אנרגיה מכנית ושימורה — פרק 4

W = F·d·cosθ | E_k = m·v²2 | ΣW = ΔE_k

10.0 kg

50 N

30°

0.20

8.0 m

0

0.0

0.00

0.0

F·cos30° = 50 · 0.866 = 43.30 N F·sin30° = 50 · 0.500 = 25.00 N

N = mg − F·sin30° = 98.0 − 25.0 = 73.0 N f = μN = 0.20 · 73.0 = 14.60 N

W(F) = F·cos30°·d = 43.30 · 8.0 = 346.4 J W(f) = −f·d = −14.60 · 8.0 = −116.8 J W(mg) = 0 ; W(N) = 0 ΣW = 346.4 − 116.8 = 229.6 J

ΣW = m·v²2 − 0 v² = 2·ΣWm = 2 · 229.610.0 = 45.92 m²/s² v = √45.92 = 6.78 m/s

Ug = m·g·h | Ue = k·x²2 | Wמשמר = −ΔU

2.0 kg

1.0 m

4.0 m

0.0 m

0.00

U₁ = m·g·h₁ = 2.0 · 9.8 · 1.0 = 19.6 J U₂ = m·g·h₂ = 2.0 · 9.8 · 4.0 = 78.4 J

ΔU = U₂ − U₁ = 78.4 − 19.6 = 58.8 J W = −ΔU = −58.8 J h₂ − h₁ = ΔUm·g = 58.82.0 · 9.8 = 3.0 m

U₁ = 2.0 · 9.8 · (1.0 − 4.0) = −58.8 J U₂ = 2.0 · 9.8 · (4.0 − 4.0) = 0 J ΔU = 0 − (−58.8) = 58.8 J

x = 15 cm = 0.15 m U = k·x²2 = 200 · 0.15²2 = 4.52 = 2.25 J

E = mgh + mv²2 | Ep + Ek + Q = E₀ | v = √2Ekm

3.00 m

1.0 kg

0.00

0.0

0.00

Estart = mgh + 0 = mgh Eend = 0 + mv²2

mgh = mv²2

v = √2mghm = √2gh

v = √2 · 9.8 · 5.0 = √98 = 9.9 m/s

T = m·v²R + mg·cosθ | vtop,min = √(g·R) | vbot,min = √(5·g·R)

2.0 m

1.0 kg

8.0 m/s

0.00

0.0

0.00

0.0

0.00

mg = m·v²topR

12m·v²bot = 12m·v²top + m·g·(2R)

v²top = g·R → v²bot = g·R + 4g·R = 5g·R

vtop = √(9.8 · 2.0) = √19.6 = 4.43 m/s vbot = √(5 · 9.8 · 2.0) = √98 = 9.90 m/s

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂ | ΔEk = Ek,i − Ek,f | ΔEkEk,i· 100 % | e = v₂ − v₁u₁ − u₂

2.0 kg

+6.0 m/s

4.0 kg

+0.0 m/s

0.00

0.00

0.0

0.00

p = m₁u₁ + m₂u₂ = 2.0·6.0 + 4.0·0 = 12.0 kg·m/s Ek,i = ½m₁u₁² + ½m₂u₂² = ½·2.0·6.0² = 36.0 J

v = m₁u₁ + m₂u₂m₁ + m₂ = 2.0·6.0 + 4.0·02.0 + 4.0 = 12.06.0 = 2.0 m/s

Ek,f = ½(m₁ + m₂)v² = ½·6.0·2.0² = 12.0 J ΔEk = Ek,i − Ek,f = 36.0 − 12.0 = 24.0 J

ΔEkEk,i· 100 = 24.036.0· 100 = 66.7 %

P = WΔt = F·v | η = PusePin · 100%

800 kg

1.5 m/s

14000 W

0

F = mg = 800 · 9.8 = 7840 N

Puse = WΔt = F·v = 7840 · 1.5 = 11760 W

η = PusePin = 1176014000 = 0.84 = 84%