Download Experiment 2 Energy Transformations

Experiment 2 Energy
Transformations;
8.01
Week 07D2
Today’s Reading Assignment:
MIT 8.01 Course Notes:
Chapter 14 Potential Energy and Conservation of Energy
Sections 14.5-14.7,14.9
Announcements
Exam 2 Thursday Oct 24 7:30-9:30 See Announcements Page
for Room Assignments
Problem Set 6 due Week 8 Tuesday at 9 pm in box outside
26-152
Potential Energy and Force
In one dimension, the potential difference is
B
U (x) − U (x0 ) = − ∫ Fx dx
A
Force is the derivative of the potential energy
dU
Fx = −
dx
1 2
U spring ( x) = kx
2
Examples: (1) Spring Potential Energy:
dU
d ⎛1 2⎞
Fx , spring = −
= − ⎜ kx ⎟ = −kx
dx
dx ⎝ 2
⎠
Gm1m2
(2) Gravitational Potential Energy:
U grav (r ) = −
Fr , gravity
Gm1m2
dU
d ⎛ Gm1m2 ⎞
=−
= − ⎜−
⎟=−
dr
dr ⎝
r ⎠
r2
r
Table Problem: Potential Energy
Diagram
A body of mass m is moving
along the x-axis. Its potential
energy is given by the function
U(x) = b(x2-a2) 2 where b = 2 J/
m4 and a = 1 m .
a) On the graph of U vs. x, sketch
the force F vs. x.
b) What is an analytic expression
for F(x)?
Energy Diagram
Choose zero point for potential energy:
U (x = 0) = 0
Potential energy function:
1 2
U (x) = kx , U (x = 0) = 0
2
Mechanical energy is represented by a
horizontal line since it is a constant
E
mech
1 2 1 2
= K(x) + U(x) = mvx + kx
2
2
Kinetic energy is difference between
mechanical energy and potential energy
(independent of choice of zero point)
K(x) = E mech − U(x)
Graph of Potential energy function
U(x) vs. x
Concept Question: Energy Diagram 1
A particle with total
mechanical energy E has
position x > 0 at t = 0
1)  escapes to infinity in the
negative x-direction
2)  oscillates around a
3)  oscillates around b
4)  periodically revisits a and b
Concept Question: Energy Diagram 2
A particle with total
mechanical energy E has
position x1< x < x2 at t = 0
1)  escapes to infinity in the
negative x-direction
2)  oscillates around a
3)  oscillates around b
4)  periodically revisits a and b
Table Problem: Energy Diagram
The figure above shows a graph of potential energy U(x) verses position for a
particle executing one dimensional motion along the x-axis. The total mechanical
energy of the system is indicated by the dashed line. At t =0 the particle is
somewhere between points A and G. For later times, answer the following
questions.
a)  At which point will the magnitude of the force be a maximum?
b)  At which point will the kinetic energy be a maximum?
c)  At how many of the labeled points will the velocity be zero?
d)  At how many of the labeled points will the force be zero?
Table Problem: Experiment 2 CartSpring on an Inclined Plane
A cart of mass m slides down a plane that is inclined at an angle θ from
the horizontal. The cart starts out at rest. The center of mass of the cart
is a distance d from an unstretched spring with spring constant k that
lies at the bottom of the plane. Assume that the inclined plane has a
coefficient of kinetic friction µ. Find an equation whose solution
describes how far the spring will compress when the cart first comes to
rest.
Experiment 2
Energy Transformation