Physics 111: Lecture 10 Todays Agenda

1
Physics 111 Lecture 10Todays Agenda

• Review of Work
• Work done by gravity near the Earths surface
• Examples
• pendulum, inclined plane, free fall
• Work done by variable force
• Spring
• Problem involving spring friction

2
Review Constant Force
Work, W, of a constant force F acting through
a displacement ?r is W F? ?r F ?r cos(?)
Fr ?r
F
?r
?
Fr
displacement
3
Review Sum of Constant Forces
Suppose FNET F1 F2 and the displacement is
S. The work done by each force is W1 F1? ?r
W2 F2? ?r
FTOT
F1
?r
WNET W1 W2 F1? ?r F2? ?r
(F1 F2 )? ?r WNET FNET ? ?r
F2
4
Review Constant Force...

• W F? ?r
• No work done if ? 90o.
• No work done by T.
• No work done by N.

T
v
v
N
5
Work/Kinetic Energy Theorem

• Net Work done on object
• change in kinetic energy of object
• WF ?K 1/2mv22 - 1/2mv12

WF F?x
m
6
Work done by gravity

• Wg F? ?r mg ?r cos ?
• -mg ?y
• Wg -mg ?y
• Depends only on ?y !

m
mg
?r
?
-?y
m
7
Work done by gravity...
W NET W1 W2 . . . Wn

• Depends only on ?y,
• not on path taken!

m
mg
?y
j
8
Lecture 10, Act 1Falling Objects
Falling objects

• Three objects of mass m begin at height h with
  velocity 0. One falls straight down, one slides
  down a frictionless inclined plane, and one
  swings on the end of a pendulum. What is the
  relationship between their velocities when they
  have fallen to height 0?

(a) Vf gt Vi gt Vp (b) Vf gt Vp gt Vi
(c) Vf Vp Vi
9
Lecture 10, Act 1Solution
Only gravity will do work Wg mgH 1/2 mv22 -
1/2 mv12 1/2 mv22
does not depend on path !!
10
Lifting a book with your handWhat is the total
work done on the book??

• First calculate the work done by gravity
• Wg mg? ?r -mg ?r
• Now find the work done bythe hand
• WHAND FHAND? ?r FHAND ?r

FHAND
?r
v const a 0
mg
11
Example Lifting a book...
Textbook

• Wg -mg ?r
• WHAND FHAND ?r
• WNET WHAND Wg
• FHAND ?r - mg ?r
• (FHAND - mg) ?r
• 0 since ?K 0 (v const)
• So WTOT 0!!

FHAND
?r
v const a 0
mg
12
Example Lifting a book...

• Work/Kinetic Energy Theorem says W ?K
• Net Work done on object change in kinetic
  energy of object
• In this case, v is constant so ?K 0and so W
  must be 0, as we found.

FHAND
?r
v const a 0
mg
13
Work done by Variable Force (1D)

• When the force was constant, we wrote W F ?x
• area under F vs. x plot
• For variable force, we find the areaby
  integrating
• dW F(x) dx.

F
Wg
x
?x
14
Work/Kinetic Energy Theorem for a Variable Force
dv
F
dx
F
dx
dv
dv
dv
v
(chain rule)
dx


dx
dt
dx
dt
dv
dx
v
dx
v dv
v22
v12
v22
v12
15
1-D Variable Force Example Spring

• For a spring we know that Fx -kx.

F(x)
x2
x1
x
relaxed position
-kx
F - k x1
F - k x2
16
Spring...

• The work done by the spring Ws during a
  displacement from x1 to x2 is the area under the
  F(x) vs x plot between x1 and x2.

F(x)
x2
x1
x
Ws
relaxed position
-kx
17
Spring...
Spring

• The work done by the spring Ws during a
  displacement from x1 to x2 is the area under the
  F(x) vs x plot between x1 and x2.

F(x)
x2
x1
x
Ws
-kx
18
Lecture 10, Act 2Work Energy

• A box sliding on a horizontal frictionless
  surface runs into a fixed spring, compressing it
  a distance x1 from its relaxed position while
  momentarily coming to rest.
• If the initial speed of the box were doubled and
  its mass were halved, how far x2 would the spring
  compress ?

(a) (b)
(c)
x
19
Lecture 10, Act 2Solution

• Again, use the fact that WNET DK.

In this case, WNET WSPRING -1/2 kx2 and
?K -1/2 mv2
In the case of x1
so kx2 mv2
x1
v1
m1
m1
20
Lecture 10, Act 2Solution
So if v2 2v1 and m2 m1/2
x2
v2
m2
m2
21
Problem Spring pulls on mass.

• A spring (constant k) is stretched a distance d,
  and a mass m is hooked to its end. The mass is
  released (from rest). What is the speed of the
  mass when it returns to the relaxed position if
  it slides without friction?

m
relaxed position
stretched position (at rest)
m
d
after release
m
v
back at relaxed position
m
vr
22
Problem Spring pulls on mass.

• First find the net work done on the mass during
  the motion from x d to x 0 (only due to the
  spring)

stretched position (at rest)
m
d
relaxed position
m
i
vr
23
Problem Spring pulls on mass.

• Now find the change in kinetic energy of the mass

stretched position (at rest)
m
d
relaxed position
m
i
vr
24
Problem Spring pulls on mass.

• Now use work kinetic-energy theorem Wnet WS
  ?K.

stretched position (at rest)
m
d
relaxed position
m
i
vr
25
Problem Spring pulls on mass.

• Now suppose there is a coefficient of friction ?
  between the block and the floor
• The total work done on the block is now the sum
  of the work done by the spring WS (same as
  before) and the work done by friction Wf. Wf
  f.?r - ?mg d

?r
stretched position (at rest)
m
d
relaxed position
f ?mg
m
i
vr
26
Problem Spring pulls on mass.

• Again use Wnet WS Wf ?K Wf -?mg d

?r
stretched position (at rest)
m
d
relaxed position
f ?mg
m
i
vr
27
Recap of todays lecture

• Review (Text 6-1 6-2)
• Work done by gravity near the Earths surface
  (Text 11-3)
• Examples
• pendulum, inclined plane, free fall
• Work done by variable force (Text
  6-1)
• Spring
• Problem involving spring friction
• Look at textbook problems Chapter 6 7, 9, 11,
  13, 15, 17, 77