Physics 3309 Homework 10
Chapter 8
8-37. From the equations in Section 8.8 regarding Hohmann transfers:
 (1)
Substituting
gives
8-41. From the equations in Section 8.8 regarding Hohmann transfers
where
Substituting
gives
From Eq. (8.58), the time of transfer is given by
Substituting gives
Chapter 9
9-3.
By symmetry,  .
From problem 9-2, the center of mass of the cone is at  .
From problem 9-1, the center of mass of the hemisphere is at
So the problem reduces to
for 
9-5.
Calculate the torque about 
Now if the total torque is zero, we must have
or
which is the definition of the center of mass. So
9-9.
Let the axes be as shown with the projectile in the y-z plane. At the top just before the explosion, the velocity is in the y direction and has magnitude  .
where  and  are the masses of the fragments. The initial momentum is
The final momentum is
The conservation of momentum equations are
The energy equation is
or
Substituting for  and  gives
 gives
So
 travels straight down with speed =
 travels in the y-z plane
The mass  is the largest it can be when  , meaning  and the mass ratio is
9-13. From Eq. (9.9), the total force is given by
As shown in Section 9.3, the second term is zero. So the total force is
It is given that this quantity is zero.
Now consider two coordinate systems with origins at 0 and 0¢
where
 is a vector from 0 to 0¢
 is the position vector of  in 0
 is the position vector of  in 0¢
We see that 
The torque in 0 is given by
The torque in 0¢ is
But it is given that 
Thus
9-19.
The force that the tabletop exerts on the chain counteracts the force due to gravity, so that we may write the change in momentum of the center of the mass of the chain as
 (1)
We can write out what the momentum is, though:
 (2)
which has a time derivative
 (3)
where we have used  and  . Setting this last expression equal to (1) gives us
 (4)
Although M. G. Calkin (personal communication) has found that experimentally the time of fall for this problem is consistently less than the value one would obtain in the above treatment by about 1.5%, he also finds evidence that suggests the free fall treatment is more valid if the table is energy absorbing.
9-21. Let us call  the length of rope hanging over the edge of the table, and  the total length of the rope. The equation of motion is
 
Let us look for solution of the form
Putting this into equation of motion, we find
Initial conditions are  ;  .
From these we find  .
Finally  . When  , the corresponding time is
At this point, you can go to the 3309 page,
the UH Space Physics Group
Web Site, or my personal Home Page.
Edgar A. Bering, III ,
Edgar A. Bering, III , <eabering@uh.edu>
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Homework 10
