Investigate whether the system will meet the requirements needed before the system run is made and provide an…

Question

Investigate whether the system will meet the requirements needed before the system run is made and provide an informed decision. Assess the safety of the passersby near the site.

ScanSolve AI Tutor · Accepted Answer

a) Kuchunguza kama mfumo utakidhi mahitaji kabla ya kuanza kazi.

Kwanza, tutachambua mfumo wa puli ili kupata kuongeza kasi (acceleration) na mvutano (tension) kwenye kamba.
Tuchukulie $g = 9.81 \, \text{m/s}^2$ .

Kutoka kwenye mchoro, tuna puli mbili. Puli ya juu imewekwa kwenye boriti ya dari na inashikilia Service Deck. Puli ya chini ni puli inayohamishika (movable pulley) na inashikilia uzito wa $50 \, \text{kg}$ . Uzito wa $20 \, \text{kg}$ umefungwa kwenye ncha nyingine ya kamba inayopita kwenye puli ya juu.

Hebu $m_1 = 50 \, \text{kg}$ na $m_2 = 20 \, \text{kg}$ .
Kwa puli inayohamishika iliyo na $m_1$ :
Kamba inashikilia uzito wa $m_1$ kwa pande mbili. Kwa hivyo, mvutano wa jumla unaoshikilia $m_1$ ni $2T$ , ambapo $T$ ni mvutano kwenye kamba.
Mlinganyo wa nguvu kwa $m_1$ (ukichukulia inashuka):

m_1 g - 2T = m_1 a_1

Ambapo $a_1$ ni kuongeza kasi ya $m_1$ .

Kwa uzito wa $m_2$ :
Mlinganyo wa nguvu kwa $m_2$ (ukichukulia inapanda):

T - m_2 g = m_2 a_2

Ambapo $a_2$ ni kuongeza kasi ya $m_2$ .

Uhusiano kati ya kuongeza kasi: Kwa puli inayohamishika, ikiwa $m_1$ inashuka kwa umbali $x$ , basi $m_2$ itapanda kwa umbali $2x$ . Hii inamaanisha $a_2 = 2a_1$ .
Hebu $a = a_1$ . Basi $a_2 = 2a$ .

Sasa tunaweza kuandika upya milinganyo:

$m_1 g - 2T = m_1 a$
$T - m_2 g = m_2 (2a)$

Kutoka mlinganyo (2), $T = m_2 (2a) + m_2 g = 2m_2 a + m_2 g$ .
Badilisha $T$ kwenye mlinganyo (1):

m_1 g - 2(2m_2 a + m_2 g) = m_1 a

m_1 g - 4m_2 a - 2m_2 g = m_1 a

m_1 g - 2m_2 g = m_1 a + 4m_2 a

(m_1 - 2m_2)g = (m_1 + 4m_2)a

a = \frac{(m_1 - 2m_2)g}{m_1 + 4m_2}

Sasa tuweke namba: $m_1 = 50 \, \text{kg}$ , $m_2 = 20 \, \text{kg}$ , $g = 9.81 \, \text{m/s}^2$ .

Kokotoa kuongeza kasi $a$ ya uzito wa $50 \, \text{kg}$ .
$a = \frac{(50 \, \text{kg} - 2 \times 20 \, \text{kg}) \times 9.81 \, \text{m/s}^2}{50 \, \text{kg} + 4 \times 20 \, \text{kg}}$ $a = \frac{(50 - 40) \times 9.81}{50 + 80}$ $a = \frac{10 \times 9.81}{130}$ $a = \frac{98.1}{130}$ $a \approx 0.7546 \, \text{m/s}^2$
Kuongeza kasi ya uzito wa $50 \, \text{kg}$ ni $a_1 = 0.7546 \, \text{m/s}^2$ .
Kuongeza kasi ya uzito wa $20 \, \text{kg}$ ni $a_2 = 2a_1 = 2 \times 0.7546 \, \text{m/s}^2 = 1.5092 \, \text{m/s}^2$ .
Angalia mahitaji ya mfumo.
i) Kuongeza kasi haipaswi kuzidi $2 \, \text{m/s}^2$ .
Kuongeza kasi kubwa zaidi katika mfumo ni $a_2 = 1.5092 \, \text{m/s}^2$ .
Kwa kuwa $1.5092 \, \text{m/s}^2 < 2 \, \text{m/s}^2$ , mahitaji haya yametimizwa.

ii) Kamba imeundwa kuhimili mvutano wa juu wa $300 \, \text{N}$ .
Kokotoa mvutano $T$ kwenye kamba:
$T = 2m_2 a + m_2 g$ $T = 2 \times 20 \, \text{kg} \times 1.5092 \, \text{m/s}^2 + 20 \, \text{kg} \times 9.81 \, \text{m/s}^2$ $T = 40 \times 1.5092 + 196.2$ $T = 60.368 + 196.2$ $T = 256.568 \, \text{N}$
Kwa kuwa $256.568 \, \text{N} < 300 \, \text{N}$ , mahitaji haya yametimizwa.

