Force and Potential Energy (3D)

• Combing these in vector form,

• We can write this more succinctly using the “del” operator.

• The force is the negative gradient of the potential.

Energy Diagram

• We can glean a lot of information by looking at graph of the potential energy.

Energy Diagram Example

Chapter 7 SummaryPotential Energy and Energy Conservation

• Gravitational potential energy:

• Conservation of mechanical energy

• Elastic potential energy:

• Conservative forces

• Potential energy, reversible, path-independent, zero closed loop

• Conservation of energy:

• Force and potential energy:

• Energy diagrams

• Stable minima and unstable maxima

Chapter 8 OutlineMomentum, Impulse, and Collisions

• Momentum

• Impulse

• Conservation of momentum

• Vector components

• Collisions

• Elastic and inelastic

• Center of mass

• Rocket propulsion

Momentum

• Consider the case of a collision between two cars.

• Using Newton’s laws to find the resulting motion is difficult.

• We do not fully know the exact forces involved.

• We can deal with situations such as these by considering a new concept, momentum.

• Newton’s second law:

• We call the product of mass and velocity momentum, .

Momentum

• We therefore rewrite Newton’s second law. The net force acting on a particle equals the time rate of change of momentum.

Impulse-Momentum Theorem

• We have already considered a force applied over some distance (work).

• What about a force applied for some time? This is called the impulse, .

• First consider a constant force.

• But, , so

• This is called the impulse-momentum theorem.

• The change in momentum over a time period is the impulse of the net force that acts on the particle during that interval.

Impulse

• In general, we express the impulse as the integral of the force over time.

• We can define an average force, , such that

Impulse Example

Conservation of Momentum

• Consider two bodies that interact with each other but nothing else.

• In this system, there are no external forces, only internal forces.

• This is an isolated system.

• Each body exerts a force on the other with an equal magnitude but opposite direction.

• The total momentum of the system, is constant.

Conservation of Momentum

• If the net external force on a system is zero, the total momentum of the system is constant.

• Conservation of momentum.

• This is a fundamental principle.

• Treat each vector component separately.

Conservation of Momentum Example

Types of Collisions

• We define a collision to be any strong interaction between bodies that lasts for a relatively short time.

• In an elastic collision, all of the forces between the colliding bodies are conservative, no mechanical energy is lost and the total kinetic energy is the same before and after.

• In an inelastic collision, the internal forces are not all conservative, the total kinetic after the collision is less than before.

• If the bodies stick together after the collision, it is a completely, or perfectly inelastic collision.

• Regardless, momentum is conserved!

Collision Example

Elastic Collisions in One Dimension

• For an elastic collision, both momentum and mechanical energy are conserved.

• In one dimension,

• Given the masses and initial velocities, we can solve for the final velocities.

• For the special case where one body is initially at rest, this reduces to:

Collision Example #2