2. 7
- FF F F F
F F F F F F F
ω ω
cm. F
F vcm
ω cm. +
F cm.
7.1 F F
1. F
F F (rad) r
θ
s
θ=
2. F F F
F / (rad/s)
θ= =2 t =T
= F ω=
3. F F F
2
F / (rad/s2)
=
θ ω
ω
θ
F F www.schoolDD.com 1
3. F ( F )
F
F
- F ω ω=
F F F α=
¬ , F F α F
7.2 F F (v), (ω), F F (at) F (α)
1. F (v) (ω)
v
v= ω=
r ω
v =ωr
ω
F v= ωr
2. F F (at) F (α )
v ω
r
θ at = α r
u
ω0
u= F F F
v= F F
ω0 = F
ω =
t = F F
α= =
–
=
–
=
= at/r
at = α r
at = F F F
F F www.schoolDD.com 2
4. F F F F F F
F F F ( F F
F F )
v , at
+
F
F
s (m) θ (rad)
F u (m/s) F ω0 (rad/s)
v (m/s) ω (rad/s)
F a (m/s2) F α (rad/s2)
F
F ( ) ( )
u a t v ω t
θ α
s ω
ω0
1. v = u + at 1. ω = ω0 + αt
2. s= t 2. θ = t
3. s = ut + ½ at2 3. θ = ω0t + ½ αt
2
4. v2 = u2 + 2as 4.
2 2
ω = ω0 + 2αθ
5. s = vt ( v ) 5. θ = ωt ( ω )
F 1
F F 5 F 4/ /
. 5
. F
.
. F5
F F www.schoolDD.com 3
5. f = 4/
ω
ω0 = 0
. ω=? f = 4/
ω = = 2 f
= 2 (4/ )
ω = 8 rad/s Ans
. α=?
ω = ω0 + αt
8 = 0 + α(5)
2
α = 1.6 rad/s Ans
. θ=?
= θ t
= 5
θ = 20 rad Ans
. =?
F . θ = 20 rad
1 F θ = 2 rad
θ = 20 rad F F 20/2 = 3.18 Ans
F 2
F 44 m 10 F F 70 cm
F
. F 10
. F F F
F
ω = ?, α = ?
ω0 = 0
u=0 v = ?, a = ?
r= 0.70
44
t = 10
F F www.schoolDD.com 4
6. . ω = ?, v = ? t = 10
θ = t ----1
θ F F 44 m F F F
F 1 F F F = 2 r = 2(22/7)(0.70) = 4.4 m.
F 44/4.4 = 10
FF 1 F θ = 2 rad
θ = 10 x θ = 10(2 ) = 20 rad
F θ = 20 , ω0 = 0, t = 10 1, 20 = 10
ω = 4 rad/s Ans
v = ωr
= 4 (0.7)
v = 2.8 m/s Ans
. α = ?, a = ?
ω = ω0 + αt
4 = 0 + α(10)
2
α = 0.4 rad/s Ans
a = αr
= 0.4 (0.7)
a = 0.28 m/s2 Ans
7.3 F
F F F
F F (N m)
=Fr
r
m
F
F = mat at = αr
F = mαr (r) = mr2 α
=Iα
I = mr2 = F
F F www.schoolDD.com 5
7. F I F F F
2
(kg m2)
A B
IA = = m1r12 + m2r22 + m3r32
m2
r2 r1 m1
r3
m3
r3
IB = IA + L2 = IA + (m1+ m2+ m3) L2
L
A B
L=
¬ F I F F FF F
F 3
20 N F 2 kg 0.5 m F F F 4
F I F = ½ mr2
. F
. F F F
. F 4
. F F
ω
r = 0.5
+
F = 20
. =?
= Fr
= 20(0.5)
= 10 N m Ans
. α = ?, at = ?
= Iα
F F www.schoolDD.com 6
8. = (½ mr2) α
10 = (½ x 2 x 0.52) α
2
α = 40 rad/s Ans
at = α r
= 40(0.5)
at = 20 m/s2 Ans
. ω = ?, v = ? t=4 s
ω = ω0 + αt
= 0 + 40(4)
ω = 160 rad/s Ans
v = ωr
= 160(0.5)
v = 80 m/s Ans
. =?
