(Yohanes Surya bagian B 1.39) / I.E Irodov 1.73

Pada sistem dibawah ini hitung percepatan benda {{m}_{1}}. Anggap benda {{m}_{2}} bergerak ke bawah. Percepatan gravitasi g.

ys1.39

Keterangan:

a= percepatan benda 1 dan 2 terhadap katrol

{{a}_{0}}= percepatan katrol turun ke bawah terhadap tanah / percepatan benda 0 ke kanan

{{a}_{1}}= percepatan benda 1 terhadap tanah

{{a}_{2}}= percepatan benda 2 terhadap tanah

 

Aturan tanda besaran vektor:

Ke atas negatif

Ke bawah positif

 

Analisis benda 0

ys1.39a

T={{m}_{0}}{{a}_{0}}~~\ldots \left( 1 \right)

 

Analisis benda 1

ys1.39b

{{m}_{1}}g-{{T}_{1}}={{m}_{1}}(-{{a}_{1}})~~\ldots \left( 2 \right)

-{{a}_{1}}=-a+{{a}_{0}}~~\ldots \left( 3 \right)

 

Analisis benda 2

ys1.39c

{{m}_{2}}g-{{T}_{1}}={{m}_{2}}{{a}_{2}}~~\ldots \left( 4 \right)

{{a}_{2}}=a+{{a}_{0}}~~\ldots \left( 5 \right)

 

Analisis katrol

ys1.39d

T=2{{T}_{1}}~~\ldots \left( 6 \right)

Solusi:

Variabel yang tidak diketahui ada 6 yaitu: {{T}_{1}},~T,~a,{{a}_{0}},~{{a}_{1}},~{{a}_{2}} di mana {{a}_{1}} adalah variabel yang ingin kita tentukan. Selesaikan keenam persamaan di atas:

{{m}_{1}}g-{{T}_{1}}={{m}_{1}}\left( {-{{a}_{1}}} \right)~~\ldots \left( 2 \right)

{{m}_{1}}g-\left( {{{m}_{2}}g-{{m}_{2}}{{a}_{2}}} \right)={{m}_{1}}\left( {-{{a}_{1}}} \right)~~~\leftarrow \left( 4 \right)

{{m}_{1}}g-{{m}_{2}}g+{{m}_{2}}{{a}_{2}}={{m}_{1}}\left( {-{{a}_{1}}} \right)

{{m}_{1}}g-{{m}_{2}}g+{{m}_{2}}\left( {a+{{a}_{0}}} \right)={{m}_{1}}\left( {-{{a}_{1}}} \right)~~~\leftarrow \left( 5 \right)

{{m}_{1}}g-{{m}_{2}}g+{{m}_{2}}a+{{m}_{2}}{{a}_{0}}={{m}_{1}}\left( {-{{a}_{1}}} \right)

{{m}_{1}}g-{{m}_{2}}g+{{m}_{2}}\left( {{{a}_{1}}+{{a}_{0}}} \right)+{{m}_{2}}{{a}_{0}}={{m}_{1}}\left( {-{{a}_{1}}} \right)~~~\leftarrow \left( 3 \right)

{{m}_{1}}g-{{m}_{2}}g+{{m}_{2}}{{a}_{1}}+{{m}_{2}}{{a}_{0}}+{{m}_{2}}{{a}_{0}}={{m}_{1}}\left( {-{{a}_{1}}} \right)

{{m}_{1}}g-{{m}_{2}}g+2{{m}_{2}}{{a}_{0}}={{m}_{1}}\left( {-{{a}_{1}}} \right)-{{m}_{2}}{{a}_{1}}

{{m}_{1}}g-{{m}_{2}}g+2{{m}_{2}}\left( {\frac{T}{{{{m}_{0}}}}} \right)={{m}_{1}}\left( {-{{a}_{1}}} \right)-{{m}_{2}}{{a}_{1}}~~\ldots \leftarrow \left( 1 \right)

{{m}_{1}}g-{{m}_{2}}g+2{{m}_{2}}\left( {\frac{{2{{T}_{1}}}}{{{{m}_{0}}}}} \right)={{m}_{1}}\left( {-{{a}_{1}}} \right)-{{m}_{2}}{{a}_{1}}~~~\leftarrow \left( 6 \right)

{{m}_{1}}g-{{m}_{2}}g+\frac{{4{{m}_{2}}}}{{{{m}_{0}}}}\left( {{{m}_{1}}g+{{m}_{1}}{{a}_{1}}} \right)={{m}_{1}}\left( {-{{a}_{1}}} \right)-{{m}_{2}}{{a}_{1}}~~~\leftarrow \left( 2 \right)

{{m}_{1}}g-{{m}_{2}}g+\frac{{4{{m}_{1}}{{m}_{2}}g}}{{{{m}_{0}}}}+\frac{{4{{m}_{1}}{{m}_{2}}{{a}_{1}}}}{{{{m}_{0}}}}={{m}_{1}}\left( {-{{a}_{1}}} \right)-{{m}_{2}}{{a}_{1}}

{{m}_{1}}g-{{m}_{2}}g+\frac{{4{{m}_{1}}{{m}_{2}}g}}{{{{m}_{0}}}}={{m}_{1}}\left( {-{{a}_{1}}} \right)-{{m}_{2}}{{a}_{1}}-\frac{{4{{m}_{1}}{{m}_{2}}{{a}_{1}}}}{{{{m}_{0}}}}

{{m}_{1}}g-{{m}_{2}}g+\frac{{4{{m}_{1}}{{m}_{2}}g}}{{{{m}_{0}}}}={{m}_{1}}\left( {-{{a}_{1}}} \right)-{{m}_{2}}{{a}_{1}}-\frac{{4{{m}_{1}}{{m}_{2}}{{a}_{1}}}}{{{{m}_{0}}}}~~

{{m}_{0}}{{m}_{1}}g-{{m}_{0}}{{m}_{2}}g+4{{m}_{1}}{{m}_{2}}g=-{{m}_{0}}{{m}_{1}}{{a}_{1}}-{{m}_{0}}{{m}_{2}}{{a}_{1}}-4{{m}_{1}}{{m}_{2}}{{a}_{1}}~~\ldots \times {{m}_{0}}

\left( {{{m}_{0}}{{m}_{1}}-{{m}_{0}}{{m}_{2}}+4{{m}_{1}}{{m}_{2}}} \right)g=-{{a}_{1}}\left( {{{m}_{0}}{{m}_{1}}+{{m}_{0}}{{m}_{2}}+4{{m}_{1}}{{m}_{2}}} \right)

-{{a}_{1}}=\frac{{{{m}_{0}}{{m}_{1}}-{{m}_{0}}{{m}_{2}}+4{{m}_{1}}{{m}_{2}}}}{{{{m}_{0}}{{m}_{1}}+{{m}_{0}}{{m}_{2}}+4{{m}_{1}}{{m}_{2}}}}\times g

-{{a}_{1}}=\frac{{({{m}_{1}}-{{m}_{2}}){{m}_{0}}+4{{m}_{1}}{{m}_{2}}}}{{({{m}_{1}}+{{m}_{2}}){{m}_{0}}+4{{m}_{1}}{{m}_{2}}}}\times g

{{a}_{1}}=\frac{{({{m}_{2}}-{{m}_{1}}){{m}_{0}}-4{{m}_{1}}{{m}_{2}}}}{{({{m}_{1}}+{{m}_{2}}){{m}_{0}}+4{{m}_{1}}{{m}_{2}}}}\times g

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