2. F ˆ F www.schoolDD.com 1
F
vcmF
F
7
- F F F F F
F F F F F F ˈ F
7.1 F F
1. ીሬሬറ F ˈ ˈ
F F ˈ (rad)
2. ሬሬሬറ F F ˈ F
F ˈ / (rad/s)
3. F હሬሬറ F ˈ F
F ˈ / 2
(rad/s2
)
cm.
ω
ω
ω
+
cm.
sr
θ
θ =
ܛ
ܚ
࣓ሬሬሬറ =
ીሬሬറ
ܜ
θ =
ଶగ୰
୰
= 2ߨ t = T
F ω =
ଶగ
ࢻሬሬറ =
ઢሬሬሬറ
ઢܜ
ωሬሬറθሬറ
ω
θ
cm.
3. F ˆ F www.schoolDD.com 2
u
ω
F ( F )
F ˈ
- F ω ω =
୲
F F F α =
ன
୲
¬ , F F α ˈ F
7.2 F F (v), (ω), F F (at) F (α)
1. F (v) (ω)
2. F F (at) F (α )
u = F F F
v = F F
ω0 = F
ω =
t = F F
α =
ன
୲
=
ன ି னబ
୲
=
୴/୰ – ୳/୰
୲
= ሺ
୴ – ୳
୲
ሻ
1
r
= at/r
at = α r
at = F F F
v
F
ωr
ω
v = ω r v =
ଶగ୰
ω =
ଶగ
F v = ω r
v
ω0
θ at = α rr
4. F ˆ F www.schoolDD.com 3
+
v, at
α
ω t
F F F F F F ˈ
F F F ( F F
F F )
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 ( ) ( )
1. v = u + at 1. ω = ω0 + αt
2. s = ሺ
୳ ା ୴
ଶ
ሻ t 2. θ = ሺ
னబ ା ன
ଶ
ሻ t
3. s = ut + ½ at2
3. θ = ω0t + ½ αt2
4. v2
= u2
+ 2as 4. ω
2
= ω0
2
+ 2αθ
5. s = v t ( v ) 5. θ = ω t ( ω )
F 1
ʽ F F 5 F 4/ ߨ /
. 5
. F
.
. ʽ F 5
a t vu
s ω0
θ
ω
5. F ˆ F www.schoolDD.com 4
. ω = ? f = 4/ ߨ
ω = ଶ
= 2ߨf
= 2ߨ(4/ ߨ)
ω = 8 rad/s Ans
. α = ?
ω = ω0 + αt
8 = 0 + α(5)
α = 1.6 rad/s2
Ans
. θ = ?
θ = ሺ
னబ ା ன
ଶ
ሻ t
= ሺ
ା ଼
ଶ
ሻ 5
θ = 20 rad Ans
. = ?
F . θ = 20 rad
1 F θ = 2ߨ rad
θ = 20 rad F F 20/2ߨ = 3.18 Ans
F 2
ˆˉ F 44 m 10 F F 70 cm
F
. F 10
. F F F
F
v= ?, a = ?
ω0 = 0
t= 10
44
u= 0
ω
ω0 = 0
f = 4/ߨ
r= 0.70
ω = ?, α = ?
6. F ˆ F www.schoolDD.com 5
. ω = ?, v = ? t = 10
θ = ሺ
னబ ା ன
ଶ
ሻ t ----1
θ F F 44 m F F F
F 1 F F F = 2ߨr = 2(22/7)(0.70) = 4.4 m.
F ˈ 44/4.4 = 10
F F 1 F θ = 2ߨ rad
θ = 10 x θ = 10(2ߨ) = 20ߨ rad
F θ = 20ߨ, ω0 = 0, t = 10 1, 20ߨ = ሺ
ା ன
ଶ
ሻ 10
ω = 4ߨ rad/s Ans
v = ωr
= 4ߨ(0.7)
v = 2.8ߨ m/s Ans
. α = ?, a = ?
ω = ω0 + αt
4ߨ = 0 + α(10)
α = 0.4ߨ rad/s2
Ans
a = α r
= 0.4ߨ(0.7)
a = 0.28ߨ m/s2
Ans
7.3 F
F ࣎ F ˈ F
F F ˈ (N m)
m
F
߬റ
r
࣎ = F r
F = mat at = αr
F ߬ = mαr (r) = mr2
α
࣎ = I α
I = mr2
= F
7. F ˆ F www.schoolDD.com 6
r3
F I ˈ F ˈ F F
ˈ 2
(kg m2
)
¬ F I F F F F F
F 3
20 N F 2 kg 0.5 m F F F ˈ 4
F I F = ½ mr2
. F
. F F F
. F 4
. F F
. ߬ = ?
߬ = Fr
= 20(0.5)
߬ = 10 N m Ans
. α = ?, at = ?
