Ch 06: Circular Motion & Gravitation

Concept: Intro to Centripetal Forces

Circular motion involves forces that act toward the center of the circle, called centripetal forces. These differ from linear forces, which act along straight lines. Understanding the distinction is crucial for solving problems involving objects moving in circles.

• Linear Forces: Act along X and Y axes; solved using .

• Centripetal Forces: Always point toward the center of the circle; direction is radially inward.

• Formula:

• Example: A block tied to a string slides in a circle; tension provides the centripetal force.

Concept: Vertical Centripetal Forces

When objects move in vertical circles, the speed is not constant due to gravity. Forces at the top and bottom of the circle differ, and both centripetal and gravitational forces must be considered.

• At the top:

• At the bottom:

• Example: Calculating the normal force on a person at the top and bottom of a vertical loop.

Concept: Flat Curve

For objects moving around horizontal curves, friction provides the centripetal force needed to keep the object in circular motion. The maximum speed before slipping depends on the coefficient of static friction.

• Formula:

• Maximum speed:

• Example: Finding the maximum speed a car can take a curve without skidding.

Concept: Banked Curve

Banked curves allow vehicles to turn without relying solely on friction. The banking angle provides a component of the normal force that acts as centripetal force.

• Formula (no friction):

• Example: Calculating the speed for a car to take a banked curve without sliding.

Concept: Universal Law of Gravitation

Newton's Law of Universal Gravitation states that every mass attracts every other mass with a force proportional to the product of their masses and inversely proportional to the square of the distance between them.

• Formula:

• Gravitational constant:

• Example: Calculating the gravitational force between two spheres.

Concept: Center-of-Mass Distance

For gravitational calculations, the distance is measured between the centers of mass of the two objects. For large objects like planets, use their radii plus the altitude above the surface.

• Formula:

• Example: Finding the gravitational force on a satellite above Earth's surface.

Concept: Gravitational Forces in 2D

When multiple masses are arranged in two dimensions, gravitational forces are vectors and must be added using vector addition techniques.

• Steps: Calculate individual forces, break into components, sum components.

• Example: Net gravitational force on a mass in a triangle or rectangle arrangement.

Concept: Acceleration Due to Gravity

The acceleration due to gravity at a distance from a planet is given by Newton's Law of Gravity. On the surface, it simplifies to .

• Any distance:

• On surface:

• Example: Comparing gravity at the top of Mount Everest to Earth's surface.

Concept: Satellite Motion

Satellites orbit planets due to the balance between gravitational force and centripetal force. The shape and speed of the orbit depend on the mass of the planet and the radius of the orbit.

• Orbital speed:

• Example: Calculating the speed and height for the International Space Station.

Concept: Orbital Period of a Satellite

The orbital period is the time it takes for a satellite to complete one orbit. It is related to the radius of the orbit and the mass of the central body.

• Formula:

• Example: Finding the orbital period of a satellite around Earth.

Concept: Kepler's Third Law

Kepler's Third Law states that the square of the orbital period is proportional to the cube of the orbital radius for any circular orbit.

• Formula:

• Application: Used to compare orbits of planets and moons.

Concept: Black Holes

Black holes are regions of space with extremely high mass and density, resulting in a gravitational field so strong that not even light can escape. The Schwarzschild radius defines the event horizon.

• Schwarzschild radius:

• Example: Calculating the size of a black hole given its mass.

Summary Table: Key Equations

Additional info:

• These notes include worked examples and practice problems for each concept, reinforcing understanding and application.

• Key constants: , .

• Vector addition is essential for multi-object gravitational problems.