College Physics

1
General Physics (PHY 2130)
Lecture VI

• Momentum (cont.)
• inelastic collisions
• 2-dimensional collisions
• Rotational kinematics
• Angular displacement
• Angular speed and acceleration
• Newtons law of gravity
• Keplers laws

http//www.physics.wayne.edu/apetrov/PHY2130/
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Lightning Review

• Last lecture
• Momentum and impulse
• elastic and inelastic collisions
• Energy
• energy is conserved
• energy stored in a spring
• Review Problem A compact car and a large truck
  collide head on and stick together. Which
  undergoes the larger momentum change?
• 1. Car
• 2. Truck
• 3. The momentum change is the same for both
  vehicles
• 4. Cant tell without knowing the final
  velocity of combined mass

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Types of Collisions

• Momentum is conserved in any collision
• what about kinetic energy?
• Inelastic collisions
• Kinetic energy is not conserved
• Some of the kinetic energy is converted into
  other types of energy such as heat, sound, work
  to permanently deform an object
• Perfectly inelastic collisions occur when the
  objects stick together
• Not all of the KE is necessarily lost

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Perfectly Inelastic Collisions

• When two objects stick together after the
  collision, they have undergone a perfectly
  inelastic collision
• Suppose, for example, v2i0. Conservation of
  momentum becomes

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Perfectly Inelastic Collisions

• What amount of KE lost during collision?

lost in heat/gluing/sound/
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More Types of Collisions

• Elastic collisions
• both momentum and kinetic energy are conserved
• Actual collisions
• Most collisions fall between elastic and
  perfectly inelastic collisions

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More About Elastic Collisions

• Both momentum and kinetic energy are conserved
• Typically have two unknowns
• Solve the equations simultaneously

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Elastic Collisions

• Using previous example (but elastic collision is
  assumed)

For perfectly elastic collision
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Problem Solving Tips

• If the collision is inelastic, KE is not
  conserved
• If the collision is elastic, KE is conserved

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Rocket Propulsion

• The basic equation for rocket propulsion is
• Mi is the initial mass of the rocket plus fuel
• Mf is the final mass of the rocket plus any
  remaining fuel
• The speed of the rocket is proportional to the
  exhaust speed

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Thrust of a Rocket

• The thrust is the force exerted on the rocket by
  the ejected exhaust gases
• The instantaneous thrust is given by
• The thrust increases as the exhaust speed
  increases and as the burn rate (?M/?t) increases

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Two-dimensional Collisions

• For a general collision of two objects in
  three-dimensional space, the conservation of
  momentum principle
• implies that the total momentum of the system
  in each direction is conserved
• Use subscripts for identifying the object,
  initial and final, and components

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Example

• What would happen after the collision?

Stationary
It is also possible for two bodies to undergo
scattering
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Example
Assume m1m2 and v1i5 m/s

• What would happen after the collision?

Stationary
It is also possible for two bodies to undergo
scattering
For this problem assume that q f 60
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Example
Given masses m1m2 velocity v1i5 m/s
v2i0 m/s angles q f 60 Find
v1f ? v2f ?
Use momentum conservation in each direction (x
and y)

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ConcepTest 1
A boy stands at one end of a floating raft that
is stationary relative to the shore. He then
walks to the opposite end, towards the shore.
Does the raft move (assume no friction)? 1.
No, it will not move at all 2. Yes, it will move
away from the shore 3. Yes, it will move towards
the shore
Please fill your answer as question 1 of
General Purpose Answer Sheet
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ConcepTest 1
A boy stands at one end of a floating raft that
is stationary relative to the shore. He then
walks to the opposite end, towards the shore.
Does the raft move (assume no friction)? 1.
No, it will not move at all 2. Yes, it will move
away from the shore 3. Yes, it will move towards
the shore
Convince your neighbor!
Please fill your answer as question 2 of
General Purpose Answer Sheet
18
ConcepTest 1
A boy stands at one end of a floating raft that
is stationary relative to the shore. He then
walks to the opposite end, towards the shore.
Does the raft move (assume no friction)? 1.
No, it will not move at all 2. Yes, it will move
away from the shore 3. Yes, it will move towards
the shore

Note Since momentum is conserved in the
boy-raft system and neither was moving at first,
the raft must move in the direction opposite to
the boys.
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• Rotational Motion
• and
• The Law of Gravity

