Download Lecture Notes 6

1
2
KE = mv
Work-Kinetic Energy
Theorem
(
2
 
WF ≡ ∫ F • dx
change in the kinetic
energy of an object
net work done on the
particle
)=(
)
€
Wnet = ΔKE = KE f − KE i
€ Work is the dot product of F and d
Note:
Wg ≡
∫

  
Fg • dx = Fg • d = mgΔy
Fg = (mg) − ˆj
( )
where
and g = 9.81m / s 2
If an object is displaced upward (Δ y positive), then the work done
by the gravitational force on the object is negative.
€
If an object is displaced downward (Δy negative), then the work
done by the gravitational force on the object is positive.
Wspring =
€
∫


1
Fspring ( x) • dx = − k x 22 − x12
2
(
)
Checkpoint
Checkpoint #2
For the situation (Figure), the initial and final positions, respectively,
along the x axis for the block are given below. Is the work done by the
spring force on the block positive, negative or zero?
(a) -3 cm, 2 cm.
1.  Positive
2.  Negative
3.  Zero
1
1
W s = kx i2 − kx 2fi
2
2
€
Checkpoint
Checkpoint #2
For the situation (Figure), the initial and final positions, respectively,
along the x axis for the block are given below. Is the work done by the
spring force on the block positive, negative or zero?
(b) 2 cm, 3 cm.
1.  Positive
2.  Negative
3.  Zero
1
1
W s = kx i2 − kx 2fi
2
2
€
Checkpoint
Checkpoint #2
For the situation (Figure), the initial and final positions, respectively,
along the x axis for the block are given below. Is the work done by the
spring force on the block positive, negative or zero?
(c) -2 cm, 2 cm.
1.  Positive
2.  Negative
3.  Zero
1
1
W s = kx i2 − kx 2fi
2
2
€
Sample Problem 7-8
A block of mass m slides across a horizontal
frictionless counter with speed v0. It
runs into and compresses the spring
with spring constant k. When the block
is momentarily stopped by the spring,
by what distance d is the spring
compressed?
Work by Spring force:
Work-Kinetic Energy theorem:
€
€
Wspring = 12 kx i2 − 12 kx 2f
Wnet = ΔKE
HW #12
Problem
A block of mass m is dropped onto a spring. The block
becomes attached to the spring and compresses it by
distance d before momentarily stopping.
While the spring is compressed, what work is done on the block by:
a) the gravitational force on it
b) the spring force?
c) What is the speed of the block just before it hits the spring?
d) From what height h was the box dropped?
e) How high will it go back.
Ch. 7 Problems
HW#7: A block of mass m is attached to one end of a
spring with spring constant k, whose other end is fixed.
The block is initially at rest at the position where the
spring is unstretched (x=0) when a constant horizontal
force F in the positive x direction is applied. A plot of the
resulting kinetic energy of the block versus its position x is
shown. Find the equation relating the F to Ks. Problem
HW#8: A block of mass m is attached to one end of a spring with
spring constant k, whose other end is fixed. The block is initially at
rest at the position where the spring is unstretched (x=0) when a
constant horizontal force F in the positive x direction is applied. a)  Where will it stop?
b)  What is the work done by the applied force?
c)  What is the work done by the spring?
d)  Where is the block when the Kinetic Energy is max.
e)  Value of Max. Kinetic energy.
Power
• Power = the rate at which work is done by a force.
• Average power is work W done in time Δt
W
Pave =
Δt
• The instantaneous rate of doing work (instantaneous power)
€
P=
dW
dt
• Units: Watt [W]
1 W = 1J/s 1 horsepower = 1 hp = 746 W
1 kW-hour = 3.6 MJ
• Power from the time-independent force and velocity:
 

d
F
•
x

d
x
(
)
( )  
dW
P=
=
=F•
= F •v
dt
dt
dt
€
Instantaneous power ! Power
• The rate at which work is done by a force is power.
• Average power is work W done in time Δt
P = Pave =
Work
time
• Units: Watt [W]
1 W = 1J/s €
1 horsepower = 1 hp = 746 W
•  Power from the force and velocity:
→ →
W F• d
P=
=
t
t
but:
€
so:
€
v=
d
t
P = F ⋅v
Example: Crate of Cheese
An initially stationary crate of cheese (mass m) is pulled via a cable a
distance d up a frictionless ramp of angle θ where it stops.
(a)  How much work WN is done on the crate by the Normal during the
→
→
lift? WN = F N • d = 0
(b)  How much work Wg is done on the crate by the gravitational force
→
→
during the lift?
Wg = F g • d = −(mg sin θ )d = −mgh
(c)  How much work WT is done on the crate by the Tension during the
→
→
lift?
WT = F T • d = (mg sin θ )d = mgh
(d)  If the speed of the moving crate were increased, how would the above
→
→
answers change? What about the power?
P = F net • v
Chapter 8: Potential Energy & Conservation of Energy
• Chapter 7: What happens to the KE of when work is done on it.
[ KE: “energy of motion”
W: energy transfer via force ]
Conservative vs non-conservative forces
•  Can you get back what you put in?
Win = -Wout
•  What happens when you reverse time?
Properties of Conservative Forces
• Net work done by a conservative force on an object moving around every
closed path is zero.
b
Wab =
∫


F (x)• dx
a
Wab,1 = Wab,2
€
Conservative forces
€ - gravitational force€
- spring force
&
Wab,1 = −Wba,2
Properties of Non-conservative Forces
• Net work done by a non-conservative force on an object moving around
every closed path is non-zero.


Wab = ∫ F (x) • dx
b
a
Wab,1 ≠ Wab, 2
€
Non-conservative forces
- kinetic frictional force
€
(noise, heat,…) Potential energy
Potential energy: Energy U which describes the configuration (or spatial
arrangement) of a system of objects that exert conservative forces on each
other. It’s the stored energy in system.
ΔU = −W
Definition
Gravitational Potential energy: [~associated with the state of separation]
yf
ΔU grav = − ∫ (−mg)dy = mg( y f − yi ) = mgΔy
yi
€
If U grav (y = 0) ≡ 0
then U grav (y) = mgy
ΔUgrav ↑ if going up
ΔUgrav ↓ if going down
Elastic Potential energy: [~associated with the state of compression/tension
of elastic object]
ΔU spring = 12 kx 2f − 12 kx i2
€
€
If
€
U spring (x = 0) ≡ 0
then
U spring (x) = 12 kx 2
ΔUspring ↑ if x goes ↑ or ↓ (any displacement)
€
Don’t forget…
Work done by force (general)
 
  
W = ∫ F ( x ) • dx = F • d
Work by Gravitational force:
 
Wg = Fg • d
€
Wspring = 12 kx i2 − 12 kx 2f
Work by Spring force:
€
Wnet = ΔKE = KE f − KE i
Work-Kinetic Energy theorem:
€
Potential energy (if conservative force):
ΔU grav = mgΔy
W = −ΔU
€ If U grav (y = 0) ≡ 0 then U grav (y) = mgy
ΔU spring = 12 kx 2f − 12 kx i2
€
€
€
If U spring (x = 0) ≡ 0
then U spring (x) = 12 kx 2