F t = 4 s, θ = t
= 4
= 320 rad
1 F θ = 2 rad
θ = 320 rad F F 320/2 = 50.8 Ans
F 4
1 kg 2 kg F F 4 kg 0.5 m F F
F F F (I F = ½ mr2)
F
F
m=4
r = 0.5 r = 0.5
+ a + a
T1 T2 F F
T1 T2 F I F F
a m1 m2 F
1 kg
2 kg m1 1 kg
2 kg m2
m1g m2g
F F F T1 T2
F F www.schoolDD.com 7
9. = F Fr
F = (T2- T1) r
= Iα
F Iα = (T2- T1) r ----1
m1 m2
= ma
F T1 m1g = m1a --> T1 = m1a + m1g
m2g T2 = m2a --> T2 = m2g m2a
T 2 - T1 = (m2 - m1) g (m1 + m2) a
F F T2 - T1 = (2 - 1) 10 (1 + 2) a
T 2 - T1 = 10 3 a
F a = αr
T2 - T1 = 10 - 3α r = 10 - 3α (0.5)
T2 - T1 = 10 1.5 α ----2
F T2 - T1 I = ½ mr2 1
F ½ mr2 α = (10 1.5 α) r
½ (4)(0.52) α = (10 1.5 α) (0.5)
2
α = 4 rad/s Ans
a = αr
= 4(0.5)
a = 2 m/s2 Ans
F α 2, T2 - T1 = 10 1.5 (4)
T 2 - T1 = 4 N
= (T2- T1) r
= 4 (0.5)
= 2 N m Ans
F 5
F F I = ½ mr2 r F F F F F
F F F F
T
F
r F
+
a
F F www.schoolDD.com 8
10. mg
F F F T
= Tr
= Iα
F T = I α/ r
F F
= ma
F mg T = ma
F T = I α/ r a = αr
F mg I α/ r = m αr
mg ½ mr2 α/ r = m αr
α =
= Iα
F = ½ mr2
= mgr N m Ans
F 6
m r F F F F F .
F F F ( I = mr2)
N
F
a
F
mg sin37
f mg mg cos37
37° 37°
F F F F
= fr
= Iα
F f = I α/ r ----1
= ma
F mg sin37 f = ma
F f = I α/ r a = αr
F F www.schoolDD.com 9
11. F mg sin37 I α/ r = m αr
mg sin37 mr2 α/ r = m αr
α =
F I α 1, f = mr2( /r
f = mg sin37
F f = µN = µ mg cos37
µ mg cos37 = mg sin37
µ = x x
=
µ = 0.21 Ans
7.4 F Ek
1. F
ω
I + EK = ½ I ω2
+
2. F ( )
ω
v EK = ½ mv2 + ½ I ω2
I +
+
F 7
20 kg 1m F 5/ π /
2
F (I = mr )
ω
v
I
F F www.schoolDD.com 10
12. F Ek = ½ mv2 + ½ Iω2 ----1
F f = 5/ π
ω = 2 π f = 2 π (5/ π ) = 10 rad/s
v = ωr = 10(1) 10 m/s
I = mr2 = 20(12) = 20 kg m2
F 1, Ek = ½ 20(102) + ½ 20(102)
Ek = 2x103 J Ans
F 8
(v) m r F h
(I = mr2)
u=0
. F
h . F
v
. v=? ( F F f = 0)
F =
mgh = ½ mv2
v = Ans
. v=?
F =
mgh = ½ mv2 + ½ Iω2
mgh = ½ mv2 + ½ mr2( 2
gh = +
v = Ans
F F
F F www.schoolDD.com 11
13. 7.5 W
W = Fs = Fr
F W= s F =θ
W= θ
r s W= θ P= W= θ
θ
m F P= F =ω
F P= ω
F P F F (w)
P= ω
F 9
F 72 km/hr F F F 0.5 m F 1x 10 4 kg
-m2 F F F F
F F
4
ω0 = 0
ω r = 0.5, I = 1x10
u=0 v = 20
F F F F
v = 72 km/hr = 20 m/s
ω = = = 40 rad/s
2
ω2 = ω0 + 2αθ
402 = 0 + 2αθ
αθ = 800
W = θ = Iαθ
= 1x104(800)
W = 8x106 J Ans
F F www.schoolDD.com 12
14. F 10
F F F 1000 N m F 120/ π / F F
ω
ω = 2 π f = 2 π (120/(60 π ))
= 1000 ω = 4 rad/s
P = ω
= 1000(4)
P = 4000 W Ans
7.6 L
I
F
L = Iω
ω F kg m2/s
= Iα = I
=
=
= F =
F
F =
F F www.schoolDD.com 13
15. I1ω1 = I2ω2
F 11
F 4 rad/s ( I =
2.5 kg m 2 , I = 2.0 kg m 2 )
ω1 = 4 ω2 = ?
I1 = 2.5
I2 = 2.0
F
I1ω1 = I2ω2
2.5(4) = 2.0 ω2
ω2 = 5 rad/s Ans I
F
FF F F F F
F (s) ⇔ (θ)
F (v) ⇔ (ω)
F F (a) ⇔ F (α)
(m) ⇔ F (I)
( F) ⇔ F ( )
(p) ⇔ ( L)
F
F = ma F = Iα
p = mv L = Iω
F= F =
W = Fs W= θ
P = Fv P= ω
F Ek = ½ mv2 F E k = ½ I ω2
F F www.schoolDD.com 14
