41.
a) Gt₁-T-Ff₁=m₁a<=>m₁gsinα-T-Ff₁=m₁a
N₁-Gn₁=0=>N₁=Gn₁=m₁gcosα
Ff₁=μ₁N₁=μ₁m₁gcosα
m₁gsinα-T-μ₁m₁gcosα=m₁a (1)
Gt₂+T-Ff₂=m₂a<=>m₂gsinα+T-Ff₂=m₂a
N₂-Gn₂=0=>N₂=Gn₂=m₂gcosα
Ff₂=μ₂N₂=μ₂m₂gcosα
m₂gsinα+T-μ₂m₂gcosα=m₂a (2)
(1)+(2)=>a(m₁+m₂)=gsinα(m₁+m₂)-gcosα(μ₁m₁+μ₂m₂) |:(m₁+m₂)=>a=gsinα-gcosα(μ₁m₁+μ₂m₂)/(m₁+m₂)=10*0,5-10*0,866(4*0,28+8*0,11)/(4+8)=5-8,66*2/12=5-1,44=3,56 m/s²
b) (2)=>T=m₂a-m₂gsinα+μ₂m₂gcosα=8*3,56-8*10*0,5+0,11*8*10*0,866=28,48-40+7,62=36,1-40=-3,9 N
c) Ff₁=Ff₂=0
m₁gsinα-T'=m₁a'
m₂gsinα+T'=m₂a'
___________"+"
a'(m₁+m₂)=sinαg(m₁+m₂)|:(m₁+m₂)=>a'=gsinα=10*0,5=5 m/s²
T'=m₂a'-m₂gsinα=8*5-8*10*0,5=40-40=0
42.
a) pentru M
Gt-T-Ff=Ma
N-Gn=0=>N=Mgcosα
Ff=μN=μMgcosα
Mgsinα-T-μMgcosα=Ma
pentru m
T-mg=ma=>T=m(a+g)
Mgsinα-T-μMgcosα=Ma <=>Mgsinα-ma-mg-μMgcosα=Ma <=>a(M+m)=Mgsinα-mg-μMgcosα<=>a(3+1)=3*10*0,5-10-0,11*3*10*0,866<=>4a=15-10-2,85<=>4a=2,15=>a=2,15/4=0,53 m/s²
b) T=m(a+g)=1(0,53+10)=10,53 N
c) R²=T²+T²+2T²cos(90-α)=2T²+2T²cos60°=2*10,53²+2*10,53²*0,5=221,76+110,88=332,64=>R=±√332,64=±18,1 N
R>0=>T=18,1 N
43.
a) pentru M
-Gt+T-Ff=Ma
N-Gn=0=>N=Mgcosα
Ff=μN=μMgcosα
-Mgsinα+T-μMgcosα=Ma
pentru m
-T+mg=ma=>T=m(g-a)
-Mgsinα+T-μMgcosα=Ma <=>-Mgsinα-ma+mg-μMgcosα=Ma <=>a(M+m)=-Mgsinα+mg-μMgcosα<=>a(1+5)=-1*10*0,866+5*10-0,2*1*10*0,5<=>6a=-8,66+50-1<=>6a=40,34=>a=40,34/6=6,72 m/s²
b) T=m(g-a)=5(10-6,72)=5*3,28=16,4 N
c) a=Δv/Δt=v/t=>v=at
vm=(v+v₀)/2=v/2=at/2
d=vm*t=at²/2=6,72*2²/2=6,72*2=13,44 m
d) a=0
pentru M
-Gt+T-Ff=0
N-Gn=0=>N=Mgcosα
Ff=μN=μMgcosα
-Mgsinα+T-μMgcosα=0
pentru m
-T+mg=0=>T=mg
-Mgsinα+T-μMgcosα=0<=>-Mgsinα+mg-μMgcosα=0<=>-1*10*0,866+10m-0,2*1*10*0,5=0<=>-8,66+10m-1=0=>10m=9,66=>m=9,66/10=0,966 kg
44. sinα=h/l=6/10=0,6
sin²α+cos²α=1=>cos²α=1-0,6²=1-0,36=0,64=>cosα=±√0,64=±0,8
α∈C1=>sinα>0 si cosα>0=>cosα=0,8
a) pentru m₁
T₁=m₁g
pentru m
Gt-T₁-Ff=0=>mgsinα-m₁g-Ff=0
N-Gn=0=>N=mgcosα
Ff=μN=μmgcosα
mgsinα-m₁g-Ff=0<=>mgsinα-m₁g-μmgcosα=0 |:g=>0,6m-10-0,8μm=0 (1)
pentru m₂
T₂=m₂g
pentru m
-Gt+T₂-Ff=0=>-mgsinα+m₂g-Ff=0
N-Gn=0=>N=mgcosα
Ff=μN=μmgcosα
-mgsinα+m₂g-Ff=0<=>-mgsinα+m₂g-μmgcosα=0|:g=>-0,6m+20-0,8μm=0 (2)
(1)-(2)=>1,2m-30=0=>m=30/1,2=25 kg
b) (1)+(2)=>10-1,6μ*25=0<=>10=40μ=>μ=1/4=0,25
c) T₁'=m₁g/2
Gt-Ff-T'₁=ma<=>mgsinα-m₁g/2-Ff=ma
N-Gn=0=>N=mgcosα
Ff=μN=μmgcosα
mgsinα-m₁g/2-Ff=ma<=>mgsinα-m₁g/2-μmgcosα=ma|:m=>a=10*0,6-10*10/25*2-0,25*10*0,8=6-2-2=2 m/s²
