דינמיקה
✦ בנוי על ידי פהד גאנם ✦
العربية
☀ מצב בהיר
2.1
▾
✕
F = k·x | x =
mg
k
50 N/m
1.0 kg
0.00
0.00
0.00
0.00
0.00
0.00
m = 200 g = 0.200 kg | x = 4.0 cm = 0.040 m
mg = 0.200 · 9.8 = 1.96 N
k =
mg
x
=
1.96
0.040
= 49 N/m
m₂ = 500 g = 0.500 kg → m₂g = 4.9 N
x₂ =
m₂g
k
=
4.9
49
= 0.100 m
x₂ = 0.100 m = 10.0 cm
2.2
▾
✕
ΣFx = F₁·cosθ₁ + F₂·cosθ₂ | ΣFy = F₁·sinθ₁ + F₂·sinθ₂ | |ΣF| = √(ΣFx² + ΣFy²)
30 N
0°
40 N
90°
0.0
0.0
0.0
0.0
0.0
0.00
0.0
0.0
ΣFx = 6.0 N ΣFy = 8.0 N
|ΣF| = √(6.0² + 8.0²) = √(36 + 64) = √100 = 10.0 N
tanθ =
ΣFy
ΣFx
=
8.0
6.0
= 1.333 → θ = 53.1°
2.5
▾
✕
N = mg − F·sinθ | a =
F·cosθ − μN
m
10.0 kg
0.40
30°
0 N
0
0
0
F·cosθ
0
F·sinθ
0
0
0.00
0.00
0
F·cos30° = 0.866·F | F·sin30° = 0.500·F | mg = 10·9.8 = 98 N
N + F·sin30° = mg → N = 98 − 0.500·F
F·cos30° = μN = 0.40 · (98 − 0.500·F)
0.866·F = 39.2 − 0.200·F → 1.066·F = 39.2
F =
μmg
cosθ + μ·sinθ
=
39.2
1.066
= 36.8 N
2.4
▾
✕
N = mg | f
s
≤ μ
s
·N | f
k
= μ
k
·N | a =
F − μ
k
·N
m
5.0 kg
0.50
0.30
0 N
0
0
0
0
0
0
0.00
0.00
0
N = mg = 5.0 · 9.8 = 49 N
f_s,max = μs·N = 0.50 · 49 = 24.5 N
20 N < 24.5 N → f = 20 N , a = 0
f_k = μk·N = 0.30 · 49 = 14.7 N
a =
F − f_k
m
=
30 − 14.7
5.0
= 3.06 m/s²
2.6
▾
✕
ΣF = m·a | N = mg | a =
F − μmg
m
12 N
3.0 kg
0.20
0
0
0
0.00
0.00
0.00
0.00
N = mg = 3.0 · 9.8 = 29.4 N
f = μN = 0.20 · 29.4 = 5.88 N
ΣF = F − f = 12 − 5.88 = 6.12 N
a =
ΣF
m
=
6.12
3.0
= 2.04 m/s²
v = v₀ + a·t = 0 + 2.04 · 4.0 = 8.16 m/s
2.8
▾
✕
N = mg·cosα | a =
F − mg·sinα − f
m
30°
2.0 kg
0.30
0 N
0
mg·sinα
0
mg·cosα
0
0
0
0
0.00
0.00
0
mg·sin30° = 2·9.8·0.500 = 9.8 N
mg·cos30° = 2·9.8·0.866 = 16.97 N
N = 16.97 N → f_max = μN = 0.30 · 16.97 = 5.09 N
9.8 N > 5.09 N
a =
mg·sinα − μN
m
=
9.8 − 5.09
2.0
= 2.36 m/s²
2.8
▾
✕
a =
m₂g − m₁g·sinα − μ·m₁g·cosα
m₁ + m₂
| T = m₂( g − a )
30°
2.0 kg
3.0 kg
0.20
0
0
0
0
m₁g·sinα
0
m₂g
0
0.00
0.00
0
m₁g·sin30° = 2·9.8·0.500 = 9.8 N
N = m₁g·cos30° = 2·9.8·0.866 = 16.97 N
f_max = μN = 0.20 · 16.97 = 3.39 N
m₂g − T = m₂·a → 29.4 − T = 3a
29.4 − 9.8 = 19.6 N > 3.39 N
T − m₁g·sinα − μN = m₁·a → T − 9.8 − 3.39 = 2a
29.4 − 9.8 − 3.39 = (2 + 3)·a
a =
m₂g − m₁g·sinα − μN
m₁ + m₂
=
29.4 − 9.8 − 3.39
2 + 3
= 3.24 m/s²
T = m₂(g − a) = 3·(9.8 − 3.24) = 19.7 N
2.8
▾
✕
N − mg = ma → N = m(g + a)
70 kg
2.0 m/s²
0
N
g
0
0
0
0
0.00
0.0
0.0
mg = 60 · 9.8 = 588 N
↑ = (+) | ↓ = (−)
N − mg = ma → N = m(g + a)
N₁ = 60 · (9.8 + 2.0) = 60 · 11.8 = 708 N
N₂ = 60 · (9.8 − 2.0) = 60 · 7.8 = 468 N
708
9.8
= 72.2 kg |
468
9.8
= 47.8 kg
2.9
▾
✕
x = v₀·cosθ·t | y = h₀ + v₀·sinθ·t −
1
2
g·t²
20 m/s
30°
0 m
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
v₀ₓ = 20·cos30° = 20·0.866 = 17.32 m/s
v₀ᵧ = 20·sin30° = 20·0.500 = 10.00 m/s
h_max =
v₀ᵧ²
2g
=
100
2·9.8
= 5.10 m
t =
2·v₀ᵧ
g
=
20
9.8
= 2.041 s
R = v₀ₓ · t = 17.32 · 2.041 = 35.35 m
2.10
▾
✕
a =
v²
r
| F =
m·v²
r
| T =
2πr
v
0.5 kg
1.2 m
3.0 m/s
0.00
0.00
0.00
0.00
0.000
m = 0.50 kg | r = 1.2 m | v = 3.0 m/s
a =
v²
r
=
(3.0)²
1.2
= 7.5 m/s²
F = m·a = 0.50 · 7.5 = 3.75 N
T =
2πr
v
=
2π · 1.2
3.0
= 2.51 s
→
🏠
♿