172
4 Rotational Dynamics
Fig. 4.27
4.43 (a) For pure sliding equation of motion is
ma = −μmg
or a = −μg
(1)
v = v 0 − μgt
(2)
At the instant pure rolling sets in
Torque I α = F R
(3)
2
5
m R
2
α = μmg R
∴ α =
5
2
μg
R
(4)
ω = αt =
5
2
μgt
R
(5)
Using (5) in (2)
v = v 0 −
2
5
ω R = v 0 −
2
5
v
∴ v =
5
7
v 0
(b) v = v 0 − μgt
t =
v 0 − v
μg
=
v 0 − (5/7) v 0
μg
=
2v 0
7μg
(c) v 2 = v 2
0 + 2as
4 Rotational Dynamics
Fig. 4.27
4.43 (a) For pure sliding equation of motion is
ma = −μmg
or a = −μg
(1)
v = v 0 − μgt
(2)
At the instant pure rolling sets in
Torque I α = F R
(3)
2
5
m R
2
α = μmg R
∴ α =
5
2
μg
R
(4)
ω = αt =
5
2
μgt
R
(5)
Using (5) in (2)
v = v 0 −
2
5
ω R = v 0 −
2
5
v
∴ v =
5
7
v 0
(b) v = v 0 − μgt
t =
v 0 − v
μg
=
v 0 − (5/7) v 0
μg
=
2v 0
7μg
(c) v 2 = v 2
0 + 2as