### 02 Mechanical Energy

```Ch 14: Mechanical Energy
M Sittig
AP Physics B
Summer Course 2012
2012年AP物理B暑假班
Work


Impulse was F·Δt.
Another useful measure if F·d, which is
called Work.
Work
W  F  d∥
Work (N ·m
or Joules (J))
Force (N)
Distance over
which the
force is
applied (m)
Example Problem

Find the work done in pushing the object
shown above for 10 m.
Practice Problem

A car is traveling to the right with a speed v. The
conductor presses the brakes and the car starts skidding
to a stop in a distance d. The work done on the car by the
frictional force F is: (Assume that the frictional force is
constant).
A. W = 0
B. W = F*d
C. W = -F*d
D. W = F*v
E. W = -F*v
Practice Problem

A rope is pulling a box up an incline that makes an angle
θ with respect to the horizontal. The rope is parallel to
the incline surface and it exerts a constant force of
magnitude F as shown. After traveling a distance d, what
is the work done by the sum of the gravitational force
and the force of tension F on the box?
A. W = F*d
B. W = -F*d
C. W = 0
D. W = (F-mg*sin θ)d
E. W = (F+mg*sin θ)d
Work and Energy




Doing work on an object changes its kinetic
energy. This can be proved with the Third
Kinetics Equation ***.
By doing work, we change an object’s
energy. But also…
By giving up energy, an object can do work!
This is summarized in the Work-Energy
Theorem.
Work-Energy Theorem
W  KE
Work (J)
Change in
Kinetic
Energy (J)
Kinetic Energy
2
1
KE 
mv
2
Kinetic
Energy (J)
Mass (kg)
Velocity
(m/s)
Example Problem
Practice Problem


A block of mass m is dragged along a horizontal
surface by a constant force F applied at an
angle θ above the horizontal as shown. The
speed of the block is constant and equals v. The
block undergoes a displacement d.
Find the work done on the block by force F
during this process.
Practice Problem


A block of mass m is dragged along a horizontal
surface by a constant force F applied at an
angle θ above the horizontal as shown. The
speed of the block is constant and equals v. The
block undergoes a displacement d.
Find the work done on the block by the force of
friction during this process.
Practice Problem


A block of mass m is dragged along a horizontal
surface by a constant force F applied at an
angle θ above the horizontal as shown. The
speed of the block is constant and equals v. The
block undergoes a displacement d.
Find the change in the kinetic energy of the
block during this process.
Practice Problem

A 0.113 kg ball is thrown straight up from
1.81 m above the ground. Its initial vertical
speed is 11.20 m/s. A short time later, it hits
the ground. Calculate the total work done by
the force of gravity during that time.
Potential Energy



Kinetic Energy is energy due to motion.
When the ball is thrown up, the kinetic
energy disappears (vf=0), but the ball moves
apart from the Earth. It now has the ability
to do work (falling by gravity) due to its
position.
Potential Energy is stored energy due to
position.
Potential Energy due to gravity
PEg  mgh
Height (m)
Gravitational
Potential
Energy (J)
Mass (kg)
Gravitational
Field Strength
(m/s2)
Potential Energy: Note



Potential energy is relative: book on a table,
above the floor, on the 3rd floor.
Always choose a zero level, or just calculate
difference in potential energy.
This is why the AP Exam has the equation:
U g  mgh
Practice Problem

Calculate the change in potential energy of a
94.7 kg man when he takes an elevator from
the first floor to 20th floor, if the distance
between floors is 3.81 m.
Conservation of Energy


For reasons beyond this course (symmetry
of time; see Wikipedia) energy can be
transformed from one form to another, but
cannot be created or destroyed.
This is called the law of conservation of
energy.
Conservation of Energy

For problems only considering mechanical
energy:
KEi  PEi  KE f  PEf
Example Problem
Practice Problem

A 67.0 kg diver is 4.70 m above the water,
falling at speed of 7.60 m/s. Calculate her
kinetic energy as she hits the water. (Neglect
air friction)
Practice Problem

A block of mass m is at rest at the top of a ramp
of vertical height h. The block starts to slide
down the frictionless ramp and reaches a
speed v at the bottom. If the same block were
to reach a speed 3 v at the bottom, it would
need to have slid down a frictionless ramp of
vertical height _____.
Practice Problem

Three balls of equal mass are
fired simultaneously with equal
speeds from the same height h
above the ground. Ball 1 is fired
straight up, ball 2 is fired
straight down, and ball 3 is fired
horizontally. Rank in order from
largest to smallest their speeds
v1, v2, and v3 at the instant
right before they hit the ground.
(Neglect friction.)
Conservation of Energy

For problems where other forms of energy
might be important (eg friction generates
heat, sound energy):
KEi  PEi  KE f  PEf  W
Example Problem
Practice Problem

A 30.0 kg child slides
down a long slide in a
playground. She starts
from rest at a height h of
18.00 m. When she is
partway down the slide, at
a height h of 8.00 m, she is
moving at a speed of 8.30
m/s. Calculate the
mechanical energy lost
due to friction (as heat,
etc.).
Practice Problem


A small block of mass m is placed at the bottom of an
inclined plane. After a quick push, the block acquires
a speed v directed up the plane. The block then
slides along the plane until it stops and remains at
rest. The vertical displacement of the block is h.
During the ascent of the block, the work done by the
force of friction is…
Springs

Another form of potential energy, by
position of the end of the spring relative to
its rest position.
Force of a spring
Fs  kx
Force of a
spring (N)
Length (m)
Spring
constant
(N/m)
Also known as Hooke’s Law.
Potential Energy of a spring
2
1
PEs 
kx
2
Potential
Energy stored
in a spring (J)
Spring
constant
(N/m)
Length (m)
Example Problem
Practice Problem

shoots a dart to a maximum
height of 24 meters. The same
dart is again shot upward, but
this time the spring is
compressed only half as far
before firing. Neglecting
friction and assuming an ideal
spring, how far up does the
dart go this time?
Practice Problem

A spring-loaded gun shoots a dart with a
speed of 4 m/s. If the spring is compressed
twice as far, what will the ball’s speed be?
Practice Problem

A small block of mass 2 kg slides down from
the top of a rough inclined plane that has a
vertical height 2.5 m. The block strikes a
horizontal spring with force constant 300 N/m
and compresses it a distance 0.12 m before it
comes to a momentary stop. How much energy
is lost in this process (due to friction)?
Power



Sometimes energy is spent quickly,
sometimes slowly.
Unit of Watts
Power is a rate. What is the difference
between 200 W and 400 W?
Power
Power (W)
W
P
t
Work (J)
Time (s)
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