1
Physics 2210. Homework assignment 2.
B 6.4.5.
r
(t)
=
i
cost
+
j
sint
+
k
t
.
v
(t) = ˙
r
(t)
=
−
i
sint
+
j
cost
+
k
.
a
(t)
=
˙
˙
r
(t)
=
−
i
cost
−
j
sint
.
Therefore,
v
(t)
=
2
, and 
a
(t)=1, and both are constant.
The particle spirals forward in
k
direction (i.e., circular motion in the
i
and
j
plane plus a
uniform motion in
k
direction.
B 6.4.6.
It is straightforward to prove that force
F
and velocity
v
are perpendicular to each other.
F
=
q(
v
×
B
)
, therefore,
F
must be perpendicular to velocity
v
and
B.
Using the hint from the book,
d
dt
(v
2
) =
d
dt
(
v
⋅
v
) = 2
d
v
dt
⋅
v
=
2
1
m
(m
d
v
dt
)
⋅
v
=
2
1
m
F
⋅
v
=
0
,
where we used the Newton’s second law and that the dot product
F
and
v
is zero because
these two vectors are perpendicular to each other. This equation implies that v
2
does not
vary with time, and hence velocity has a constant magnitude. This then suggests that the
magnitude of the force
F
is a constant, as F=qvB.
We can also argue that, based on that
F
is perpendicular to velocity
v
, the force does not
do work to the particle and therefore, the kinetic energy 1/2mv
2
for the particle must be
constant with time, and hence v does not vary with time.
B 8.1.6.
Let m be the mass and r be the radius for the ball at time t, according to the condition
given, we have
dm/dt=4
π
r
2
k,
where k is the proportion constant.
Let the ball density be
ρ
, we have m=4/3
π
r
3
ρ
, and
dr/dt=a,
where a=k/
ρ
.
Solving this 1
st
order differential equation leads to
r=at+C, where C is an integration constant and is 0.5 cm, using the initial condition
r(t=0)=r
o
=0.5 cm.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 '09
 ZHONG
 Cartesian Coordinate System, Work, Velocity, Sin

Click to edit the document details