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Angular Displacement

• Recall for linear motion
• displacement, velocity, acceleration
• Need similar concepts for objects moving in
  circle (CD, merry-go-round, etc.)
• As before
• need a fixed reference system (line)
• use polar coordinate system

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Angular Displacement

• Every point on the object undergoes circular
  motion about the point O
• Angles generally need to be measured in radians
• Note

length of arc
radius
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Angular Displacement

• The angular displacement is defined as the angle
  the object rotates through during some time
  interval
• Every point on the disc undergoes the same
  angular displacement in any given time interval

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Angular Velocity

• The average angular velocity (speed), ?, of a
  rotating rigid object is the ratio of the angular
  displacement to the time interval

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Angular Speed

• The instantaneous angular velocity (speed) is
  defined as the limit of the average speed as the
  time interval approaches zero
• Units of angular speed are radians/sec (rad/s)
• Angular speed will be
• positive if ? is increasing (counterclockwise)
• negative if ? is decreasing (clockwise)

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Angular Acceleration

• What if object is initially at rest and then
  begins to rotate?
• The average angular acceleration, a, of an object
  is defined as the ratio of the change in the
  angular speed to the time it takes for the object
  to undergo the change
• Units are rad/s²
• Similarly, instant. angular accel.

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Notes about angular kinematics

• When a rigid object rotates about a fixed
  axis, every portion of the object has the same
  angular speed and the same angular acceleration
• i.e. q,w, and a are not dependent upon r,
  distance form hub or axis of rotation

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Examples
1. Bicycle wheel turns 240 revolutions/min. What
is its angular velocity in radians/second?

2. If wheel slows down uniformly to rest in 5
seconds, what is the angular acceleration?

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Examples

• Given
• 1. Angular velocity
• 240 rev/min
• 2. Time t 5 s
• Find
• q ?

3. How many revolution does it turn in those 5
sec?
Recall that for linear motion we had Perhaps
something similar for angular quantities?

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Analogies Between Linear and Rotational Motion
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Relationship Between Angular and Linear Quantities

• Displacements
• Speeds
• Accelerations

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Relationship Between Angular and Linear Quantities

• Displacements
• Speeds
• Accelerations

• Every point on the rotating object has the same
  angular motion
• Every point on the rotating object does not have
  the same linear motion

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ConcepTest 2
A ladybug sits at the outer edge of a
merry-go-round, and a gentleman bug sits halfway
between her and the axis of rotation. The
merry-go-round makes a complete revolution once
each second.The gentleman bugs angular speed
is 1. half the ladybugs. 2. the same as the
ladybugs. 3. twice the ladybugs. 4. impossible
to determine
Please fill your answer as question 3 of
General Purpose Answer Sheet
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ConcepTest 2
A ladybug sits at the outer edge of a
merry-go-round, and a gentleman bug sits halfway
between her and the axis of rotation. The
merry-go-round makes a complete revolution once
each second.The gentleman bugs angular speed
is 1. half the ladybugs. 2. the same as the
ladybugs. 3. twice the ladybugs. 4. impossible
to determine
Convince your neighbor!
Please fill your answer as question 4 of
General Purpose Answer Sheet
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ConcepTest 2
A ladybug sits at the outer edge of a
merry-go-round, and a gentleman bug sits halfway
between her and the axis of rotation. The
merry-go-round makes a complete revolution once
each second.The gentleman bugs angular speed
is 1. half the ladybugs. 2. the same as the
ladybugs. 3. twice the ladybugs. 4. impossible
to determine
Note both insects have an angular speed of 1
rev/s
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Centripetal Acceleration

• An object traveling in a circle, even though it
  moves with a constant speed, will have an
  acceleration (since velocity changes direction)
• This acceleration is called centripetal
  (center-seeking).
• The acceleration is directed toward the center of
  the circle of motion

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Centripetal Acceleration and Angular Velocity

• The angular velocity and the linear velocity are
  related (v ?r)
• The centripetal acceleration can also be related
  to the angular velocity

Similar triangles!
Thus
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Total Acceleration

• What happens if linear velocity also changes?
• Two-component acceleration
• the centripetal component of the acceleration is
  due to changing direction
• the tangential component of the acceleration is
  due to changing speed
• Total acceleration can be found from these
  components

slowing-down car
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Vector Nature of Angular Quantities

• As in the linear case, displacement, velocity and
  acceleration are vectors
• Assign a positive or negative direction
• A more complete way is by using the right hand
  rule
• Grasp the axis of rotation with your right hand
• Wrap your fingers in the direction of rotation
• Your thumb points in the direction of ?

