Download Chapter 19 Outline The First Law of Thermodynamics - Help-A-Bull

Force and Potential Energy (3D)
• Combing these in vector form,
𝜕𝑈
𝜕𝑈
𝜕𝑈
𝑭=−
𝒙+
𝒚+
𝒛
𝜕𝑥
𝜕𝑦
𝜕𝑧
• We can write this more succinctly using the “del” operator.
𝜕
𝜕
𝜕
𝛁= 𝒙
+𝒚
+𝒛
𝜕𝑥
𝜕𝑦
𝜕𝑧
• The force is the negative gradient of the potential.
𝑭 = −𝛁𝑈
Energy Diagram
• We can glean a lot of information by looking at graph of
the potential energy.
Energy Diagram Example
Chapter 7 Summary
Potential Energy and Energy Conservation
• Gravitational potential energy: 𝑈g = 𝑚𝑔ℎ
• Conservation of mechanical energy
𝐸tot = 𝐾1 + 𝑈g1 = 𝐾2 + 𝑈g2 = constant
• Elastic potential energy: 𝑈el = 12𝑘𝑥 2
• Conservative forces
• Potential energy, reversible, path-independent, zero closed loop
• Conservation of energy: ∆𝐾 + ∆𝑈 + ∆𝑈int = 0
• Force and potential energy: 𝑭 = −
• Energy diagrams
• Stable minima and unstable maxima
𝜕𝑈
𝒙
𝜕𝑥
+
𝜕𝑈
𝒚
𝜕𝑦
+
𝜕𝑈
𝒛
𝜕𝑧
Chapter 8 Outline
Momentum, 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:
𝑑𝒗 𝑑
𝑭=𝑚
=
𝑚𝒗
𝑑𝑡 𝑑𝑡
• We call the product of mass
and velocity momentum, 𝒑.
𝒑 = 𝑚𝒗
Momentum
• We therefore rewrite
Newton’s second law. The
net force acting on a particle
equals the time rate of
change of momentum.
𝑑𝒑
𝑭=
𝑑𝑡
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, 𝑱.
• First consider a constant force.
𝑱 = Σ𝑭 Δ𝑡
• But, Σ𝑭 =
𝚫𝑭
𝛥𝑡
, so 𝑱 = Δ𝒑
• This is called the impulse-momentum theorem.
𝑱 = 𝒑2 − 𝒑1
• 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.
𝑡2
𝑱=
Σ𝑭 𝑑𝑡
𝑡1
• We can define an average
force, 𝑭ave , such that
𝑱 = 𝑭ave Δ𝑡
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.
𝑭𝐵 on 𝐴 + 𝑭𝐴 on 𝐵 = 0
𝑑𝒑𝐴 𝑑𝒑𝐵 𝑑(𝒑𝐴 + 𝒑𝐵 )
+
=
=0
𝑑𝑡
𝑑𝑡
𝑑𝑡
• The total momentum of the
system, 𝑷 = 𝒑𝐴 + 𝒑𝐵 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.
𝑷 = 𝑚𝐴 𝒗𝐴 + 𝑚𝐵 𝒗𝐵 + ⋯ = constant
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,
𝑚𝐴 𝑣𝐴1𝑥 + 𝑚𝐵 𝑣𝐵1𝑥 = 𝑚𝐴 𝑣𝐴2𝑥 + 𝑚𝐵 𝑣𝐵2𝑥
1
1
1
1
2
2
2
2
𝑚
𝑣
+
𝑚
𝑣
=
𝑚
𝑣
+
𝐴
𝐵
𝐴
𝐴1𝑥
𝐵1𝑥
𝐴2𝑥
2
2
2
2𝑚𝐵 𝑣𝐵2𝑥
• 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:
2𝑚𝐴
𝑣𝐵2𝑥 =
𝑣𝐴1𝑥
𝑚𝐴 + 𝑚𝐵
Collision Example #2