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

Circular Motion, Orbits, and Gravity

Uniform Circular Motion

Uniform circular motion describes the movement of an object along a circular path at a constant speed. Although the speed remains unchanged, the direction of the velocity vector continuously changes, resulting in acceleration.

• Velocity in Circular Motion: The instantaneous velocity is always tangent to the circle at any point.

• Centripetal Acceleration: The acceleration is directed toward the center of the circle and is given by: $a = \frac{v^2}{r}$

• Key Point: Even with constant speed, the changing direction means the object is accelerating.

• Example: A car driving around a circular track at constant speed experiences centripetal acceleration toward the center.

Period, Frequency, and Speed

Period and frequency are fundamental quantities for describing circular motion. The period is the time for one complete revolution, while frequency is the number of revolutions per second.

• Period (T): Time to complete one revolution.

• Frequency (f): Number of revolutions per second. $f = \frac{1}{T}$

• Speed in Circular Motion: The speed of an object moving in a circle of radius $r$ is: $v = \frac{2\pi r}{T}$ or $v = 2\pi r f$

• Relation to Centripetal Acceleration: $a = (2\pi f)^2 r = \left(\frac{2\pi}{T}\right)^2 r$

• Example: A point on the rim of a rotating wheel travels a distance $2\pi r$ in one period $T$.

Dynamics of Uniform Circular Motion

Newton's second law applies to circular motion, requiring a net force toward the center of the circle to maintain the motion.

• Net Force: The net force required for circular motion is: $\vec{F}_{\text{net}} = m\vec{a} = \frac{mv^2}{r}$ (toward center of circle)

• Source of Centripetal Force: This force can be provided by tension, friction, or the normal force, depending on the situation.

• Example: On a carnival ride, the net force keeping riders in circular motion is provided by the seat or the walls of the ride.

Conceptual Examples: Forces on a Car

Real-world examples illustrate the application of circular motion principles.

• Example 6.4: At the bottom of a dip, the normal force from the road is greater than the car's weight, making passengers feel heavier.

• Example 6.5: When turning a corner, the static friction force between the tires and the road provides the necessary centripetal acceleration. Kinetic friction does not apply unless the tires skid.

Maximum Speed for a Car to Turn a Corner

The maximum speed at which a car can safely turn is determined by the maximum static friction force.

• Maximum Static Friction: $f_s = \mu_s n = \mu_s mg$

• Maximum Speed Condition: $f_s = \frac{mv^2}{r}$ Rearranging for $v$: $v_{\text{max}} = \sqrt{\mu_s g r}$

• Example: For rubber tires ($\mu_s = 1.0$) on pavement, $g = 9.8\,\text{m/s}^2$, $r = 20\,\text{m}$: $v_{\text{max}} = \sqrt{(1.0)(9.8)(20)} = 14\,\text{m/s}$ (about 30 mph)

Apparent Weight in Circular Motion

Apparent weight is the normal force felt by a person in a non-inertial frame, such as on a roller coaster. It can differ from true weight due to acceleration.

• At the Bottom of a Loop: Apparent weight is greater than true weight. $w_{\text{app}} = n = w + \frac{mv^2}{r}$

• At the Top of a Loop: Apparent weight can be less than true weight. $w_{\text{app}} = n = w - \frac{mv^2}{r}$

• Critical Speed: The minimum speed to maintain contact at the top of the loop: $v_c = \sqrt{g r}$

• Example: On a roller coaster, passengers feel heavier at the bottom and lighter at the top of a loop.

Orbital Motion

Orbital motion occurs when a projectile moves fast enough that its trajectory matches the curvature of the planet, resulting in continuous free fall around the planet.

• Closed Trajectory: When the curve of the trajectory matches the curve of the planet, the object enters orbit.

• Orbital Speed: For an object just skimming the surface of a planet: $v_{\text{orbit}} = \sqrt{g R_e}$ For Earth: $v_{\text{orbit}} \approx 7900\,\text{m/s}$

• Orbital Period: $T = \frac{2\pi r}{v_{\text{orbit}}} = 2\pi \sqrt{\frac{r}{g}}$ For a low Earth orbit: $T \approx 84$ minutes

• Weightlessness: Astronauts in orbit experience weightlessness because they are in continuous free fall.

Gravity and Universal Gravitation

Gravity is a universal force described by Newton's law of gravitation, which states that every mass attracts every other mass with a force proportional to their masses and inversely proportional to the square of the distance between them.

• Inverse-Square Law: The gravitational force decreases with the square of the distance.

• Newton's Law of Gravity: $F = G \frac{m_1 m_2}{r^2}$ Where $G = 6.67 \times 10^{-11}\,\text{N} \cdot \text{m}^2 / \text{kg}^2$

• Example: Doubling the distance between two masses reduces the force by a factor of 4.

Gravity on Other Worlds

The weight of an object depends on the gravitational acceleration of the planet or moon it is on.

• Weight on the Moon: $w = m g_{\text{moon}}$

• Using Newton's Law: $F_{\text{moon on } m} = \frac{G M_{\text{moon}} m}{R_{\text{moon}}^2}$

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

• Example: Your mass remains the same on other planets, but your weight changes according to the local value of $g$.

Gravity and Orbits

Newton's second law and the law of gravitation together describe the motion of satellites and planets in orbit.

• Gravitational Force Provides Centripetal Acceleration: $F_{\text{gravity}} = \frac{G M m}{r^2} = m \frac{v^2}{r}$

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

• Orbital Period: $v = \frac{2\pi r}{T}$ Combining with orbital speed: $T^2 = \frac{4\pi^2}{G M} r^3$ (Kepler's Third Law for circular orbits)

• Example: The period of a satellite depends on the radius of its orbit and the mass of the central body.

Summary Table: Key Equations in Circular Motion and Gravity

Quantity	Equation	Description
Centripetal Acceleration	$a = \frac{v^2}{r}$	Acceleration toward center in circular motion
Period	$T = \frac{1}{f}$	Time for one revolution
Speed in Circular Motion	$v = 2\pi r f$	Speed for frequency $f$ and radius $r$
Maximum Speed (Friction)	$v_{\text{max}} = \sqrt{\mu_s g r}$	Max speed before sliding on a flat curve
Apparent Weight (Bottom)	$w_{\text{app}} = w + \frac{mv^2}{r}$	Normal force at bottom of loop
Apparent Weight (Top)	$w_{\text{app}} = w - \frac{mv^2}{r}$	Normal force at top of loop
Critical Speed	$v_c = \sqrt{g r}$	Minimum speed to stay in contact at top
Newton's Law of Gravity	$F = G \frac{m_1 m_2}{r^2}$	Gravitational force between two masses
Orbital Speed	$v = \sqrt{\frac{G M}{r}}$	Speed for circular orbit of radius $r$
Orbital Period	$T^2 = \frac{4\pi^2}{G M} r^3$	Kepler's Third Law (circular orbits)
Free-Fall Acceleration	$g_{\text{planet}} = \frac{G M_{\text{planet}}}{R_{\text{planet}}^2}$	Acceleration due to gravity on a planet

Summary of Applications

• Apparent Weight: Changes in circular motion due to acceleration; important in amusement park rides and vehicle dynamics.

• Orbital Motion: Satellites and planets follow predictable paths governed by gravity and circular motion equations.

• Weightlessness: Astronauts in orbit experience weightlessness because they are in continuous free fall around Earth.