Welcome to Physics I !!!

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Physics I95.141LECTURE 1711/8/10
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Outline/Administrative Notes

• Notes
• HW Review Session on 11/17 shifted to 11/18.
• Last day to withdraw with a W is 11/12 (Friday)

• Outline
• Ballistic Pendulums
• 2D, 3D Collisions
• Center of Mass and translational motion

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Ballistic Pendulum

• A device used to measure the speed of a
  projectile.

m
h
M
Mm
vo
v1
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Ballistic Pendulum
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Mm
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Ballistic Pendulum
Mm
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Exam Prep Problem

• You construct a ballistic pendulum out of a
  rubber block (M5kg) attached to a horizontal
  spring (k300N/m). You wish to determine the
  muzzle velocity of a gun shooting a mass (m30g).
  After the bullet is shot into the block, the
  spring is observed to have a maximum compression
  of 12cm. Assume the spring slides on a
  frictionless surface.
• A) (10pts) What is the velocity of the block
  bullet immediately after the bullet is embedded
  in the block?
• B) (10pts) What is the velocity of the bullet
  right before it collides with the block?
• C) (5pts) If you shoot a 15g mass with the same
  gun (same velocity), how far do you expect the
  spring to compress?

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Exam Prep Problem

• k300N/m, m30g, M5kg, ?xmax12cm
• A) (10pts) What is the velocity of the block
  bullet immediately after the bullet is embedded
  in the block?

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Exam Prep Problem

• k300N/m, m30g, M5kg, ?xmax12cm
• B) (10pts) What is the velocity of the bullet
  right before it collides with the block?

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Exam Prep Problem

• k300N/m, m30g, M5kg, ?xmax12cm
• C) (5pts) If you shoot a 15g mass with the same
  gun (same velocity), how far do you expect the
  spring to compress?

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Collisions

• In the previous lecture we discussed collisions
  in 1D, and the role of Energy in collisions.
• Momentum always conserved!
• If Kinetic Energy is conserved in a collision,
  then we call this an elastic collision, and we
  can write
• Which simplifies to
• If Kinetic Energy is not conserved, the collision
  is referred to as an inelastic collision.
• If the two objects travel together after a
  collision, this is known as a perfectly inelastic
  collision.

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Collision Review

• Imagine I shoot a 10g projectile at 450m/s
  towards a 10kg target at rest.
• If the target is stainless steel, and the
  collision is elastic, what are the final speeds
  of the projectile and target?

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Collision Review

• Imagine I shoot a 10g projectile at 450m/s
  towards a 10kg target at rest.
• If the target is wood, and projectile embeds
  itself in the target, what are the final speeds
  of the projectile and target?

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Additional Dimensions

• Up until this point, we have only considered
  collisions in one dimension.
• In the real world, objects tend to exist (and
  move) in more than one dimension!
• Conservation of momentum holds for collisions in
  1, 2 and 3 dimensions!

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2D Momentum Conservation

• Imagine a projectile (mA) incident, along the
  x-axis, upon a target (mB) at rest. After the
  collision, the two objects go off at different
  angles
• Momentum is a vector, in order for momentum to be
  conserved, all components (x,y,z) must be
  conserved.

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2D Momentum Conservation

• Imagine a projectile (mA) incident, along the
  x-axis, upon a target (mB) at rest. After the
  collision, the two objects go off at different
  angles

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Conservation of Momentum (2D)

• Solving for conservation of momentum gives us 2
  equations (one for x-momentum, one for
  y-momentum).
• We can solve these if we have two unknowns
• If the collision is elastic, then we can add a
  third equation (conservation of kinetic energy),
  and solve for 3 unknowns.

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Example problem

• A cue ball travelling at 4m/s strikes a billiard
  ball at rest (of equal mass). After the
  collision the cue ball travels forward at an
  angle of 45º, and the billiard ball forward at
  -45º. What are the final speeds of the two balls?

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Example Problem II

• Now imagine a collision between two masses
  (mA1kg and mB2kg) travelling at vA2m/s and vB
  -2m/s along the x-axis. If mA bounces back at an
  angle of -30º, what are the final velocities of
  each ball?

