Fall 2013
Physics 172 – Recitation 7
Using Energy Representations
Solution
Purpose: The purpose of this recitation is to give you experience working with graphical
representation of the energy of multi-particle system. The graphical representations show
how the energy of a system as a function of the relative position of the different objects.
We can represent the energy of many systems in this manner.
Readings: Chapter 6.8-6.12
Learning Objectives:
6.8.1 Calculate the potential energy for a system of two or more particles due to
gravitational or electrical interactions
6.8.3 Write down the multiparticle energy principle
6.9.1 Relate force to potential energy
6.9.2 Write down the formula for gravitational potential energy
6.10.1 Describe how the potential energy of system varies with separation of its
component objects
6.11.1 Graphically represent the kinetic energy, potential energy and sum of kinetic and
potential energy
6.12.1 Identify bound or unbound systems from their energy diagrams
Challenge Problem:
Draw graphical representations for the following multi-particle systems.
• A system of two electrons that are initially far apart, moving toward each other
(that is, their initial velocities are nonzero and they are heading straight at each
other).
• A system of two stars that are initially at rest with respect to each other but
thereafter move under the influence of their mutual gravitational attraction.
• A system of a proton and an electron that are initially far apart, moving towards
each other (that is, their initial velocities are nonzero and they are heading straight
at each other)
Your graphs should include the following:
• Identification of the type of interaction (electrical or gravitational).
• Identification if the type interaction is attraction or repulsive.
• Indication of whether the system is bound or not.
• Graph of the potential energy as a function of separation.
• Graph of the kinetic energy as a function of separation.
• Graph of the total energy as a function of separation.
• State any assumptions or reasonable approximations you needed to make to
produce these graphs.
•
•
Provide justification for all answers.
State any assumptions or reasonable approximations you needed to make to
produce these graphs.
In section 6.11, the following instructions are given for producing these energy
representations:
• Draw U vs r for the particular interaction (electric or gravitational).
• At some r where you happen to know K, plot the point (r, K).
• Add that value of K(r) to the value U(r) at the same separation r.
• Plot K+U at that r, then draw a horizontal line through that point. That lin
represents the system’s energy (ignoring its rest mass energy)
• Find another value of K at another position r.
Situation 1: Two electrons are initially very far apart, moving toward each other.
Type of Interaction: This situation represents an electrical interaction since we are
dealing with two charged particles.
Attractive or Repulsive? This interaction is repulsive in nature. We know that like
charges repel and opposite charges attract with electrical interactions.
Approximation and simplifying assumptions: No other objects interacting with the
system.
Justification: The K+U energy of the two-electron system is positive because it has
nonzero kinetic energy when the electrons are far apart and the system’s potential energy
is zero. We assume the system is isolated so that its energy will remain constant.
Because the electrons exert repulsive forces on each other, they will slow as they
approach each other and come to rest at some separation. At that point the system’s
kinetic energy is zero and the its total energy will be equal to its electrostatic potential
energy. The system is unbound since K + U ≥ 0 . Question: How can you determine the
distance of closest approach?
€
Situation 2: A system of two stars that are initially at rest with respect to each other but
thereafter move under the influence of their mutual gravitational attraction.
Type of Interaction: This situation represents a gravitational interaction since we are
dealing with two large massive objects.
Attractive or Repulsive? This interaction is attractive, since gravitational interactions
are always attractive.
Approximation and simplifying assumptions: No other objects interacting with the
system. The rest masses remain constant.
Justification: At the initial separation, r3 in the figure, K= 0, but U is less than zero since
the two stars are move under the influence of their mutual attraction. As r decreases K
increases and U decreases. The total energy of the two-star system is constant and K+U
is constant. The system is bound since K +U < 0 .
Situation 3: A system of a proton and an electron are initially very far apart, moving
towards each other.
Type of Interaction: This situation represents an electrical interaction since we are
dealing with two charged particles.
Attractive or Repulsive? This interaction is attractive in nature. We know that like
charges repel and opposite charges attract with electrical interactions.
Approximation and simplifying assumptions: No other objects interacting with the
system.
Justification: The K+U energy of the proton-electron system is positive because it has
nonzero kinetic energy when the particles are far apart and the system’s potential energy
is zero. We assume the system is isolated so that its energy will remain constant.
Because the particles exert attractive forces on each other, they will speed up as they
approach each other. So, the system’s kinetic energy increases. Since system’s total
energy remains constant its electrostatic potential energy must decrease (Since it was
initially zero, this means it becomes negative). The system is unbound since K + U ≥ 0 .
€