Gravitation and Circular Motion: Centripetal Acceleration, Newton's Law of Universal Gravitation, and Orbital Motion

Circular Motion and Centripetal Acceleration

Uniform Circular Motion

Uniform circular motion occurs when an object moves in a circle at constant speed. Although the speed is constant, the direction of velocity changes continuously, resulting in acceleration toward the center of the circle.

• Centripetal Acceleration (arad): The acceleration that points toward the center of the circle and is responsible for changing the direction of velocity.

• Formula:

$a_{rad} = \frac{v^2}{r}$

• Centripetal Force (Fc): Not a new type of force, but the net force causing circular motion. It is always directed toward the center of the circle.

• Formula:

$F_{net} = m a_{rad} = \frac{m v^2}{r}$

• Key Point: Real forces (such as tension, gravity, friction) provide the centripetal force required for circular motion.

Proportionality in Circular Motion

When a car drives around a circular curve of radius R at speed v, if the speed doubles, the radius must increase by a factor of 4 to maintain the same acceleration, since $a_{rad} \propto \frac{v^2}{r}$.

• Example: If $v \rightarrow 2v$, then $a_{rad} \rightarrow 4a_{rad}$ unless $r \rightarrow 4r$.

Applications: Car on a Flat Curve

When a car moves along a flat curve, the angle of a hanging object (like a fuzzy die) in the car can indicate the magnitude of the centripetal acceleration and thus the speed of the car.

• Key Point: The greater the angle, the higher the speed and centripetal acceleration.

• Related Examples: Banked curve problems, spring rides, and swing rides all involve circular motion and centripetal forces.

Vertical Circular Motion

In vertical loops (such as a car on a track), the normal force from the track varies at the top and bottom of the loop due to gravity.

• At the top: Both gravity and normal force point downward.

• At the bottom: Normal force points upward, gravity downward.

• Minimum speed: Required at the top to keep the car on the track, where normal force can be zero.

Newton's Law of Universal Gravitation

Fundamental Law

Newton's law describes the gravitational force between any two masses. It is an inverse square law, meaning the force decreases with the square of the distance between the centers of the masses.

• Formula:

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

• Where:

  • G: Gravitational constant, $6.673 \times 10^{-11} \text{ N} \cdot \text{m}^2 / \text{kg}^2$

  • m1, m2: Masses of the two objects

  • r: Distance between the centers of the masses

• Direction: The force on each mass points toward the other mass.

Spheres as Point Masses

For spherical masses, the gravitational force can be calculated as if all the mass were concentrated at the center of the sphere. This approximation holds as long as the spheres are not deformed and have uniform density.

• Key Point: Earth is not a perfect sphere, but the approximation is sufficiently accurate for most calculations.

Calculating Gravitational Force Near Earth's Surface

Near Earth's surface, the gravitational force on an object of mass m is:

$F_g = G \frac{M_E m}{R_E^2}$

• Where:

  • ME: Mass of Earth

  • RE: Radius of Earth

• Standard value: $g = 9.8 \text{ m/s}^2$

• Weight: $w = m g$

Newton's Third Law and Gravitational Force

The force you exert on the Earth is equal in magnitude and opposite in direction to the force the Earth exerts on you, consistent with Newton's Third Law.

• Formulas:

Force by Earth on you: $w_{E \rightarrow you} = G \frac{M_E m}{R_E^2}$

Force by you on Earth: $w_{you \rightarrow E} = G \frac{m M_E}{R_E^2}$

$w_{E \rightarrow you} = w_{you \rightarrow E}$

Weight on Other Planets

If a planet has the same mass as Earth but twice the radius, the weight of an object on its surface is:

$w = G \frac{M_E m}{(2 R_E)^2} = \frac{1}{4} G \frac{M_E m}{R_E^2}$

• Key Point: Weight decreases with the square of the radius if mass is constant.

Orbital Motion and Gravity

Combining Laws of Motion and Gravity

Orbital motion is analyzed by combining Newton's laws of motion with the law of universal gravitation. Two simplifying assumptions are often made:

• Orbits are circular: Most planetary orbits are slightly elliptical, but circular orbits are easier to analyze mathematically.

• Mass difference: The mass of the central body (e.g., Earth) is much greater than the orbiting object (e.g., satellite), allowing us to treat the central body as stationary.

Orbital Speed and Period

For a satellite of mass m orbiting a planet of mass M at radius r:

• Newton's Second Law: $\sum F = m a$

• Gravitational force provides centripetal acceleration:

$\frac{G M m}{r^2} = \frac{m v^2}{r}$

Solving for orbital speed:

$v^2 = \frac{G M}{r}$

• Orbital period (time for one complete orbit):

$T_{orb}^2 = \frac{(2\pi)^2}{G M} r^3$

• Key Point: The orbital speed depends on the mass of the central body and the radius of the orbit, not on the mass of the satellite.

Geosynchronous Orbits

Satellites in geosynchronous orbits have the same orbital period as Earth's rotation. The orbital radius for such satellites depends only on the mass of Earth and not on the satellite's mass.

• Key Point: Two satellites of different masses will orbit at the same distance if they are both geosynchronous.

Relating Circular Motion and Projectiles

Projectile Motion and Circular Paths

When projectiles are fired, their paths can be analyzed in terms of circular motion at the peak of their trajectory. The radius of curvature and radial acceleration can be compared for different projectiles.

• Horizontal velocity components: For projectiles fired simultaneously, the horizontal component of velocity can be compared to determine which projectile travels farther or faster horizontally.

Summary Table: Key Equations

Concept	Equation (LaTeX)	Description
Centripetal Acceleration	$a_{rad} = \frac{v^2}{r}$	Acceleration toward center in circular motion
Centripetal Force	$F_{net} = m a_{rad} = \frac{m v^2}{r}$	Net force required for circular motion
Universal Gravitation	$F_g = G \frac{m_1 m_2}{r^2}$	Gravitational force between two masses
Weight Near Earth's Surface	$w = m g$	Gravitational force on mass near Earth
Orbital Speed	$v^2 = \frac{G M}{r}$	Speed of satellite in circular orbit
Orbital Period	$T_{orb}^2 = \frac{(2\pi)^2}{G M} r^3$	Time for one complete orbit

Additional info: Some context and explanations have been expanded for clarity and completeness, including the physical meaning of equations and the connection between circular motion and gravitational force.