IAP Exercises Chapter 3 – Answers
3.1 Water tank
(a) ℎ(𝑡) =
̂
𝐴𝑉
𝐴0 𝜔
[1 − cos(𝜔𝑡)] + ℎ0
3.2 Leaking water tank 1
ℎ(𝑡) = ℎ0 −
𝐴𝑈0
𝑡
𝐴0
3.3 Leaking water tank 2
2
ℎ(𝑡) = [√ℎ0 −
𝐴
𝐴0
𝑔𝑡
√ 𝑡]
2
3.6 Order of a differential equation
(a) First order
(b) Second order
(c) Second order
(d) Fourth order
(e) Two coupled first order differential equations gives a second-order
differential equation →
𝑑2𝑥
𝑑𝑡 2
+𝑥 =0
(f) Three coupled first order differential equations gives a third-order
differential equation →
3.7 Linear versus non-linear
(a) Linear
(b) Non-linear
(c) Non-linear
(d) Linear
(e) Linear
(f) Non-linear
𝑑3𝑥
𝑑𝑡 3
+2
𝑑𝑥
𝑑𝑡
+ 2𝑥 = 0
(g) Non-linear
(h) Non-linear
(i) Non-linear
1
1
2
2
(j) Linear → 𝑥(𝑡) = 𝑒 𝐶 exp (− 𝑡 2 ) = 𝑥0 exp⁡(− 𝑡 2 )
(k) Linear
(l) Non-linear → 𝜃̈ + sin(𝜃) = 0
(m) Non-linear → 𝑥̈ + 𝑥 + (𝑥 2 − 1)𝑥̇ = 0
3.8 Separation of variables
𝛼
− 𝑡
(a) 𝑇̅(𝑡) = 𝑇0 𝑒 𝐶𝑤
(b) 𝑥(𝑡) = 𝑒 𝑡
(c) 𝑁(𝑡) = 𝑁0 𝑒 𝛼𝑡
(d) 𝑣(𝑡) = ⁡ −
1
1+𝛽𝑡
3.9 Drag
(a)
𝑑𝑉
𝑑𝑡
𝐷
= − 𝑉2
𝑀
(b) 𝑇⁡~⁡
𝑀
𝐷𝑉0
𝑉
(c) 𝑉(𝑡) = 𝐷𝑉00
𝑀
𝑡+1
3.10 Decelerating bike
(a) 𝑀
𝑑𝑉
𝑑𝑡
1
= ⁡ −𝑘𝑉 2
1
3
(b) 𝑘⁡⁡ [𝑘𝑔⁡𝑀2 ⁡𝑠 −2 ]
(c) 𝑉(𝑡) = (√𝑉0 −
𝑘
2𝑀
3.11 Railway turntable
(a) 𝑘 ∶ ⁡ [𝑘𝑔⁡𝑚−1 ⁡𝑠 −2 ]
(c) 𝜃̇ = 𝜔
2
𝑡)
3.12 Kirchhoff 1
(a) 𝐼1 = 2.0⁡A
(b) 𝐼2 = −3.0⁡A
3.13 Kirchhoff 3
(a) 𝐼1 = 1.38⁡A
(b) 𝑄 = 𝐶⁡∆𝑉𝑐𝑎𝑝 = 66.0⁡𝜇C
3.14 Kirchhoff 3
(a) 𝑉 − 𝐼1 𝑅1 − 𝐼2 𝑅2 = 0
(b) 𝐼1 =
𝑉
𝑅1
(c) 𝐼1 = 𝐼𝑣 + 𝐼2 + 𝐼3
𝑅2 𝑅3 +𝑅𝑣 𝑅3 +𝑅𝑣 𝑅2
(d) 𝐼1 = 𝑉 (
𝑅𝑣 𝑅2 𝑅3 +𝑅1 𝑅2 𝑅3 +⁡𝑅1 𝑅𝑣 𝑅3 +⁡𝑅1 𝑅2 𝑅𝑣
)
(e) 𝑅𝑣 = 𝑅1 = 𝑅2 = 𝑅3
3.15 Charging a capacitor
(a)
𝑑𝑞
𝑑𝑡
=−
1
𝑅𝐶
(𝑞 − 𝐶𝜉)
𝑡
(b) 𝑞(𝑡) = 𝐶𝜉 (1 − 𝑒 −𝑅𝐶 )
(d) For example: after 𝑡 = 3𝑅𝐶 → 𝑞(𝑡 = 3𝑅𝐶) ≅ 0.95⁡𝐶𝜉
3.16 Langmuir model
(a) 𝑁⁡[𝑚−2 ]⁡, 𝜃⁡[−]⁡, 𝛤⁡[𝑚−2 𝑠 −1 ]⁡
1
(b) 𝜃(𝑡 + ∆𝑡) − 𝜃(𝑡) = 𝛤(1 − 𝜃)∆𝑡
𝑁
(c)
𝑑𝜃
𝑑𝑡
𝛤
= (1 − 𝜃)⁡
𝑁
𝛤
(d) 𝜃(𝑡) = 1 − 𝑒 −𝑁𝑡
(e)
𝑑𝜃
𝑑𝑡
𝛤
= (1 − 𝜃) − 𝑘𝜃⁡⁡
𝑁
3.17 Solar sail
(b) 𝑝(𝑡 + ∆𝑡) − 𝑝(𝑡) = ⁡𝜂𝑝𝑝 · 𝛤𝐴𝛥𝑡⁡
(c) 𝐹𝑠 = ⁡
𝑑𝑝
𝑑𝑡
(d) 𝐹𝑠 = ⁡𝑚
= 𝜂𝑝𝑝 𝛤𝐴⁡
𝑑2𝑥
𝑑𝑡 2
= 𝜂𝑝𝑝 𝛤𝐴
(e) 𝑥(𝑡) = 𝑥0 +
𝜂𝑝𝑝 𝛤𝐴 2
𝑡
2𝑚
3.18 Characteristic timescales
𝐶
(a) 𝜏 = 𝑤
𝛼
(b) 𝜏⁡~⁡𝑅𝐶
𝑚
(c) 𝜏⁡~ ⁡
𝛼
(d)
𝑑𝑁
𝑑𝑡
= 𝐶𝑁 with 𝐶 > 0.
𝑡 = ⁡𝜏 ln(2)