This document discusses concepts in mechanics including: 1. Conditions for static equilibrium, including that the net force and net torque must equal zero. 2. Analysis of forces in different mechanical systems using free body diagrams and applying Newton's laws and principles of torque. 3. Problem solving techniques for calculating unknown forces, torques or accelerations given force diagrams and relevant equations of motion.

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2. 8
F
8.1. F F 1
F F ( F )
- F F F F
- F F F
- F F F F F
F F F F
8.2.
F F F
1. F F F
=0 = F
=0
=0 =
- F v --> a = 0 = ma = 0
- F ω --> α = 0 = Iα = 0
2. F F
=0 =
8.3. F F
F M F
F (N m)
F F
O+
O
F
O
MO = F
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3. 2 F F F v
F F
1. F F F F F
2. F ( F) F F, =0
F
N
F2 F1 T m
m m mg sinθ
m θ
mg mg
F
=0 =0 =0 =0
N = mg F 1 = F2 T = mg F = mg sinθ
3 F F F
1. F F
2. F F F, =0
3. F F F F, =0
F1 F2
a b T2
θ
F1 T1 θ
A B C α
F2 F3 F3 m
=0 =0 mg
F 1 = F2 + F3 F1 = F2 cosθ + F3 cosα
=0 =0
=0
F1(a) = F3(a+b) T2 cosθ = T1
F2 sinθ = F3 sinα
=0 =0
F2(a) = F3(b) T2 sinθ = mg
=0
F2(a+b) = F1(a)
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4. 8.4. F F
F F F F F
F F F
F F F
F
F
F
MC = F
F 1
F 10 kg 2m F F 2 F 20 kg F F
F F 0.40 m F F F F
2.0
T1 0.4 1.0 T2
A B
m W
mg
F B
=
mg(1.6) + W(1.0) = T1(2.0)
20(10)(1.6) + 100(1.0) = 2T1
T1 = 210 N Ans
= 0
T1 + T2 = mg + W =
210 + T2 = 20(10) + 100
T2 = 90 N Ans
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5. F 2
10 kg 5m 53 F .
F 0.4 F 50 kg F F FF
F
B N
B
5.0 5.0
4.0
mg h
W
53° fA 53°
A 1.5
NA x F
F
F F x F A F F F
fA
= 0
NA = mg + W =
= 50(10) + 100
NA = 600 N
fA = µ NA
= 0.4(600)
fA = 240 N
= 0
fA = NB = F
NB = 240 N
F A F
=
NB(4.0) = W(1.5) + mg(x)
240(4.0) = 100(1.5) + 50(10)(x)
x = 1.62
= tan53
h = x tan53 = 1.62(4/3)
h = 2.16 m Ans
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6. F 3
F 20 kg F
T sinθ F
T
F
θ A θ A
mg cosθ T cosθ
θ
mg mg sinθ
B B
F B F F B
=
Tsinθ(AB) = mg sinθ
T = =
T = 100 N Ans
F 4
16 kg F 10 kg F F F
F
F
F
T T sin37
A 37° B A T cos37 37° B
Nx
m N Ny m
W
mg
F A F F F
=
Tsin37(AB) = mg(AB) + W
T( ) = 10(10) + 160
T = 300 N Ans
= 0
Nx = T cos37 = F
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7. = 300(4/5)
Nx = 240 N
= 0
Ny + T sin37 = mg + W =
Ny + 300 (3/5) = 10(10) + 160
Ny = 80 N
N =
=
N= 80 N Ans
F 5
F F F 60 kg 0.50 m 0.20 m F
F F F F F F
F
F 0.5 F
0.3
F
O
0.2 0.2
N 0.4
mg
F O
=
F(0.3) = mg(0.4)
F = mg(0.4)/0.3 = 60(10)(0.4/0.3)
F = 800 N Ans
F 6
F F 1m 2m 15 kg F F . 0.5 F = 60 N
F 1.5 m F F F
1.0 1.0 F
F
F F
2.0 1.5 2.0 1.5
f mg
N
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8. F F F F
= 0
f= F = F
µN = F
0.5(15)(10) = F
F = 75 N
F F F F 75 N F F
F F F F F F F
=
F(1.5) = mg(1/2)
F = mg(1/2)/1.5 = 15(10)(1/2)/1.5
F = 50 N
F F F F 50 N F F F
F = 60 N F F F F Ans
F 7
F F F F . F F
0.75 F F F F F F F
N F
a a F
mg sinθ
f mg mg cosθ
θ θ
θ F F F
= 0
f = mg sinθ
µN = mg sinθ
0.75 mg cosθ = mg sinθ
0.75 = tanθ , θ = 37°
F θ = 37° Ans
θ F F F F F F F
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9. =
mg sinθ (a/2) = mg cosθ (a/2)
tanθ = 1.0 , θ = 45°
F F θ = 45° Ans
8.5
F F
F F 3
mg
mg mg
N N N
. . F .
1.
F F FF F
. F F cm. N mg F F F
F F
2. F
F F F FF FF
. cm. F F F F
3.
F F
. cm. F F N mg F F F F
8.6 F
F F F F F
F F F F F
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10. F 8
F F F F F F F F F F
150 N F
F F
F 1 F
W
0.02 0.15
0.02 0.15
R
F
F F
=
W (0.02) = F (0.15)
150 (0.02) = F (0.15)
F = 20 N Ans
F F 20 N F F 150 N
¬ F F F F F F F
F 9
F 120 kg F F F 2m F F 240
F F F F F F
F
x 2-x F
mg
F F
=
mg (x) = F (2-x)
120(10) x = 240 (2-x)
x = 0.33 m Ans
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11. F 10
F F F F F 40 kg
F F
R= 1.0 =
r = 0.2 mg (0.2) = F (1.0)
F
40(10)(0.2) = F (1.0)
m F = 80 N Ans
mg
8.7 F
F
F F F F F F F
F F F F
C
F
B D
A L
F F
L0 ΔL
ΔL
O
F
1. F OA F F F ∆ L A F F
F ∆L A F F F A F F
∆L F
2. F AB F F F F F F FF
F B F F F F B F F
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12. 3. F B F F F F F F
C( ) F F F
F D F F
- F OB F F F F F
F
- F BD F F F
8.7.1. F
1. F F
2. F F
3. F F F
( F )
F
F F F F F
8.7.2. F
F (stress ) F F F F
F F F ( N/m2 )
A
F F
F =
A
F F
F
( strain ) F F
F F
L0 ΔL
F F
=
F F
ΔL
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13. 8.7.3.
E F F F F F
2
(N/m )
E=
- F E F
F 11
200 N F F F F 10 mm 1.0 m F 1.02 m
F F F
L0 =1.0 ΔL =0.02
F =200 F = 200
L =1.02
. F F A = d2 = 3.14(0.01)2 = 3.14x10-4
F = = = 6.37x105 N/m2 Ans
= = = 0.02 Ans
F E= = = 3.18x107 N/m2 Ans
F 12
50 kg 2m F 0.40 mm2 F 2x1010
F
T
m
mg
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14. T = mg = 50(10) = 500 N
= = = = 1.25x109 N/m2
= =
E= = = 2.0x1010 N/m2
ΔL = 0.125 m
L = L0 + ΔL = 0.125 + 2.0
L = 2.125 m Ans
F 13
F F F ( ) ( ) F
(E )
( x107 N/m2)
20
10
( x10-3)
0 2 4
E= =
E= = 5.0x1010 N/m2 Ans
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