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
