Physics 2211: Lecture 34 Todays Agenda

1
Physics 2211 Lecture 34Todays Agenda

• Some final comments on rotational motion
• Newtons Law of Gravitation
• In general near Earths surface
• Potential energy
• Escape velocity

2
Some Final Comments on , , and .

• Fundamental definition of angular momentum
  
• is from the origin of coordinate system
  to particle.
• is the angular momentum about the
  origin.
• A component of , say the k-component
  , is the angular momentum about the k-axis.

• Fundamental definition of torque
  
• is from the origin of coordinate system
  to particle.
• is the torque about the origin due to
  .
• A component of , say the k-component
  , is the torque about the k-axis.
• System of particles

3
Some Final Comments on , , and .

• Relationship between torque and angular momentum
  
• is the external torque calculated
  about the origin.

• Conservation of angular momentum Since
  ,
• if the external torques about a given axis
  (k axis) are zero
• ( ), then the total angular
  momentum of the system
• about that axis (k axis) is constant in time.

4
Some Final Comments on , , and .

• The total angular momentum of a system is the sum
  of
• the angular momentum of the center of mass about
  the origin and
• the angular momentum of the system about the
  position of the center of mass.

• For the k component of
• The k-comp. of angular momentum of a system is
  the sum of
• the angular momentum of the center of mass about
  the k-axis and
• the angular momentum of the system about an axis
  parallel to the k-axis through the center of
  mass.

5
Moment of Inertia

• For a discrete collection of point masses
• ri is measured from the axis.

• For a continuous solid object
• r is measure from the axis.

dm
r
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Angular Momentum and Moment of Inertia

• In general,

• Example

but
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Angular Momentum and Moment of Inertiafor Rigid
Bodies

• If a rigid body has an axis of symmetry
  (principal axis of inertia) about which the body
  is rotating, then

• Example

rotating cone
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Angular Momentum and Moment of Inertiafor Rigid
Bodies

• In general,

where
is the inertia tensor

• A set of orthogonal axes (principal axes) can be
  found for any rigid body such that
  . Thus,

Where are the diagonal elements
(principal moments of inertia) of .

9
Newton the Moon

• What is the acceleration of the Moon due to its
  motion around the Earth?
• Newton knew
• T 27.3 days 2.36 x 106 s (period 1 month)
• R 3.84 x 108 m (distance to moon)
• RE 6.35 x 106 m (radius of earth)

R
RE
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Newton the Moon

• Calculate angular velocity
• So ? 2.66 x 10-6 s-1.
• Now calculate the acceleration.
• a ?2R 0.00272 m/s2 0.000278 g
• direction of a points at the center of the Earth
  (-r ).

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Newtons Law of Gravitation

• Newton found that amoon / g 0.000278
• He also noticed that RE2 / R2 0.000273
• This inspired him to propose the Universal Law
  of Gravitation

R
RE
where G 6.67 x 10 -11 m3 kg-1 s-2
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Newtons Law of Gravitation
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Newtons Law of Gravitation
m
z
M
y
x
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Newtons Law of Gravitation

• The magnitude of the gravitational force
  exerted on an object having mass m1 by another
  object having mass m2 a distance r12 away is
• The direction of is attractive, and lies
  along the line connecting the centers of the
  masses.

m1
m2
r12
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Gravity near the Earths surface

• Near the Earths surface
• r12 RE
• Wont change much if we stay near the Earth's
  surface.
• i.e. since RE gtgt h, RE h RE.

m
h
M
RE
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Gravity near the Earths surface

• Near the Earths surface...

???
g

• So mg ma
• a g

All objects accelerate with acceleration g,
regardless of their mass!
Where
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Gravitational Potential Energy
r
M
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Gravitational Potential Energy

• Integrate dWg to find the total work done by
  gravity in a bigdisplacement

r2
r1
M
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Gravitational Potential Energy
m
r2
r1
M
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Gravitational Potential Energy Near the Earths
Surface

• Let R1 RE and R2 RE y

• So DU mgy

21
Maximum height

• A projectile of mass m is launched straight up
  from the surface of the earth with initial speed
  v0. Neglecting air resistance, what is the
  maximum distance from the center of the earth
  RMAX it reaches before falling back down.

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Maximum height

• System mass-earth
• ?E ?K ?U 0

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Maximum height
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Escape Velocity

• If we want the projectile to escape to infinity
  we need to make the denominator in the above
  equation zero

We call this value of v0 the escape velocity, vesc
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How far is infinity?

• What is the speed needed to send a spaceship to
  the moon?

• So,as far as vesc is concerned (to 1 accuracy),
• the moon is at infinity!

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Recap of todays lecture

• Some final comments on rotational motion
• Newtons Law of Gravitation
• In general near Earths surface
• Potential energy
• Escape velocity
• Read Chapter 11 in Tipler