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
xy
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
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3.
Comment on the validity of these statements about forces and torques.
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 Spring '07
 Mueller
 Physics, Force, Friction, Mass, Duke Marine Laboratory

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