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Problems 261

PROBLEMS

that the bowstring pushes against the arrow. Suppose that
8.1 Potential Energy of a when the archer stretches the “spring” 0.52 m, he must exert a
Conser vative Force † force of 160 N to hold the arrow in this position. If he now
releases the arrow, what will be the speed of the arrow when
1. The spring used in the front suspension of a Triumph sports the “spring” reaches its equilibrium position? The mass of the
4
car has a spring constant k 3.5 10 N/m. How far must arrow is 0.020 kg. Pretend that the “spring” is massless.
we compress this spring to store a potential energy of 100 J?
*12. A mass m hangs on a vertical spring of a spring constant k.
2. A particle moves along the x axis under the influence of a vari- (a) How far will this hanging mass have stretched the spring
3
able force F 2x 1 (where force is measured in newtons and
x from its relaxed length?
distance in meters). Show that this force is conservative; that is,
show that for any back-and-forth motion that starts and ends at (b) If you now push up on the mass and lift it until the spring
the same place (round trip), the work done by the force is zero. reaches its relaxed length, how much work will you have
done against gravity? Against the spring?
3. Consider a force that is a function of the velocity of the parti-
cle (and is not perpendicular to the velocity). Show that the *13. A particle moving in the x–y plane experiences a conservative
work for a round trip along a closed path can then be different force
from zero. F byi bx j
4. The force acting on a particle moving along the x axis is given where b is a constant.
4
by the formula F K/x , where K is a constant. Find the cor-
x (a) What is the work done by this force as the particle moves
responding potential-energy function. Assume that U (x) 0
from x 0,y 0 to x x, y y?
2
2
1
1
for x .
(Hint: Use a path from the origin to the point x , y con-
2 2
5. A 50-g particle moving along the x axis experiences a force sisting of a segment parallel to the x axis and a segment
3
3
F Ax , where A 50 N/m . Find the corresponding parallel to the y axis.)
x
potential-energy function. If the particle is released from rest
(b) What is the potential energy associated with this force?
at x 0.50 m, what is its speed as it passes the origin?
Assume that the potential energy is zero when the particle
6. The force on a particle confined to move along the positive x is at the origin.
axis is constant, F F , where F 25 N. Find the corre-
x 0 0 *14. The four wheels of an automobile of mass 1200 kg are sus-
sponding potential-energy function. Assume U (x) 0 at
pended below the body by vertical springs of constant
x 0. 4
k 7.0 10 N/m. If the forces on all wheels are the same,
7. A particular spring is not ideal; for a distance x from equilib- what will be the maximum instantaneous deformation of the
3
rium, the spring exerts a force F 2x x , where x is in springs if the automobile is lifted by a crane and dropped on
x
meters and F is in newtons. What is the potential-energy the street from a height of 0.80 m?
x
function for this spring? How much energy is stored in the
*15. A rope can be regarded as a long spring; when under tension,
spring when it is stretched 1.0 m? 2.0 m? 3.0 m?
it stretches and stores elastic potential energy. Consider a
8. The force on a particle moving along the x axis is given by nylon rope similar to that which snapped during a giant
tug-of-war at a school in Harrisburg, Pennsylvania (see
F x a
0
Problem 23 of Chapter 5). Under a tension of 58000 N
F 0 a x a (applied at its ends), the rope of initial length 300 m stretches
x f
to 390 m. What is the elastic energy stored in the rope at this
F x a
0
tension? What happens to this energy when the rope breaks?
where F is a constant. What is the potential-energy function *16. Among the safety features on elevator cages are spring-loaded
0
for this force? Assume U (x) 0 for x 0. brake pads which grip the guide rail if the elevator cable
9. Consider a particle moving in a region where the potential should break. Suppose that an elevator cage of 2000 kg has
2
4
energy is given by U 2x x , where U is in joules and x is in two such brake pads, arranged to press against opposite sides
5
meters. What is the position-dependent force on this particle? of the guide rail, each with a force of 1.0 10 N. The friction
coefficient for the brake pads sliding on the guide rail is 0.15.
10. The force on an electron in a particular region of space is
Assume that the elevator cage is falling freely with an initial
given by F F sin (ax) i, where F and a are constants (this
0 0 speed of 10 m/s when the brake pads come into action. How
force is achieved with two oppositely directed laser beams).
long will the elevator cage take to stop? How far will it travel?
What is the corresponding potential-energy function?
How much energy is dissipated by friction?
*11. A bow may be regarded mathematically as a spring. The
archer stretches this “spring” and then suddenly releases it so
† For help, see Online Concept Tutorial 10 at www.ww norton.com/physics