This document discusses concepts related to rotational kinematics and dynamics including: 1. Rotational kinematics equations relating angular displacement (θ), angular velocity (ω), angular acceleration (α), and time (t). 2. Rotational dynamics equations relating torque (τ), moment of inertia (I), angular acceleration (α), and angular velocity (ω). 3. Examples calculating values like angular velocity, angular acceleration, linear velocity, torque, power, work, and kinetic energy for rotating objects using the rotational kinematics and dynamics equations.

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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
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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
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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
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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
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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
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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
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16. F F www.schoolDD.com 15