߬ = I α
L
A
A
m2
m3
m1
r3
r1r2
IA = ∑ ܚܕ
IB = IA + ሺ∑ ܕሻ L2
L =
B
B
= m1r1
2
+ m2r2
2
+ m3r3
2
= IA + (m1+ m2+ m3) L2
ω
F = 20
+
r = 0.5
8. F ˆ F www.schoolDD.com 7
= (½ mr2
) α
10 = (½ x 2 x 0.52
) α
α = 40 rad/s2
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 = ሺ
னబ ା ன
ଶ
ሻ t
= ሺ
ା ଵ
ଶ
ሻ 4
= 320 rad
1 F θ = 2ߨ rad
θ = 320 rad F F 320/2ߨ = 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 F T1 T2
m1
T2
T1
aa
m2
+
r = 0.5
1 kg
2 kg
+
r = 0.5
1 kg
2 kg
m1g m2g
F
F
T1 T2 F F
F I F F
a m1 m2 F
m = 4
9. F ˆ F www.schoolDD.com 8
߬ = F F r
F ߬ = (T2- T1) r
߬ = I α
F I α = (T2- T1) r ----1
m1 m2
∑ F = ma
F T1 m1g = m1a --> T1 = m1a + m1g
m2g T2 = m2a --> T2 = m2g m2a
T2 - T1 = (m2 - m1) g (m1 + m2) a
F F T2 - T1 = (2 - 1) 10 (1 + 2) a
T2 - T1 = 10 3 a
F a = α r
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)
α = 4 rad/s2
Ans
a = α r
= 4(0.5)
a = 2 m/s2
Ans
F α 2, T2 - T1 = 10 1.5 (4)
T2 - 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
a
T
+
r
F
F
10. F ˆ F www.schoolDD.com 9
f
F F F T
߬ = Tr
߬ = I α
F T = I α/ r
F 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 =
ଶ
ଷ୰
߬ = I α
F ߬ = ½ mr2 ଶ
ଷ୰
߬ =
ଵ
ଷ
mgr N m Ans
F 6
m r F F F F F .
F F F ( I =
ଶ
ହ
mr2
)
F F F F
߬ = f r
߬ = I α
F f = I α/ r ----1
∑ F = ma
F mg sin37 f = ma
F f = I α/ r a = αr
mg
37° 37°
a
mg cos37
mg sin37
mg
N
F
F
11. F ˆ F www.schoolDD.com 10
F mg sin37 I α/ r = m αr
mg sin37
ଶ
ହ
mr2
α/ r = m αr
α =
ହ ୱ୧୬ଷ
୰
F I α 1, f =
ଶ
ହ
mr2
(
ହ ୱ୧୬ଷ
୰
ሻ/ r
f =
ଶ
mg sin37
F f = µN = µ mg cos37
µ mg cos37 =
ଶ
mg sin37
µ =
ଶ
x
ଷ
ହ
x
ହ
ସ
=
ଷ
ଵସ
µ = 0.21 Ans
7.4 F Ek
1. F
2. F ( )
F 7
20 kg 1 m F 5/π /
F (I = mr2
)
v
ω
I +
+
EK = ½ I ω2
ω
I +
+
EK = ½ mv2
+ ½ I ω2
v
ω
I
12. F ˆ F www.schoolDD.com 11
F Ek = ½ mv2
+ ½ Iω
2
----1
F f = 5/π
ω = 2π f = 2π (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(10
2
)
Ek = 2x103
J Ans
F 8
(v) m r F h
(I = ଶ
ହ
mr2
)
. v = ? ( F F f = 0)
F ∑ Eଵ = ∑ Eଶ
mgh = ½ mv2
v = ඥ2gh Ans
. v = ?
F ∑ Eଵ = ∑ Eଶ
mgh = ½ mv2
+ ½ Iω
2
mgh = ½ mv2
+ ½ ଶ
ହ
mr2
(
୴
୰
ሻ
2
gh = ୴మ
ଶ
+ ୴మ
ହ
v = ට
ଵ୦
Ans
F F
u = 0
v
h
. F
. F
13. F ˆ F www.schoolDD.com 12
7.5 W
F P ˈ F ˈ F (w)
F 9
F 72 km/hr F F F 0.5 m F 1x 104
kg
- m2
F F F F
F F F F
v = 72 km/hr = 20 m/s
ω =
୴
୰
= ଶ
.ହ
= 40 rad/s
ω2
= ω0
2
+ 2αθ
402
= 0 + 2αθ
αθ = 800
W = ߬ θ = Iαθ
= 1x104
(800)
W = 8x106
J Ans
m
F
r W = ࣎ θθθθ
W = Fs ߬ = Fr
F W = ሺ
ఛ
୰
ሻ s F
ୱ
୰
= θ
W = ߬ θ
θ
s
P = ࣎ ω
P =
୲
W = ߬ θ
F P =
ఛ
୲
F
୲
= ω
P = ߬ ω
u = 0
ω0 = 0
v = 20
ω r = 0.5, I = 1x104
F F
14. F ˆ F www.schoolDD.com 13
F 10
F F F 1000 N m F 120/π / F F
P = ߬ ω
= 1000(4)
P = 4000 W Ans
7.6 L
߬ = I α = I ሺ
ன ି னబ
୲
ሻ
= ୍ன ି ୍னబ
୲
߬ =
୲
F
∑ ۺ F = ∑ ۺ
ω
߬ = 1000
ω = 2π f = 2π (120/(60 π ))
ω = 4 rad/s
I
ω
L = Iω
ˈ F
F ˈ kg m2
/s
࣎ =
ઢۺ
ઢܜ F =
15. F ˆ F www.schoolDD.com 14
I1ω1 = I2ω2
F 11
F 4 rad/s ( I =
2.5 kg m2
, I = 2.0 kg m2
)
F
I1ω1 = I2ω2
2.5(4) = 2.0 ω2
ω2 = 5 rad/s Ans I
F
F F 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 ߬ = I α
p = mv L = I ω
F = ୮
୲
F ߬ =
୲
W = Fs W = ߬ θ
P = Fv P = ߬ω
F Ek = ½ mv2
F Ek = ½ I ω
2
I1 = 2.5
ω1 = 4 ω2 = ?
I2 = 2.0