Chapter 11 Systems of Particles
*What is a "point" particle?
The Quicktime movie, dive.mov, is in the movies folder on the CDROM that came with the textbook. It is a 5 second video of a person diving into a swimming pool with one-half rotation of the body.
page 11-2 Center of Mass
Where is the center of mass of a meter stick?
Figure 1 For your own interest divide the radius of the moon’s orbit "around" the earth by the value of the radius of the earth’s orbit around the sun.
In Figure 1, is the dotted path around the sun a circle?
In Figure 1, is the dotted path around the sun an ellipse?
What does the dotted path in Figure 1 represent?
Approximately how many stars are in a globular cluster?
Why is the detailed computer calculation of the motions of the stars in a globular cluster difficult?
If the stars are randomly distributed in the globular cluster where is the center of mass?
How can you experimentally determine the location of the center of mass for an object?
If a marble was launched at the first dash on the parabola in Figure 3 and with the same velocity that the rod had, how would the path of the marble compare to the dashed path? Does your answer depend on the masses of the rod and marble?
page 11-3 Center of Mass Formula
Let M be the total mass of a system of separate particles.
To determine the X coordinate of the center of mass:
1. For each object in the system, multiply the mass of the object by the X coordinate of the object.
2. Add up the mX values for all of the objects
3. Divide this sum by M to find the X coordinate of the center of mass
Repeat this procedure to find the Y and Z coordinates if necessary.
Suppose a system consists of three objects on an X axis: 1) 2 kg at X=-2 meters 2) 3 kg at X=0 meters 3) 5 kg at X = 3 meters. Calculate the X coordinate of the center of mass.
For both examples above, how far is the center of mass from the 5 kg object?
What is the purpose in having you do these two calculations?
page 11-4 Dynamics of the Center of Mass
The derivative of position with respect to time produces velocity. Equation 3 should remind you of a physics concept. Name the concept.
The derivative of velocity with respect to time produces acceleration. Is it true that the net force on object 1 is equal to the mass of object 1 times the acceleration of object 1?
What is true regarding the sum of the internal forces for a system of particles?
What does "com" stand for in equation 8?
What does Fext represent?
*Write your answer to Exercise 4 before you come to class.
page 11-6 Newton’s Third Law
When two bodies interact the force on object one due to object two is equal in magnitude but opposite in direction compared to the force on object two due to object one.
page 11-7 Conservation of Linear Momentum
Recall p = mv.
What questions do you have on the derivation of equation 12?
The text describes a globular cluster drifting through empty space and sets Fext equal to zero. Describe another situation that also has Fext = 0.
page 11-8 Momentum Version of Newton’s Second Law
This is given in equation 12.
Equation 12 is actually the original Newton’s Second Law. F=ma is a form of the second law that is useful in some problems. What assumption is being made when you use F=ma?
Work Exercise 5.
page 11-9 Collisions, Impulse
In Figure 10b, what is the direction of the force applied by the force detector on the aircart?
What is the direction of the force applied by the aircart on the force detector?
Do these two forces have the same magnitude?
Then, is the net force on the aircart zero?
Does the force applied by the force detector on the aircart have a constant magnitude?
page 11-10 PHY151 students are not responsible for the calculus derivation. Suppose that the force of the collision can be represented by an average value of force.
Then Impulse = Faverage * elapsed time for the collision or F Δt F represents the average force.
The impulse is equal to the change in the momentum for the object F Δt = Δp
Consider two collisions:
1) A 0.4kg tomato is thrown into a wall at 10 m/s. It hits the wall and slides down.
2) A 0.4kg rubber ball is thrown into a wall at 10 m/s. It hits the wall and rebounds at a speed of 10m/s.
Let the time of collision be the same for both 1) and 2). Which collision involved the largest force of the wall on the object?
Let the time of the collision be 0.06 seconds. Calculate the force of the wall on the object for both 1) and 2).
1) 2)
Suppose that I stand on a chair at a height of 67 cm. I then slide horizontally off the chair and land on the floor. Calculate my speed the instant before my feet hit the floor.
If I bend my knees during the landing such that a time of 0.7 seconds elapses before I come to rest, calculate the average force of the floor on my feet.
If I lock my knees during the landing such that a time of 0.03 seconds elapses before I come to rest, calculate the average force of the floor on my feet.
Which landing is more uncomfortable?
Why do basketball shoes have padded insoles?
Why do automobile manufacturers put shock absorbers between the bumper and the frame of the car?
Why do airbags minimize injuries in auto collisions?
You should read the following sections but they will not be discussed in class:
Calibration of the Force Detector
The Impulse Measurement
Change in Momenum
page 11-13 Momentum Conservation during Collisions
What is often true about the impulsive forces in collisions compared to the external force acting on the objects?
Why don’t the external forces contribute significantly to the change in velocities of the objects during the collision?
page 11-14 Collisions and Energy Loss
After some collisions the KE and PE of the system is less (or more) than the original sum of KE and PE. The original energy may be accounted for by describing the sound energy or light energy that leaves the collision, or in the deformation of the objects, etc. Even though the amount of mechanical energy (KE + PE) changes, momentum will still be conserved (as long as the net external force = 0).
What questions do you have on the derivation of equation 35?
Do Exercise 10. Write down why you think the result in each case is "reasonable."
M = 0 M = infinity
page 11-15 Why is momentum conserved during the collision?
Why is momentum not conserved as the pendulum swings upward after the collision?
Why can energy be described as conserved after the collision?
Suppose for Figure 18 and 19, the incoming object has a mass of 120 grams and a speed of 12 m/s. Let the target object be made of clay with a mass of 400 grams. The incoming object sticks to the clay at the time of the collision. Calculate the maximum height for the clay as it swings upward after the collision.
page 11-16 Collisions that Conserve Momentum and Energy
Elastic collision: KE before the collision = KE after the collision
*Write out a definition of an inelastic collision.
Atomic and subatomic collisions are usually elastic. They can be inelastic if the target or incoming particles split, or store or release energy during the collision.
page 11-17 Elastic Collisions
Momentum and KE are conserved in an elastic collision.
What questions do you have about the derivation of v1 * v2 = 0?
What assumption was made in the derivation?
Consider the elastic collision of two masses. If the masses are allowed to be different the results for the velocities of the two objects are:
V1 after = (m1 – m2)*V1 /(m1 + m2) + 2*m2*V2/(m1 + m2)
V2 after = 2*m1*V1 / (m1 + m2) - (m1 – m2)*V2 / (m1 + m2)
Let V2 = 0 (stationary target)
Case 1 Let m1 = m2 Describe V1 Describe V2
Case 2 Let m1 >> m2 Describe V1 Describe V2
Case 3 Let m1 << m2 Describe V1 Describe V2
page 11-19 Discovery of the Atomic Nucleus
Radioactivity will be discussed in more detail in the second semester of physics. You should look for the application of elastic collisions as you read this section.
page 11-20 Neutrinos
As you read this section look for the connection of the material to the conservation laws.
page 11-21 Neutrino Astronomy
What is the incentive for making measurements of neutrinos coming from the sun?
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