Download Elastic Potential Energy (Warning: Algebra required)

Elastic Potential Energy (Warning: Algebra required)
1. Consider a mass m attached to a horizontal spring of
spring constant k resting on a frictionless table. All
measurements are from the equilibrium rest point. An
external applied force F is gradually applied to
compress the spring until the force is just balanced by
the spring force. Write a relation between F, k, and the
maximum compression distance X. (Use a capital “X”
for the maximum compression distance.)
2. Draw a Force-Displacement graph for this motion
and from this calculate the work done on the spring.
Write this elastic potential energy in terms of the
maximum force and the compression distance.
3. The mass is then released. Because the system is
frictionless, energy is conserved. Draw a diagram
showing the types of energy in the system when the
spring is at maximum compression, at its equilibrium
position, and at maximum extension.
4. What is the elastic energy of the spring at these
three positions? What does this mean for the
maximum compression and maximum extension
distances?
Now consider a
vertical spring of
unextended length L.
A mass m is attached
to the spring and
released. The mass
oscillates up and
down through a
distance 2A (A stands
for amplitude), and
eventually comes to rest after a long time due to
friction at distance A below the unextended length L.
All distances x are measured from the unextended end
of the spring.
7. What is the total energy of the spring-mass system
just prior to release (elastic + kinetic + gravitational
potential).
8. What is the total energy of the system when the
mass comes to rest a distance 2A below the
unextended length. Use this result to calculate the size
of the amplitude of oscillation A in terms of k and m.
5. What is the kinetic energy of the mass at these three
positions. From this, calculate the speed of the mass
as it passes through the equilibrium position.
9. What is the total energy of the system when the
mass is a distance A below the unextended length. Use
this result to calculate the velocity at this point. What
is the kinetic energy at this point. Relate this to the
initial elastic energy – what do you notice.
6. Write the total energy of the spring – mass system
when the mass is at some arbitrary position x. Equate
this to the total initial energy. From this, write a
formula for the velocity of the mass at any position x.
Because the term in the square root can’t be negative,
what can you conclude about the sizes of X and x.
10. After friction has acted for a long time, the mass
comes to rest a distance A below the unextended
length. Write the force balance equation to derive a
relation for A in terms of m,g, and k. What is the total
energy of the system now, noting that friction has
removed the kinetic component.
Answers: (1) F = k X (2) E = ½ Fmax X = ½ k X2 (4) E = ½ k
X2, E = 0, E = ½ k X2 (5) 0, KE = ½ m v2, 0, v=Root(k/m) x
(6) Energy = ½ k X2 = ½ k x2 + ½ m v2, v = ± ROOT(2/m
[1/2 k { X2 – x2 } ] )
Answers: (7)E = 0J (8) E = ½ k (2A)2 - m g 2A, A = m g /
k (9) E = ½ k A2 + ½ m v2 – m g A,
v=A
2
2
Root(k/m), KE = ½ m v = ½ k A (10) A = m g / k, E = m g A + ½ k A2