Physics 111: Lecture 6 Todays Agenda

1
Physics 111 Lecture 6Todays Agenda

• Recap
• Problems...problems...problems!!
• Accelerometer
• Inclined plane
• Motion in a circle

2
Review

• Discussion of dynamics.
• Review Newtons 3 Laws
• The Free Body Diagram
• The tools we have for making solving problems
• Ropes Pulleys (tension)
• Hookes Law (springs)

3
Review Pegs Pulleys

• Used to change the direction of forces
• An ideal massless pulley or ideal smooth peg will
  change the direction of an applied force without
  altering the magnitude The tension is the same
  on both sides!

massless rope
F1 -T i
F1 F2
ideal peg or pulley
F2 T j
4
Review Springs

• Hookes Law The force exerted by a spring is
  proportional to the distance the spring is
  stretched or compressed from its relaxed
  position.
• FX -kx Where x is the displacement
  from the equilibrium and k is the constant of
  proportionality.

relaxed position
FX 0
x
5
Lecture 6, Act 1Springs

• A spring with spring constant 40 N/m has a
  relaxed length of 1 m. When the spring is
  stretched so that it is 1.5 m long, what force is
  exerted on a block attached to the end of the
  spring?

x 0
x 0
x 1
x 1.5
k
k
(a) -20 N (b) 60 N
(c) -60 N
6
Lecture 6, Act 1Solution

• Recall Hookes law
• FX -kx Where x is the displacement from
  equilibrium.

FX - (40) ( .5) FX - 20 N
(a) -20 N (b) 60 N
(c) -60 N
7
Problem Accelerometer

• A weight of mass m is hung from the ceiling of a
  car with a massless string. The car travels on a
  horizontal road, and has an acceleration a in the
  x direction. The string makes an angle ? with
  respect to the vertical (y) axis. Solve for ? in
  terms of a and g.

a
?
i
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Accelerometer...

• Draw a free body diagram for the mass
• What are all of the forces acting?

i
9
Accelerometer...

• Using components (recommended)
• i FX TX T sin ? ma
• j FY TY - mg
• T cos ??- mg 0

TX
?
TY
T
?
m
ma
mg
10
Accelerometer...

• Using components
• i T sin ? ma
• j T cos ??- mg 0
• Eliminate T

TX
TY
T
?
m
ma
T sin ??? ma
T cos ??? mg
mg
11
Accelerometer...

• Alternative solution using vectors (elegant but
  not as systematic)
• Find the total vector force FNET

T (string tension)
T
?
mg
?
m
FTOT
mg (gravitational force)
12
Accelerometer...

• Alternative solution using vectors (elegant but
  not as systematic)
• Find the total vector force FNET
• Recall that FNET ma
• So

T (string tension)
?
T
mg
?
m
ma
mg (gravitational force)
13
Accelerometer...
Cart w/ accelerometer

• Lets put in some numbers
• Say the car goes from 0 to 60 mph in 10 seconds
• 60 mph 60 x 0.45 m/s 27 m/s.
• Acceleration a ?v/?t 2.7 m/s2.
• So a/g 2.7 / 9.8 0.28 .
• ? arctan (a/g) 15.6 deg

a
?
14
Problem Inclined plane

• A block of mass m slides down a frictionless ramp
  that makes angle ? with respect to the
  horizontal. What is its acceleration a ?

m
a
?
15
Inclined plane...

• Define convenient axes parallel and perpendicular
  to plane
• Acceleration a is in x direction only.

m
a
?
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Inclined plane...
Incline

• Consider x and y components separately
• i mg sin ? ma. a g sin ?
• j N - mg cos ? 0. N mg cos ?

ma
mg sin ?
N
?
mg cos ?
mg
17
Inclined plane...

• Alternative solution using vectors

m
N
?
mg
a g sin ???i N mg cos ???j
18
Angles of an Inclined plane
The triangles are similar, so the angles are the
same!
N
?
mg
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Lecture 6, Act 2Forces and Motion

• A block of mass M 5.1 kg is supported on a
  frictionless ramp by a spring having constant k
  125 N/m. When the ramp is horizontal the
  equilibrium position of the mass is at x 0.
  When the angle of the ramp is changed to 30o what
  is the new equilibrium position of the block
  x1?(a) x1 20cm (b) x1 25cm
  (c) x1 30cm

x1 ?
k
M
q 30o
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Lecture 6, Act 2Solution
x1
k
M
q
21
Lecture 6, Act 2Solution

• Since the total force in the x-direction must be
  0

x1
Fx,s -kx1
k
M
Fx,g Mg sinq
q
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Problem Two Blocks

• Two blocks of masses m1 and m2 are placed in
  contact on a horizontal frictionless surface. If
  a force of magnitude F is applied to the box of
  mass m1, what is the force on the block of mass
  m2?

F
m1
m2
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Problem Two Blocks

• Realize that F (m1 m2) a
• Draw FBD of block m2 and apply FNET ma

F / (m1 m2) a
F2,1
F2,1 m2 a
m2
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Problem Tension and Angles

• A box is suspended from the ceiling by two ropes
  making an angle ? with the horizontal. What is
  the tension in each rope?

?
?
m
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Problem Tension and Angles
T1

• Draw a FBD

T2
T2sin ?
T1sin ?
?
?
T1cos ?
T2cos ?
mg

• Since the box isnt going anywhere, Fx,NET 0
  and Fy,NET 0

Fy,NET T1sin ? T2sin ? - mg 0
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Problem Motion in a Circle
Tetherball

• A boy ties a rock of mass m to the end of a
  string and twirls it in the vertical plane. The
  distance from his hand to the rock is R. The
  speed of the rock at the top of its trajectory is
  v.
• What is the tension T in the string at the top of
  the rocks trajectory?

v
T
R
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Motion in a Circle...

• Draw a Free Body Diagram (pick y-direction to be
  down)
• We will use FNET ma (surprise)
• First find FNET in y direction
• FNET mg T

y
mg
T
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Motion in a Circle...

• FNET mg T
• Acceleration in y direction
• ma mv2 / R
• mg T mv2 / R
• T mv2 / R - mg

v
y
mg
T
F ma
R
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Motion in a Circle...
Bucket

• What is the minimum speed of the mass at the top
  of the trajectory such that the string does not
  go limp?
• i.e. find v such that T 0.
• mv2 / R mg T
• v2 / R g
• Notice that this doesnot depend on m.

v
mg
T 0
R
30
Lecture 6, Act 3Motion in a Circle
Track w/ bump

• A skier of mass m goes over a mogul having a
  radius of curvature R. How fast can she go
  without leaving the ground?

(a) (b)
(c)
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Lecture 6, Act 3Solution

• mv2 / R mg - N
• For N 0

v
N
mg
R
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Recap of Todays lecture

• Example Problems
• Accelerometer
• Inclined plane (Text example 6-1)
• Motion in a circle (Text 5-2, 9-1)
• Look at textbook problems Chapter 4 47
  Chapter 5 51, 95