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Forces Causing Centripetal Acceleration

• Newtons Second Law says that the centripetal
  acceleration is accompanied by a force
• F stands for any force that keeps an object
  following a circular path
• Force of friction (level and banked curves)
• Tension in a string
• Gravity

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Example1 level curves

• Consider a car driving at 20 m/s (45 mph) on a
  level circular turn of radius 40.0 m. Assume the
  cars mass is 1000 kg.
• What is the magnitude of frictional force
  experienced by cars tires?
• What is the minimum coefficient of friction in
  order for the car to safely negotiate the turn?

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Example1

• Given
• masses m1000 kg
• velocity v20 m/s
• radius r 40.0m
• Find
• f?
• m?

1. Draw a free body diagram, introduce coordinate
frame and consider vertical and horizontal
projections

2. Use definition of friction force
Lesson m for rubber on dry concrete is 1.00!
rubber on wet concrete is 0.2!

driving too fast
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ConcepQuestion
Is this static or kinetic friction is the car
does not slide or skid? 1. Static 2. Kinetic
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Example2 banked curves

• Consider a car driving at 20 m/s (45 mph) on a
  30 banked circular curve of radius 40.0 m.
  Assume the cars mass is 1000 kg.
• What is the magnitude of frictional force
  experienced by cars tires?
• What is the minimum coefficient of friction in
  order for the car to safely negotiate the turn?

A component of the normal force adds to the
frictional force to allow higher speeds
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Example2

• Given
• masses m1000 kg
• velocity v20 m/s
• radius r 40.0m
• angle a 30
• Find
• f?
• m?

1. Draw a free body diagram, introduce coordinate
frame and consider vertical and horizontal
projections

2. Use definition of friction force
Lesson by increasing angle of banking, one
decreases minimal m or friction with which one
can take curve!

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Horizontal Circle

• The horizontal component of the tension causes
  the centripetal acceleration

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Forces in Accelerating Reference Frames

• Distinguish real forces from fictitious forces
• Centrifugal force is a fictitious force
• Real forces always represent interactions between
  objects

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Newtons Law of Universal Gravitation

• Every particle in the Universe attracts every
  other particle with a force that is directly
  proportional to the product of the masses and
  inversely proportional to the square of the
  distance between them.

• G is the universal gravitational constant
• G 6.673 x 10-11 N m² /kg²
• This is an example of an inverse square law

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Gravitation Constant

• Determined experimentally
• Henry Cavendish
• 1798
• The light beam and mirror serve to amplify the
  motion

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Example
Question Calculate gravitational attraction
between two students 1 meter apart

Extremely small
Compare
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Applications of Universal Gravitation 1 Mass of
the Earth

• Use an example of an object close to the surface
  of the earth
• r RE

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Applications of Universal Gravitation 2
Acceleration Due to Gravity

• g will vary with altitude

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Gravitational Potential Energy

• PE mgy is valid only near the earths surface
• For objects high above the earths surface, an
  alternate expression is needed
• Zero reference level is infinitely far from the
  earth

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Escape Speed

• The escape speed is the speed needed for an
  object to soar off into space and not return
• For the earth, vesc is about 11.2 km/s
• Note, v is independent of the mass of the object

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Keplers Laws

• All planets move in elliptical orbits with the
  Sun at one of the focal points.
• A line drawn from the Sun to any planet sweeps
  out equal areas in equal time intervals.
• The square of the orbital period of any planet is
  proportional to cube of the average distance from
  the Sun to the planet.

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Keplers Laws, cont.

• Based on observations made by Brahe
• Newton later demonstrated that these laws were
  consequences of the gravitational force between
  any two objects together with Newtons laws of
  motion

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Keplers First Law

• All planets move in elliptical orbits with the
  Sun at one focus.
• Any object bound to another by an inverse square
  law will move in an elliptical path
• Second focus is empty

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Keplers Second Law

• A line drawn from the Sun to any planet will
  sweep out equal areas in equal times
• Area from A to B and C to D are the same

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Keplers Third Law

• The square of the orbital period of any planet is
  proportional to cube of the average distance from
  the Sun to the planet.
• For orbit around the Sun, KS 2.97x10-19 s2/m3
• K is independent of the mass of the planet

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Keplers Third Law application

• Mass of the Sun or other celestial body that has
  something orbiting it
• Assuming a circular orbit is a good approximation