Title: College Physics
1General 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/
2Lightning 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
3Types 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
4Perfectly 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
5Perfectly Inelastic Collisions
- What amount of KE lost during collision?
lost in heat/gluing/sound/
6More Types of Collisions
- Elastic collisions
- both momentum and kinetic energy are conserved
- Actual collisions
- Most collisions fall between elastic and
perfectly inelastic collisions
7More About Elastic Collisions
- Both momentum and kinetic energy are conserved
- Typically have two unknowns
- Solve the equations simultaneously
8Elastic Collisions
- Using previous example (but elastic collision is
assumed)
For perfectly elastic collision
9Problem Solving Tips
- If the collision is inelastic, KE is not
conserved - If the collision is elastic, KE is conserved
10Rocket 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
11Thrust 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
12Two-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
13Example
- What would happen after the collision?
Stationary
It is also possible for two bodies to undergo
scattering
14Example
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
15Example
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)
16ConcepTest 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
17ConcepTest 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
18ConcepTest 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.
19- Rotational Motion
- and
- The Law of Gravity
20Angular 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
21Angular 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
22Angular 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
23Angular Velocity
- The average angular velocity (speed), ?, of a
rotating rigid object is the ratio of the angular
displacement to the time interval
24Angular 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)
25Angular 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.
26Notes 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
27Examples
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?
28Examples
- 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?
29Analogies Between Linear and Rotational Motion
30Relationship Between Angular and Linear Quantities
- Displacements
- Speeds
- Accelerations
31Relationship 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
32ConcepTest 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
33ConcepTest 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
34ConcepTest 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
35Centripetal 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
36Centripetal 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
37Total 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
38Vector 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 ?
39Forces 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
40Example1 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?
41Example1
- 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
42ConcepQuestion
Is this static or kinetic friction is the car
does not slide or skid? 1. Static 2. Kinetic
43Example2 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
44Example2
- 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!
45Horizontal Circle
- The horizontal component of the tension causes
the centripetal acceleration
46Forces in Accelerating Reference Frames
- Distinguish real forces from fictitious forces
- Centrifugal force is a fictitious force
- Real forces always represent interactions between
objects
47Newtons 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
48Gravitation Constant
- Determined experimentally
- Henry Cavendish
- 1798
- The light beam and mirror serve to amplify the
motion
49Example
Question Calculate gravitational attraction
between two students 1 meter apart
Extremely small
Compare
50Applications of Universal Gravitation 1 Mass of
the Earth
- Use an example of an object close to the surface
of the earth - r RE
51Applications of Universal Gravitation 2
Acceleration Due to Gravity
- g will vary with altitude
52Gravitational 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
53Escape 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
54Keplers 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.
55Keplers 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
56Keplers 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
57Keplers 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
58Keplers 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
59Keplers Third Law application
- Mass of the Sun or other celestial body that has
something orbiting it - Assuming a circular orbit is a good approximation