iii) Umbali ambao puli inayohamishika itasafiri baada ya sekunde 5.
Puli inayohamishika inashuka na uzito wa $50 \, \text{kg}$ . Kuongeza kasi yake ni $a_1 = 0.7546 \, \text{m/s}^2$ .
Tumia mlinganyo wa kinematiki: $s = ut + \frac{1}{2}at^2$ .
Kutoka kwenye mapumziko, $u = 0$ .
$s = 0 \times 5 \, \text{s} + \frac{1}{2} \times 0.7546 \, \text{m/s}^2 \times (5 \, \text{s})^2$ $s = \frac{1}{2} \times 0.7546 \times 25$ $s = 0.5 \times 0.7546 \times 25$ $s = 9.4325 \, \text{m}$
Umbali ambao puli inayohamishika itasafiri baada ya sekunde 5 ni $9.4325 \, \text{m}$ .

Hitimisho la a):
Mfumo unakidhi mahitaji yote yaliyotajwa. Kuongeza kasi ya juu ni $1.5092 \, \text{m/s}^2$ (chini ya $2 \, \text{m/s}^2$ ), na mvutano wa kamba ni $256.568 \, \text{N}$ (chini ya $300 \, \text{N}$ ). Puli inayohamishika itasafiri $9.4325 \, \text{m}$ baada ya sekunde 5.

b) Kutathmini usalama wa wapita njia karibu na eneo hilo.

Kuna wasiwasi kuhusu sanduku la zana kuteleza kutoka kwenye Service Deck.
Data iliyotolewa:
Kasi ya usawa ya sanduku la zana, $v_x = 2 \, \text{m/s}$ .
Urefu kutoka ardhini, $h = 20 \, \text{m}$ .
Eneo salama la ardhini lililowekwa na mhandisi ni $5 \, \text{m}$ kutoka kwenye Service Deck.
Kokotoa muda unaochukua sanduku la zana kufika ardhini.
Tumia mlinganyo wa mwendo wima: $h = v_{oy}t + \frac{1}{2}gt^2$ .
Kwa kuwa sanduku la zana linateleza kwa usawa, kasi yake ya awali wima ni $v_{oy} = 0$ .
$20 \, \text{m} = 0 \times t + \frac{1}{2} \times 9.81 \, \text{m/s}^2 \times t^2$ $20 = 4.905 t^2$ $t^2 = \frac{20}{4.905}$ $t^2 \approx 4.0775$ $t = \sqrt{4.0775}$ $t \approx 2.019 \, \text{s}$
Kokotoa umbali wa usawa ambao sanduku la zana litasafiri kabla ya kugonga ardhini.
Tumia mlinganyo wa mwendo wa usawa: $x = v_x t$ .
$x = 2 \, \text{m/s} \times 2.019 \, \text{s}$ $x = 4.038 \, \text{m}$
Hitimisho la b):
Sanduku la zana litaanguka umbali wa usawa wa $4.038 \, \text{m}$ kutoka kwenye Service Deck. Eneo salama la ardhini lililowekwa na mhandisi ni $5 \, \text{m}$ .
Kwa kuwa $4.038 \, \text{m} < 5 \, \text{m}$ , sanduku la zana litaanguka ndani ya eneo salama.
Hivyo, kwa kuzingatia sanduku la zana, wapita njia wako salama kwani eneo la kuanguka liko ndani ya eneo salama lililowekwa.

Majibu ya mwisho:

a) Mfumo unakidhi mahitaji yote. Kuongeza kasi ya juu ni $1.5092 \, \text{m/s}^2$ (chini ya $2 \, \text{m/s}^2$ ), na mvutano wa kamba ni $256.568 \, \text{N}$ (chini ya $300 \, \text{N}$ ). Puli inayohamishika itasafiri $9.4325 \, \text{m}$ baada ya sekunde 5.

b) Wapita njia wako salama kwani sanduku la zana litaanguka $4.038 \, \text{m}$ kutoka kwenye Service Deck, ambayo iko ndani ya eneo salama la $5 \, \text{m}$ .

Investigate whether the system will meet the requirements needed before the system run is made and provide an informed decision. Assess the safety of the passersby near the site.

Step-by-step solution

Kokotoa kuongeza kasi $a$ ya uzito wa $50 \, \text{kg}$ .

Angalia mahitaji ya mfumo.

Kokotoa muda unaochukua sanduku la zana kufika ardhini.

Kokotoa umbali wa usawa ambao sanduku la zana litasafiri kabla ya kugonga ardhini.

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Investigate whether the system will meet the requirements needed before the system run is made and provide an informed decision. Assess the safety of the passersby near the site.

Step-by-step solution

Kokotoa kuongeza kasi aaa ya uzito wa 50 kg50 \, \text{kg}50kg.

Angalia mahitaji ya mfumo.

Kokotoa muda unaochukua sanduku la zana kufika ardhini.

Kokotoa umbali wa usawa ambao sanduku la zana litasafiri kabla ya kugonga ardhini.

Still stuck on this one?

More Physics Questions

Kokotoa kuongeza kasi $a$ ya uzito wa $50 \, \text{kg}$ .