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 the forces and accelerations involved, the concept of apparent forces, and the gravitational principles governing orbital motion. Applications range from cars turning corners to satellites orbiting planets.

Uniform Circular Motion

Velocity and Acceleration

In uniform circular motion, an object moves at constant speed along a circular path. Although speed is constant, the velocity changes due to the changing direction, resulting in centripetal acceleration directed toward the center of the circle.

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

• Direction: Always points toward the center of the circle.

• Example: A ball swung in a circle by a string accelerates because its direction changes, not its speed.

Period, Frequency, and Speed

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

• $T = \frac{2\pi r}{v}$

• $f = \frac{1}{T}$

• $v = 2\pi r f$

Example: A table saw blade with radius 0.125 m spinning at 3600 rpm has a high edge speed and acceleration.

Dynamics of Uniform Circular Motion

Forces Producing Centripetal Acceleration

According to Newton's second law, a net force is required for circular motion. This force is not a new type, but arises from familiar sources:

• Tension: For a ball on a string.

• Friction: For a car turning on a road.

• Normal force: For a car in a dip or on a banked curve.

• Equation: $F_{net} = m a_c = m \frac{v^2}{r}$

Example: Static friction allows a car to turn; if friction is too low, the car slides straight.

Problem-Solving Approach

• Draw a free-body diagram.

• Identify forces acting toward the center.

• Apply Newton's second law along axes aligned with the circle.

• Solve for unknowns using $F_{net} = m \frac{v^2}{r}$.

Apparent Forces in Circular Motion

Centrifugal Force and Apparent Weight

In a rotating frame, passengers may feel an outward 'force' (centrifugal), but this is not a real force in Newtonian mechanics. Apparent weight changes in circular motion, such as on roller coasters:

• At the bottom of a loop: Apparent weight is greater than true weight.

• At the top: Apparent weight can be less than true weight, or zero at critical speed.

• Critical speed: $v_{critical} = \sqrt{r g}$

Centrifuges

Centrifuges use high centripetal acceleration to separate substances by density. Apparent weight in a centrifuge can be thousands of times greater than true weight.

Circular Orbits and Weightlessness

Orbital Motion

Objects in orbit are in free fall, experiencing gravity as the centripetal force. The orbital speed and period are given by:

• $v_{orbit} = \sqrt{\frac{G M}{r}}$

• $T = 2\pi \sqrt{\frac{r^3}{G M}}$

Example: The International Space Station orbits Earth about 15 times per day, feeling weightless because it is in free fall.

Newton’s Law of Gravity

Inverse-Square Law

Newton’s law states that the gravitational force between two masses is:

• $F = G \frac{m_1 m_2}{r^2}$

• G: Gravitational constant, $6.674 \times 10^{-11} \text{N m}^2 \text{kg}^{-2}$

• Doubling distance reduces force by factor of 4.

Example: The gravitational force between two people sitting next to each other is extremely small.

Gravity on Other Worlds

• Weight depends on planet’s mass and radius.

• Free-fall acceleration: $g = \frac{G M}{r^2}$

• Mass remains constant, weight changes with location.

Gravity and Orbits

Orbital Speed and Period

For satellites and planets:

• $v = \sqrt{\frac{G M}{r}}$

• $T = 2\pi \sqrt{\frac{r^3}{G M}}$

Example: Geostationary satellites orbit at a radius where their period matches Earth's rotation (24 hours).

Summary of Key Concepts

• Uniform circular motion: Constant speed, changing direction, centripetal acceleration toward center.

• Apparent weight: Changes in circular motion; weightlessness in orbit is due to free fall.

• Newton’s law of gravity: Universal, inverse-square law.

• Orbital motion: Gravity provides centripetal force; speed and period depend on mass and radius.