}{v_{\text{orbit}}}$

Newton’s Law of Gravity

Universal Gravitation

Newton’s law of gravity 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: $F = G \frac{m_1 m_2}{r^2}$

• Gravitational constant: $G = 6.67 \times 10^{-11}$ N·m2/kg2

• Doubling the distance reduces the force by a factor of four.

Gravity on Other Worlds

The weight of an object on another planet depends on the planet’s mass and radius:

• Weight: $w = m g_{\text{planet}}$

• Free-fall acceleration: $g_{\text{planet}} = \frac{G M_{\text{planet}}}{R_{\text{planet}}^2}$

Gravity and Orbits

Orbital Speed and Period

For a satellite in a circular orbit around a planet or star:

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

• Period: $T = 2\pi \sqrt{\frac{r^3}{G M}}$

Summary Table: Key Equations in Circular Motion and Gravity

Quantity	Equation (LaTeX)	Description
Centripetal acceleration	$a = \frac{v^2}{r}$	Acceleration toward center in circular motion
Centripetal force	$F = m \frac{v^2}{r}$	Net force required for circular motion
Period	$T = \frac{1}{f}$	Time for one revolution
Speed	$v = 2\pi r f$	Linear speed in circular motion
Orbital speed	$v_{\text{orbit}} = \sqrt{g r}$	Speed for orbit near planet's surface
Universal gravitation	$F = G \frac{m_1 m_2}{r^2}$	Force between two masses
Free-fall acceleration	$g = \frac{G M}{R^2}$	Acceleration due to gravity on planet
Orbital period	$T = 2\pi \sqrt{\frac{r^3}{G M}}$	Period for satellite in circular orbit

Additional info: These notes expand on the brief points in the original slides, providing full academic context, definitions, and examples for each concept. Images included are directly relevant to the explanations and reinforce key ideas in circular motion and gravity.