Download A mass slides down a frictionless ramp of height h. Its initial speed is

A mass slides down a frictionless ramp of height h. Its initial speed
is zero. Its final speed at the bottom of the ramp is v. +y
h
0
As the mass descended, its KE, PE, and total Energy:
A:
B:
C:
D:
KE
increased
decreased
increased
decreased
L22 W 10/15/14 a+er lecture PE
increased
decreased
decreased
increased
total Energy
increased
decreased
stayed the same
stayed the same
1 Trials and tribulations of friction – a skater encounters a
carpet.
https://www.youtube.com/watch?
annotation_id=annotation_3308747645&feature=iv&src_vid=SRHoPL0WtdM&v=e
pSrFV2Wtjs
Or a shorter version at:
https://www.youtube.com/watch?v=SRHoPL0WtdM
L22 W 10/15/14 a+er lecture 2 Assignments
For this week:
•  You should have read up through Ch. 6 and Ch. 7 of Wolfson and
Prof. Dubson’s notes.
•  Read Ch. 8 of Wolfson by Friday.
•  Complete HW 7 this week. CAPA 8 is now live.
•  Midterm 2 will be Thursday of next week: two old exams on D2L.
•  LA applications will close Oct 24: lacentral.colorado.edu
Today:
•  Continue:
Conservation of Energy: mechanical and total. •  Introduce:
Power, today or Friday.
L22 W 10/15/14 a+er lecture 3 1. Work:
 
 
WF = ∫ F ⋅dr (in some circumstances: F ⋅ Δr )
f
i
Component of “specific” force in direction of
displacement
2. Kinetic Energy:
3. Work – KE
Principle:
1 2
KE = mv
2
Wnet = WFnet = ΔKE = KEf - KEi
(Point-like object.)
L22 W 10/15/14 a+er lecture 4 4. Potential energy:
PE is the amount of work done on a system by an external
conservative force when KE does not change and no heat
flows (no friction, dissipation):
PEgrav = mgΔy
ΔPE = Wext when ΔKE = 0 1 2
PEspring = kx
2
The work done by some other
forces depends on the path taken:
e.g., friction. Under the action of
these forces, mechanical energy is
lost.
L22 W 10/15/14 a+er lecture 5 Claim: F"internal" = −
4. Potential energy:
dU
dx
Definition: Fexternal = force “external” to the system (e.g., my hand
lifting an eraser against gravity or
stretching a spring).
Definition: Finternal = force “internal” to the system (e.g., gravity or
the force of the spring).
Notation: U ≡ PE
dU grav
d
Fgrav = −
= − ( mgx ) = −mg ✔
dx
dx
For Example: dU elastic
d ⎛ 1 2⎞
Felastic = −
= − ⎜ kx ⎟ = −kx
⎠
dx
dx ⎝ 2
L22 W 10/15/14 a+er lecture ✔
6 5. Conservation of “Mechanical Energy”:
Emechanical = KE + PE = constant (no friction, no heat dissipation)
or KEi + PEi = KEf + PEf L22 W 10/15/14 a+er lecture 7 A hockey puck slides without friction along a frozen lake toward an
ice ramp and plateau as shown. The speed of the puck is 4m/s and the
height of the plateau is 1m. Will the puck make it all the way up the ramp? +y
0
A)  Yes B)  No C)  Impossible to tell without knowing the mass of the puck. To make it up the ramp: KEi ≥ PE f
1 2
KEi = mv = (8m)J
2
PE f = mgh = mg = (9.8m)J
∴ KEi < PE f so the puck will not make it up the ramp.
L22 W 10/15/14 a+er lecture 8 Example. A block of mass m is
released from rest at height H on a
frictionless ramp. It strikes a spring
with spring constant k at the end of
the ramp. How far will the spring
compress, x?
Ethermal = 0
KEi + PEigrav = KE f + PE elastic
f
1 2
0 + mgH = 0 + kx
2
2mgH
x=
k
L22 W 10/15/14 a+er lecture 9 A mass m is at the end of light (massless) rod
of length R, the other end of which has a
frictionless pivot so the rod can swing in a
vertical plane. The rod is initially horizontal
and the mass is pushed down with an initial
speed vo . What minimum speed is required
for the mass to pivot 270o to the vertical
position? A) v0 = gR
B) v0 = 2gR
Ei = E f
C) v0 = 2 gR
1 2
mv0 = mgR
2
D) v0 = 6gR
L22 W 10/15/14 a+er lecture PE = 0
→
v0 = 2gR
10 5. Conservation of “Mechanical Energy”:
Emechanical = KE + PE = constant (no friction, heat, dissipation)
or KEi + PEi = KEf + PEf 6. General Statement of the Conservation of Energy:
Etotal = Emechanical + Ethermal = constant
Although we’ll use this statement, it is somewhat confusing because it’s important
to remember that although thermal energy (TE) is produced as a physical system
evolves, it cannot be converted to mechanical energy (ME). Therefore, ME can
covert to TE but TE cannot convert to ME directly.
L22 W 10/15/14 a+er lecture 11 5. Conservation of “Mechanical Energy”:
Emechanical = KE + PE = constant (no friction, heat, dissipation)
or KEi + PEi = KEf + PEf 6. General Statement of the Conservation of Energy:
Etotal = Emechanical + Ethermal = constant or KEi + PEi = KEf + PEf + Ethermal (Typically: Wfriction = -Ethermal)
L22 W 10/15/14 a+er lecture Initial thermal energy is
typically considered = 0,
so Ethermal is placed on
the RHS of the eqn and
considered to be the
thermal energy produced
during the evolution of
the system.
12 A mass slides down a rough ramp of height h. Its initial speed is
zero. Its final speed at the bottom of the ramp is v. +y
h
0
As the mass descended, Mechanical Energy and Total Energy:
A:
B:
C:
D:
Mechanical Energy
increased
decreased
increased
decreased
L22 W 10/15/14 a+er lecture Total Energy
increased
decreased
stayed the same
stayed the same
13 KE + PE + Etherm = constant
A block of mass m slides down a rough ramp of height h. Its initial
speed is zero. Its final speed at the bottom of the ramp is v. m
h
Which of the following expressions would give the amount of
thermal energy, Ethermal, released from the block’s motion down
the ramp?
KE + PE = KE + PE + E
i
A) mgh +
1 2
mv
2
1 2
mv − mgh
2
E) mgh
C)
B) mgh −
D)
1 2
mv
2
L22 W 10/15/14 a+er lecture 1 2
mv
2
i
f
f
thermal
1 2 dU
0 + mgh =
mv =+−0 + Ethermal
F"internal"
2
dx
1 2
Ethermal = mgh − mv > 0
2
14 KE + PE + Etherm = constant
A block of mass m slides down a rough ramp of height h. Its initial
speed is zero. Its final speed at the bottom of the ramp is v. m
Ethermal
1 2
h
= mgh − mv > 0
2
What is the Work done by friction, Wfriction?
1 2
A) mgh − mv
2
C) mgh
L22 W 10/15/14 a+er lecture 1 2
B) mv − mgh
2
1 2
D) mv
2
W friction = −Ethermal
1 2
= mv − mgh < 0
2
15 A small track, starting at rest, slides without friction along a
frictionless rail in the shape of a loop. The maximum height of
the track is the same as the initial height of the ball. ASSUME
NO FRICTION.
Will the ball make it to the top
of the loop?
A) Yes, the ball will make it to
the top of the loop. B) No, the ball will not make it
to the top.
C) Not enough info to say.
L22 W 10/15/14 a+er lecture 16