Circular Motion and Orbital Motion: Study Notes

Circular Motion at Constant Speed

Basic Concepts

Circular motion occurs when an object moves along a circular path. If the speed is constant, the object experiences a continuous change in direction, resulting in a centripetal acceleration directed toward the center of the circle.

• Speed: The distance traveled per revolution divided by the time taken.

• Frequency (f): The number of revolutions per second.

• Period (T): The time for one complete revolution. $T = \frac{1}{f}$

• Distance per revolution: For a circle of radius r, the distance is $2\pi r$.

Centripetal Force and Acceleration

Objects in circular motion require a net force directed toward the center of the circle, called the centripetal force. This force causes the centripetal acceleration.

• Centripetal Acceleration: $a_{\text{cent}} = \frac{v^2}{r}$

• Centripetal Force: $F_{\text{cent}} = m \frac{v^2}{r}$

• Direction: The force and acceleration vectors point toward the center of the circle.

Component Analysis

When the speed is not constant, Newton's laws must be applied to each component of motion. For example, in a tetherball scenario, forces such as tension and gravity must be resolved into radial and tangential components.

• Free Body Diagram: Shows all forces acting on the object (e.g., tension, gravity).

• Resolving Forces: Use sine and cosine to resolve forces into components.

• Example: For a ball on a string of length 1.8 m, the tension provides the centripetal force.

Orbital Motion

Gravitational Orbits

Orbital motion is a form of perpetual free fall, where an object moves around a planet or star due to gravity. The centripetal force required for circular motion is provided by gravitational attraction.

• Newton's Law of Universal Gravitation: $F_{\text{grav}} = G \frac{m_1 m_2}{r^2}$

• Gravitational Constant: $G = 6.67 \times 10^{-11} \text{N} \cdot \text{m}^2 / \text{kg}^2$

• Orbital Centripetal Force: $F_{\text{cent}} = F_{\text{grav}}$

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

Newton's Third Law

For every action, there is an equal and opposite reaction. In orbital motion, the gravitational force exerted by the planet on the satellite is equal and opposite to the force exerted by the satellite on the planet.

• Action-Reaction Pair: $F_{\text{Earth on Moon}} = F_{\text{Moon on Earth}}$

Kepler's Third Law

Kepler's Third Law relates the period of orbit to the size of the orbit. The square of the period is proportional to the cube of the semi-major axis.

• Kepler's Third Law: $T^2 \propto a^3$

• Application: The mean distance of Mars to the Sun compared to Earth can be used to compare orbital periods.

Comparison Table: Orbital Properties of Earth and Mars

Planet	Mean Distance to Sun (a)	Orbital Period (T)
Earth	1 AU	1 year
Mars	1.5 AU	Additional info: $T_{\text{Mars}} = \sqrt{(1.5)^3} \approx 1.84$ years

Additional info: The orbital period increases with the cube of the distance from the Sun, as shown by Kepler's Third Law.