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Example Problem II

• Now imagine a collision between two masses
  (mA1kg and mB2kg) travelling at vA2m/s and vB
  -2m/s on the x-axis. If mA bounces back at an
  angle of -30º, what are the final velocities of
  each ball, assuming the collision is elastic?

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Simplification of Elastic Collisions

• In 1D, we showed that the conservation of Kinetic
  Energy can be written as
• This does not hold for more than one dimension!!

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Problem Solving Collisions

1. Choose your system. If complicated (ballistic
   pendulum, for example), divide into parts
2. Consider external forces. Choose a time interval
   where they are minimal!
3. Draw a diagram of pre- and post- collision
   situations
4. Choose a coordinate system
5. Apply momentum conservation (divide into
   component form).
6. Consider energy. If elastic, write conservation
   of energy equations.
7. Solve for unknowns.
8. Check solutions.

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Center of Mass

• Conservation of momentum is powerful for
  collisions and analyzing translational motion of
  an object.
• Up until this point in the course, we have chosen
  objects which can be approximated as a point
  particle of a certain mass undergoing
  translational motion.
• But we know that real objects dont just move
  translationally, they can rotate or vibrate
  (general motion) ? not all points on the object
  follow the same path.
• Point masses dont rotate or vibrate!

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Center of Mass

• We need to find an addition way to describe
  motion of non-point mass objects.
• It turns out that on every object, there is one
  point which moves in the same path a particle
  would move if subjected to the same net Force.
• This point is known as the center of mass (CM).
• The net motion of an object can then be described
  by the translational motion of the CM, plus the
  rotational, vibrational, and other types of
  motion around the CM.

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Example
F
F
F
F
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Center of Mass

• If you apply a force to an non-point object, its
  center of mass will move as if the Force was
  applied to a point mass at the center of mass!!
• This doesnt tell us about the vibrational or
  rotation motion of the rest of the object.

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Center of Mass (2 particles, 1D)

• How do we find the center of mass?
• First consider a system made up of two point
  masses, both on the x-axis.

xB
mA
xA
x0
xB
x-axis
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Center of Mass (n particles, 1D)

• If, instead of two, we have n particles on the
  x-axis, then we can apply a similar formula to
  find the xCM.

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Center of Mass (2D, 2 particles)

• For two particles lying in the x-y plane, we can
  find the center of mass (now a point in the xy
  plane) by individually solving for the xCM and
  yCM.

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Center of Mass (3D, n particles)

• We can extend the previous CM calculations to
  n-particles lying anywhere in 3 dimensions.

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Example

• Suppose we have 3 point masses (mA1kg, mB3kg
  and mC2kg), at three different points
  A(0,0,0), B(2,4,-6) and C(3,-3,6).

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Solid Objects

• We can easily find the CM for a collection of
  point masses, but most everyday items arent made
  up of 2 or 3 point masses. What about solid
  objects?
• Imagine a solid object made out of an infinite
  number of point masses. The easiest trick we can
  use is that of symmetry!

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CM and Translational Motion

• The translational motion of the CM of an object
  is directly related to the net Force acting on
  the object.
• The sum of all the Forces acting on the system is
  equal to the total mass of the system times the
  acceleration of its center of mass.
• The center of mass of a system of particles (or
  objects) with total mass M moves like a single
  particle of mass M acted upon by the same net
  external force.

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Example

• A 60kg person stands on the right most edge of a
  uniform board of mass 30kg and length 6m, lying
  on a frictionless surface. She then walks to the
  other end of the board. How far does the board
  move?

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Solid Objects (General)

• If symmetry doesnt work, we can solve for CM
  mathematically.
• Divide mass into smaller sections dm.

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Solid Objects (General)

• If symmetry doesnt work, we can solve for CM
  mathematically.
• Divide mass into smaller sections dm.

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Example Rod of varying density

• Imagine we have a circular rod (r0.1m) with a
  mass density given by ?2x kg/m3.

x
L2m
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Example Rod of varying density

• Imagine we have a circular rod (r0.1m) with a
  mass density given by ?2x kg/m3.

x
L2m