Ch. 5: Circular Motion and Gravity

Newton’s Law of Universal Gravitation

Newton's Law of Universal Gravitation describes the attractive force between any two masses in the universe. This law is fundamental to understanding planetary motion and satellite orbits.

• Universal Gravitation Formula: The gravitational force between two masses is given by:

• Special Case: Near Earth's surface, this reduces to .

• Separation Distance: is the distance between the centers of mass of the two objects.

Acceleration Due to Gravity

• General Formula: The acceleration due to gravity at a distance above a planet's surface is:

• On the Surface: If , then .

Satellites and Circular Orbits

For a satellite in a stable circular orbit, the gravitational force provides the necessary centripetal force.

• Force Balance:

• Orbital Velocity:

Ch. 6: Work and Energy

Work and the Work-Kinetic Energy Theorem

Work measures the energy transferred to or from an object via a force acting over a distance.

• Constant Force:

• Variable Force:

• Work-Energy Theorem: The net work done on an object equals its change in kinetic energy:

• Kinetic Energy:

• Change in Kinetic Energy:

Potential Energy

Potential energy is energy stored due to an object's position or configuration. Conservative forces, like gravity and springs, have associated potential energies.

• Gravitational Potential Energy:

• Elastic (Spring) Potential Energy:

• Zero Points: The zero of potential energy is chosen for convenience (e.g., for gravity, equilibrium position for springs).

Conservation of Mechanical Energy

Mechanical energy is the sum of kinetic and potential energies. In the absence of nonconservative forces (like friction), mechanical energy is conserved.

• Mechanical Energy:

• Conservation Law: If , then (energy is conserved).

• Expanded Form:

Work Done by Nonconservative Forces

• Friction: For constant friction, where is the normal force, is the coefficient of friction.

• Normal Force: On flat surfaces with no other vertical forces, .

• General Work: For any constant force, ; for variable forces, use the area under the force vs. displacement curve.

Ch. 7: Linear Momentum

Impulse and Momentum

Momentum is a vector quantity defined as the product of mass and velocity. Impulse is the product of force and the time interval over which it acts, and it changes an object's momentum.

• Momentum:

• Impulse: The area under a force vs. time graph represents impulse.

• Impulse-Momentum Theorem:

Collisions and Conservation of Momentum

During collisions, the total momentum of a system is conserved if no external net force acts on the system. Mechanical energy may or may not be conserved, depending on the type of collision.

• Newton’s Third Law: Forces between colliding objects are equal and opposite.

• Momentum Conservation: For two objects:

• Change in Momentum:

Types of Collisions

• Inelastic Collisions: Objects stick together after collision.

• Elastic Collisions: Objects bounce off and move separately.

• Explosions: One object splits into two or more.

Special Cases in Elastic Collisions

• One Object at Rest: If , use as the relative velocity.

• Equal Masses (): , (momentum is completely transferred).

• Large Mass Difference (): , .

• Small Mass (): , .

• General Solution:

2D Collisions

Momentum is a vector and must be conserved in both x and y directions in two-dimensional collisions.

• Components:

• Conservation in Each Direction:

Formula Sheet