178
4 Rotational Dynamics
4.54 r = 3t ˆ
i + 2 ˆ
j
v =
dr
dt
= 3 ˆ
i
L = r × p = m(r × v) = m(3t ˆ
i + 2 ˆ
j) × 3 ˆ
i
= 6 m( ˆ
j × ˆ
i) = −6 m ˆ
k (constant)
4.55 Angular momentum conservation gives
J = mvd = I ω
(1)
Linear momentum conservation gives
mv = Mv c
(2)
Energy conservation gives
1
2
mv
2
=
1
2
I ω
2
+
1
2
Mv
2
c
(3)
I =
Ml 2
12
(4)
Eliminating ω and v c from (1) and (2) and using (3)
1
2
mv
2
=
1
2
m 2 v 2 d 2
I
+
1
2
m 2 v 2
M
(5)
Simplifying and using (4) in (5)
d =
l
2
M − m
3m
4.56 (a) Let the initial velocity be v 0 , then at instant t the velocity
v = v 0 − at = v 0 − μgt
(1)
Torque τ = I α = F R
1
2
m R
2
α = μ mg R
α =
2μg
R
μgt =
α Rt
2
=
ω R
2