TD 4 Energie mécanique

M= 200 tonnes
F F = 60 103N
ℓ υ = 57.6km/h
υ= 57.6km/h =⇒υ=57.6
3.6= 16 m/s
Ec=1
2Mυ2=F ℓ⇒ℓ=MV 2 2F=200000 ×(16)2 2×60000 = 426.7m
a
υ t a
a=F
M=60 200 = 0,3ms−2
υ=at t ⇒t=υ
a=16 0.3≃53s
m= 100 kg
R= 1 m ω = 600 tours/mn
EC=1
2Jω2
J=1
2mR2ω=600 ×2π 60 rad/s
EC=1
4mR2ω2=1 4×100 ×(1)2×(10 ×2π)2= 98696 Joules
θ= 30
υ= 1 m/s
ℓ
g= 9,8m/s2
Em=EC+Ep
Em=EC=1
2mυ2
Em=Ep=mgh ℓ =h
sinθ θ= 30
⇒1
2mυ2=mgℓsinθ ℓ =mυ2 2mgsinθ =1 2×9.8×0.5= 0.1m
M= 1960 kg
R= 60 cm
J=1
2mR2= 0.5×1960 ×(0.6)2= 352.8kgm2

t= 0s
ω1= 300 tours/minute
W=ECf −ECi
W1=1
2Jω2
1= 0.5×352.8×(300 ×2π
60 )2= 1.74 105J
ω2= 297 tours/min
W2=1
2×352.8×(2π 60 )2
×[(300)2−(297)2]= 3.46 103J
△t= 4s
⟨P⟩=W2
△t= 866 W att
M= 50tonnes
R= 3 m.
J=1
2MR2=⇒J=1 2×50 103×(3)2= 2.25 105kgm2
ω= 900 tours/min
Ec,r =1
2Jω2ω ω =900 ×2π 60 = 94rads−1
Ec,r = 109J
P= 2 105W atts
ω= 900tours/min
△
P P =M∆ω M∆=2 105
94 = 2122 J M∆
∆
0 900 tours/minute
α
α=dω
dt =⇒dω =αdt α t = 0 ω= 0 ω=αt
α M∆= 2F R
F=Mγ M
γ=F
M=M∆ 2RM γ=Rα α =2122 2×3×50000 ×3= 2.36 10−3rads−2
t=ω
α= 40000s