1. PHYS 1443 – Section 501 Lecture #19
Monday Apr 5, 2004
Dr. Andrew Brandt
Review for test weds Apr. 7
Monday, April 5, 2004
PHYS , Spring 2004
Dr. Andrew Brandt

2. Lec. 11: Newton+Kepler =9.8 m/s2 Monday, April 5, 2004
PHYS , Spring 2004
Dr. Andrew Brandt

3. Kepler’s Laws & EllipseF1 F2 b c a
Ellipses have two different axis, major (long) and minor (short) axis, and two focal points, F1 & F2
a is the length of a semi-major axis
b is the length of a semi-minor axis
All planets move in elliptical orbits with the Sun at one focal point.
The radius vector drawn from the Sun to a planet sweeps out equal area in equal time intervals. (Angular momentum conservation)
The square of the orbital period of any planet is proportional to the cube of the semi-major axis of the elliptical orbit.
Monday, April 5, 2004
PHYS , Spring 2004
Dr. Andrew Brandt

4. Example of Kepler’s Third Law
Calculate the mass of the Sun using the fact that the period of the Earth’s orbit around the Sun is 3.16x107s, and its distance from the Sun is 1.496x1011m.
Using Kepler’s third law.
The mass of the Sun, Ms, is
Monday, April 5, 2004
PHYS , Spring 2004
Dr. Andrew Brandt

5. Lec. 12: Work and KE F M M variable force spring force work and KE
A man cleaning a floor pulls a vacuum cleaner with a force of magnitude F=50.0N at an angle of 30.0o with East. Calculate the work done by the force on the vacuum cleaner as it is displaced by 3.00m to East. F
30o M M d
variable force
spring force
work and KE
Monday, April 5, 2004
PHYS , Spring 2004
Dr. Andrew Brandt

6. Lec. 13: Potential Energy
Energy associated with a system of objects  Stored energy which has Potential or possibility to work or to convert to kinetic energy
cons of energy
grav PE
elastic PE
Monday, April 5, 2004
PHYS , Spring 2004
Dr. Andrew Brandt

7. Lec: 14 conservative+non-cons forces
A ball of mass m is attached to a light cord of length L, making up a pendulum. The ball is released from rest when the cord makes an angle qA with the vertical, and the pivoting point P is frictionless. Find the speed of the ball when it is at the lowest point, B.
Compute the potential energy at the maximum height, h. Remember where 0 is. mg m qA L T PE KE
mgh h{
Using the principle of mechanical energy conservation B
mv2/2
Monday, April 5, 2004
PHYS , Spring 2004
Dr. Andrew Brandt

8. Work Done by non-conservative Forces
Mechanical energy of a system is not conserved when any one of the forces in the system is a non-conservative force.
Two kinds of non-conservative forces:
Applied forces: Forces that are external to the system. These forces can take away or add energy to the system. So the mechanical energy of the system is no longer conserved.
If you were to carry around a ball, the force you apply to the ball is external to the system of ball and the Earth. Therefore, you add kinetic energy to the ball-Earth system.
Kinetic Friction: Internal non-conservative force that causes irreversible transformation of energy. The friction force causes the kinetic and potential energy to transfer to internal energy
Monday, April 5, 2004
PHYS , Spring 2004
Dr. Andrew Brandt

9. Lec. 15: Grav. field, escape speed, power
Average power
Instantaneous power
Monday, April 5, 2004
PHYS , Spring 2004
Dr. Andrew Brandt

10. Lec.16: Momentum,Impulse,collisions
A new concept of linear momentum can also be used to solve physical problems, especially the problems involving collisions of objects.
Linear momentum of an object whose mass is m and is moving at a velocity of v is defined as
Momentum is a vector quantity.
The heavier the object the higher the momentum
The higher the velocity the higher the momentum
Its unit is kg.m/s
Whenever two or more particles in an isolated system interact, the total momentum of the system remains constant.
Monday, April 5, 2004
PHYS , Spring 2004
Dr. Andrew Brandt

11. Example for Collisions
A car of mass 1800kg stopped at a traffic light is rear-ended by a 900kg car, and the two become entangled. If the lighter car was moving at 20.0m/s before the collision what is the velocity of the entangled cars after the collision?
The momenta before and after the collision are
Before collision
20.0m/s m2 m1
After collision
Since momentum of the system must be conserved m1 m2 vf
What can we learn from these equations on the direction and magnitude of the velocity before and after the collision?
The cars are moving in the same direction as the lighter car’s original direction to conserve momentum.
The magnitude is inversely proportional to its own mass.
Monday, April 5, 2004
PHYS , Spring 2004
Dr. Andrew Brandt

12. Lec. 17: Center of Mass
We’ve been solving physical problems treating objects as sizeless points with masses, but in realistic situation objects have shapes with masses distributed throughout the body.
Center of mass of a system is the average position of the system’s mass and represents the motion of the system as if all the mass is on the point.
The total external force exerted on the system of total mass M causes the center of mass to move at an acceleration given by as if all the mass of the system is concentrated on the center of mass.
What does above statement tell you concerning forces being exerted on the system? m1 m2 x1 x2
Consider a massless rod with two balls attached at either end.
xCM
The position of the center of mass of this system is the mass averaged position of the system
CM is closer to the heavier object
Monday, April 5, 2004
PHYS , Spring 2004
Dr. Andrew Brandt

13. Example for Center of Mass
A system consists of three particles as shown in the figure. Find the position of the center of mass of this system.
Using the formula for CM for each position vector component m1
y=2
(0,2)
(0.75,4)
rCM m2
x=1
(1,0) m3
x=2
(2,0)
One obtains If
Monday, April 5, 2004
PHYS , Spring 2004
Dr. Andrew Brandt

14. Motion of a Group of Particles
We’ve learned that the CM of a system can represent the motion of a system. Therefore, for an isolated system of many particles in which the total mass M is preserved, the velocity, total momentum, acceleration of the system are
Velocity of the system
Total Momentum of the system
Acceleration of the system
External force exerting on the system
What about the internal forces?
System’s momentum is conserved.
If net external force is 0
Monday, April 5, 2004
PHYS , Spring 2004
Dr. Andrew Brandt