🎡AP Physics 1 Unit 3 – Circular Motion and Gravitation

Key Concepts and Definitions

• Circular motion involves an object moving in a circular path at a constant speed
• Centripetal force is a force directed toward the center of a circular path that causes an object to follow a curved trajectory
• Centripetal acceleration is the acceleration directed toward the center of the circular path, causing a change in the direction of the velocity vector
• Gravitational force is the attractive force between two objects with mass, proportional to the product of their masses and inversely proportional to the square of the distance between them
• Kepler's laws describe the motion of planets around the sun and satellites around planets
  • Kepler's first law states that planets orbit the sun in elliptical paths with the sun at one focus
  • Kepler's second law states that a line segment joining a planet and the sun sweeps out equal areas during equal intervals of time
  • Kepler's third law relates the orbital period and semi-major axis of an orbit: $T^2 \propto a^3$
• Gravitational field is a region around a massive object where another object with mass experiences a gravitational force

Circular Motion Fundamentals

• Objects in circular motion have a constant speed but changing velocity due to the change in direction
• The direction of the velocity vector is always tangent to the circular path
• The acceleration vector points toward the center of the circular path, perpendicular to the velocity vector
• The magnitude of the acceleration is given by: $a_c = \frac{v^2}{r}$, where $v$ is the speed and $r$ is the radius of the circular path
• The period ($T$) is the time taken for one complete revolution, and the frequency ($f$) is the number of revolutions per unit time
  • They are related by: $f = \frac{1}{T}$
• Angular displacement ($\theta$) is measured in radians and relates to the arc length ($s$) and radius ($r$) by: $\theta = \frac{s}{r}$
• Angular velocity ($\omega$) is the rate of change of angular displacement and is related to the tangential speed ($v$) by: $v = r\omega$

Forces in Circular Motion

• Centripetal force is the net force acting on an object in circular motion, directed toward the center of the circular path
• The magnitude of the centripetal force is given by: $F_c = \frac{mv^2}{r}$, where $m$ is the mass of the object, $v$ is its speed, and $r$ is the radius of the circular path
• Various forces can act as the centripetal force, such as tension (in a string or cable), gravitational force, or friction
• In the absence of a centripetal force, an object will continue moving in a straight line due to inertia (Newton's first law)
• The centripetal force is always perpendicular to the velocity vector and does not change the object's speed, only its direction
• Banked curves (roads or tracks) provide a centripetal force through the normal force, allowing vehicles to maintain circular motion without relying solely on friction

Centripetal Acceleration

• Centripetal acceleration is the acceleration directed toward the center of the circular path, causing a change in the direction of the velocity vector
• The magnitude of centripetal acceleration is given by: $a_c = \frac{v^2}{r}$, where $v$ is the speed and $r$ is the radius of the circular path
• Centripetal acceleration is always perpendicular to the velocity vector and does not change the object's speed, only its direction
• The direction of centripetal acceleration is always toward the center of the circular path
• Centripetal acceleration is directly proportional to the square of the speed and inversely proportional to the radius of the circular path
  • Doubling the speed quadruples the centripetal acceleration, while doubling the radius reduces the centripetal acceleration by a factor of four

Gravitational Force and Fields

• Gravitational force is the attractive force between two objects with mass, given by Newton's law of universal gravitation: $F_g = G\frac{m_1m_2}{r^2}$, where $G$ is the gravitational constant, $m_1$ and $m_2$ are the masses of the objects, and $r$ is the distance between their centers
• The gravitational force is always attractive and acts along the line connecting the centers of the objects
• The strength of the gravitational force decreases with the square of the distance between the objects
• Gravitational field is a region around a massive object where another object with mass experiences a gravitational force
• The gravitational field strength ($g$) is the force per unit mass experienced by an object in the field: $g = \frac{F_g}{m}$
• The gravitational field strength at a distance $r$ from a point mass $M$ is given by: $g = G\frac{M}{r^2}$
• The gravitational potential energy ($U$) of an object with mass $m$ at a distance $r$ from a point mass $M$ is given by: $U = -G\frac{mM}{r}$

Kepler's Laws of Planetary Motion

• Kepler's laws describe the motion of planets around the sun and satellites around planets
• Kepler's first law (law of ellipses) states that planets orbit the sun in elliptical paths with the sun at one focus
  • The shape of the ellipse is described by its eccentricity ($e$), which ranges from 0 (a perfect circle) to 1 (a parabola)
• Kepler's second law (law of equal areas) states that a line segment joining a planet and the sun sweeps out equal areas during equal intervals of time
  • This means that planets move faster when they are closer to the sun and slower when they are farther away
• Kepler's third law (law of periods) relates the orbital period ($T$) and semi-major axis ($a$) of an orbit: $T^2 \propto a^3$
  • The proportionality constant depends on the mass of the central body (e.g., the sun for planets or a planet for satellites)
• Kepler's laws apply to any system with a central force that varies inversely with the square of the distance, such as the gravitational force

Orbits and Satellites

• Orbits are the paths followed by objects under the influence of a gravitational force, such as planets around the sun or satellites around planets
• Circular orbits have a constant radius and speed, with the gravitational force acting as the centripetal force: $F_g = \frac{mv^2}{r}$
• The speed of an object in a circular orbit is given by: $v = \sqrt{\frac{GM}{r}}$, where $G$ is the gravitational constant, $M$ is the mass of the central body, and $r$ is the orbital radius
• Elliptical orbits have varying speeds and distances from the central body, with the gravitational force and velocity vectors constantly changing direction
• Escape velocity is the minimum speed an object needs to escape the gravitational pull of a massive body: $v_{esc} = \sqrt{\frac{2GM}{r}}$
• Geostationary satellites orbit the Earth at an altitude of about 35,786 km above the equator, with an orbital period equal to the Earth's rotational period (1 day)
  • This allows the satellite to remain fixed above a specific point on the Earth's surface

Problem-Solving Strategies

• Identify the type of motion (circular or elliptical) and the forces involved (e.g., gravitational, tension, friction)
• Draw a free-body diagram to visualize the forces acting on the object and their directions
• Determine the relevant variables (e.g., mass, speed, radius, period) and the unknown quantity to be solved for
• Select the appropriate equations based on the given information and the unknown quantity
  • For circular motion, use equations relating centripetal force, centripetal acceleration, speed, and radius
  • For gravitational problems, use Newton's law of universal gravitation and equations for gravitational field strength and potential energy
  • For orbits, use Kepler's laws and equations for orbital speed and escape velocity
• Substitute the known values into the selected equations and solve for the unknown quantity
• Check the units of the answer to ensure they are consistent with the quantity being solved for
• Verify that the answer makes sense in the context of the problem (e.g., positive values for distances and speeds)