Assign8 - PHY 53 Summer 2010 Duke Marine Laboratory 1 Some...

Assignment 8
1. Some questions about the vector product C = A × B .
a. Prove the rules given in the notes for the magnitude and direction of C
from the definition in terms of components. Let A and B lie in the x-y
plane with A along the x -axis and B having direction with angle θ relative
to A .
b. One requires θ to be the angle between A and B that is no larger than π .
Why?
c. What are the rules for the vector products of the unit vectors ( i , j , k )?
d. Prove directly from the definition in terms of components that
A × B = − B × A . {Use the situation in (a).]
2. Consider the effects of the gravitational force on the particles of a system. The i th
particle, of mass m
i
at position r
i
experiences a gravitational force m
i
g ( r
i
) ,
where g ( r
i
) is the gravitational field at that point. If the gravitational field is
uniform , then g ( r
i
) is the same at the location of all the particles. Call it g .
a. If the gravitational field is uniform, show that the total gravitational force
on the system is simply M g , where M is the total mass.
b. The gravitational torque about the CM from the force on the i th particle is
τ
i
= ′ r
i
× m
i
g ( r
i
) , where ′ r
i
is the position of the particle relative to the CM.
Show that if the gravitational field is uniform the total gravitational torque
about the CM is zero.
PHY 53 — Summer 2010 Duke Marine Laboratory
1

3. Comment on the validity of these statements about forces and torques.