Physics 2211: Lecture 20 Today's Agenda

1
Physics 2211 Lecture 20Today's Agenda

• Problems using work / energy theorem
• Escape velocity
• Space spring
• Loop the loop
• Vertical springs

2
Lecture 20, Act 1Escape Velocity

• Two identical spaceships are awaiting launch on
  two planets with the same mass. Planet 1 is
  stationary, while Planet 2 is rotating with an
  angular velocity ?.
• Which spaceship needs more fuel to escape to
  infinity?

(a) 1 (b) 2 (c) same

• ?

(1)
(2)
3
Problem Space Spring

• A low budget space program decides to launch a
  10,000 kg spaceship into space using a big
  spring. If the spaceship is to reach a height RE
  above the surface of the Earth, what distance d
  must the launching spring be compressed if it has
  a spring constant of 108 N/m. (Ignore air
  resistance.)

4
Problem Space Spring...

• WEXT WNC 0, therefore mechanical energy is
  conserved. Since K 0 both initially and at the
  maximum height (v 0) we know
• Ubefore Uafter
• (US UG )before (US UG )after

5
Problem Space Spring
So we find

• For the numbers given, d 79.1 m
• But dont get too happy...

F kd ma a kd/m a 79.1 x 106 m/s2 a
791000 g unhappy astronaut!
a
6
Problem Loop the loop

• A mass m starts at rest on a frictionless track a
  distance H above the floor. It slides down to
  the level of the floor where it encounters a loop
  of radius R. What is H if the mass just barely
  makes it around the loop without losing contact
  with the track.

H
R
7
Problem Loop the loop

• Draw a FBD of the mass at the top of the loop

• FTOT -mg - N
• ma -mv2/R
• mg N mv2/R
• If it just makes it, N 0.
• mg mv2/R

8
Problem Loop the loop

• Now notice that KU energy is conserved gt ?K
  -?U.

• ?U -mgy -mg(H-2R)
• ?K 1/2 mv2 1/2 mRg

9
Lecture 20, Act 2Energy Conservation

• A mass starts at rest on a frictionless track a
  distance H above the floor. It slides down to
  the level of the floor where it encounters a loop
  of radius R. What is H if the normal force on
  the block by the track at the top of the loop is
  equal to the weight of the block ?

(a) 2R (b) 2.5R (c)
3R
H
R
10
Vertical Springs
(a)
(b)

• A spring is hung vertically. Its relaxed
  position is at y 0 (a). When a mass m is hung
  from its end, the new equilibrium position is ye
  (b).

y
k

• Recall that the force of a spring is Fs -kx.
  In case (b) FTOT 0 and x ye-kye - mg
  0 (ye lt 0)

y 0
y ye
11
Vertical Springs
(a)
(b)

• The potential energy of the spring-mass-earth
  system is

y
k
y 0
y ye
choose C to make U 0 at y ye
m
12
Vertical Springs
(a)
(b)

• So

y
k
y 0
which can be written
y ye
m
13
Vertical Springs
(a)
(b)
y
k

• So if we define a new y? coordinate system such
  that y? 0 is at the equilibrium position, ( y?
  y - ye ) then we get the simple result

y? 0
m
?
14
Vertical Springs
(a)
(b)

• If we choose y 0 to be at the equilibrium
  position of the mass hanging on the spring, we
  can define the potential in the simple form.
• Notice that g does not appear in this
  expression!!
• By choosing our coordinates and constants
  cleverly, we can hide the effects of gravity.

y
k
y 0
m
15
160
140
U of Spring
120
U
100
US 1/2ky2
80
60
40
20
0
-10
-8
-6
-4
-2
0
2
4
6
8
10
-20
y
-40
-60
16
160
140
U of Gravity
120
U
100
80
60
UG mgy
40
20
0
-10
-8
-6
-4
-2
0
2
4
6
8
10
-20
y
-40
-60
17
160
UNET UG US
140
U of Spring Gravity
120
U
100
US 1/2ky2
80
60
UG mgy
40
20
0
-10
-8
-6
-4
-2
0
2
4
6
8
10
-20
y
-40
-60
ye
0
shift due to mgy term
18
160
Choose C such as to show that the new equilibrium
position has zero potential energy
140
U of Spring Gravity
120
U
100
80
60
40
UNET UG US C
20
0
-10
-8
-6
-4
-2
0
2
4
6
8
10
-20
US 1/2ky2
y
-40
-60
ye
0
shift due to mgy term
19
Vertical SpringsExample Problem

• If we displace the mass a distance d from
  equilibrium and let it go, it will oscillate up
  down. Relate the maximum speed of the mass v to
  d and the spring constant k.
• Since all forces are conservative,E K U is
  constant.

y
k
y d
y 0
v
y -d
20
Vertical SpringsExample Problem
y
k
y d

• Energy is shared between the K and U terms.
• At y d or -d the energy is all potential
• At y 0, the energy is all kinetic.

y 0
v
y -d
21
Recap of todays lecture

• Problems using work / energy theorem
• Escape velocity
• Space spring
• Loop the loop
• Vertical springs
• Begin reading Chapter 8 in Tipler