Transcript Physics I Mechanics

Chapter 7
Potential Energy and Energy Conservation
7.1 Gravitational Potential Energy
Conservation of Mechanical Energy
(Gravitational Forces Only)
Example 7.1 Height of a baseball from energy
conservation
You throw a 0.145-kg baseball straight up in the air,
giving it an initial upward velocity of magnitude 20.0
m/s. Find how high it goes, ignoring air resistance.
When forces other than gravity do
work
 Example 7.2 Work and energy in throwing a baseball
In Example 7.1, suppose your hand moves up 0.50 m while
you are throwing the ball, which leaves your hand with an
upward velocity of 20.0 m/s. Again ignore air resistance. (a)
assuming that your hand exerts a constant upward force on
the ball, find the magnitude of that force. (b) Find the
speed of the ball at a point 15.0 m above the point where it
leaves your hand.
Gravitational Potential Energy for
Motion Along a Curved Path
 Conceptual Example 7.3
A batter hits two identical baseballs with the same
initial speed and height but different initial angles.
Prove that at a given height h, both balls have the same
speed if air resistance can be neglected.
7.2 Elastic Potential Energy
Situations with both gravitational and elastic potential
energy
Example 7.9 Motion with gravitational, elastic and friction forces
In a “worst-case” design scenario, a 2000-kg elevator with broken
cables is falling at 4.00 m/s when it first contacts a cushioning
spring at the bottom of the shaft. The spring is supposed to stop
the elevator, compressing 2.00 m as it does so. During the
motion a safety clamp applies a constant 17,000-N frictional force
to the elevator. As a design consultant, you are asked to
determine what the force constant of the spring should be.
Test Your Understanding of Section
7.2
 Consider the situation in Example 7.9 at the instant
when the elevator is still moving downward and the
spring is compressed by 1.00 m. The kinetic energy K,
gravitational potential energy Ugrav, and elastic
potential energy Uel at this instant, are they positive or
negative?
7.3 Conservative and
Nonconservative Forces
 Example 7.11 Conservative or nonconservative?
In a certain region of space the force on an electron is

F  Cxˆj , where C is a positive constant. The electron
moves in a counter-clockwise direction around a
square loop in the xy-plane (Fig. 7.20). The corners of
the square are at (x,y) = (0,0), (L,0), (L,L), and (0,L). 
Calculate the work done on the electron by the force F
during one complete trip around the square. Is this
force conservative or nonconservative?
The Law of Conservation of Energy
Test Your Understanding of Section 7.3
In a hydroelectric generating station, falling water is
used to drive turbines (water wheels), which in turn
run electric generators. Compared to the amount of
gravitational potential energy released by the falling
water, how much electrical energy is produced? (i) the
same; (ii) more; (iii) less.
7.4 Force and Potential Energy
 Example 7.13
A electrically charged particle is held at rest at the
point x=0, while a second particle with equal charge is
free to move along the positive x-axis. The potential
energy of the system is
U ( x) 
C
x
where C is a positive constant that depends on the
magnitude of the charges. Derive an expression for the
x-component of force acting on the movable charged
particle, as a function of its position.
Force and Potential Energy in Three
Dimensions
 Example 7.14 Force and potential energy in two
dimensions
A puck slides on a level, frictionless air-hocky table.
The coordinates of the puck are x and y. It is acted on
by a conservative force described by the potentialenergy function
1
U ( x, y ) 
2
k (x2  y2 )
Derive an expression for the force acting on the puck,
and find an expression for the magnitude of the force
as a function of position.
Test Your Understanding of Section
7.4
A particle moving along the x-axis is acted on by a
conservative force Fx. At a cer point, the force is zero.
(a) Which of the following statements about the value
of potential-energy function U(x) at that point is
correct? (i) U(x)=0; (ii) U(x)>0; (iii) U(x)<0; (iv) not
enough information is given to decide. (b) Which of
the following statements about the value of the
derivative of U(x) at that point is correct? (i)
dU(x)/dx=0; (ii) dU(x)/dx>0; (iii) dU(x)/dx<0; (iv) not
enough information is given to decide.
7.5 Energy Diagrams
Test Your Understanding of Section 7.5
The curve in Fig. 7.24b has a maximum at a point between x2 and
x3. Which statement correctly describes what happens to the
particle when it is at this point? (i) The particle’s acceleration is
zero. (ii) The particle accelerates in the positive x-direction; the
magnitude of the acceleration is less than at any other point
between x2 and x3. (iii) The particle accelerates in the positive xdirection; the magnitude of the acceleration is greater than any
other point between x2 and x3. (iv) The particle accelerates in
negative x-direction; the magnitude of the acceleration is less
than any other point between x2 and x3. (v) The particle
accelerates in the negative x-direction; the magnitude of the
acceleration is greater than any other point between x2 and x3.