Misali na tsarin layi-layi na maɓuɓɓugan ruwa

Misalai 4 na shirye-shiryen bazara masu kama da juna

1. Maɓuɓɓugan ruwa guda uku iri ɗaya, kowannensu yana da madaidaitan roba na 200 N/m, an shirya su a jere kuma a layi ɗaya kamar yadda aka nuna a cikin hoton da ke ƙasa. An dakatar da nauyin w daga ƙarshen ƙasan maɓuɓɓugan ruwa, yana ƙara tsawon maɓuɓɓugan ruwa da 1 cm. Nauyin nauyin w shine…
Tattaunawa
An san cewa:
Daidaiton kowace bazara (k)₁ = ku₂ = ku₃) = 200 N/m
Ƙara tsawon tsarin bazara (x) = 1 cm = mita 0,01
An tambaya: nauyin kaya (w)
Amsa:
Kidaya ta farko madaidaitan bazara haɗuwa.
An shirya bazara ta 1 da bazara ta 2 a layi ɗaya. Daidaitaccen ma'aunin bazara shine:
k_p = ku₁ +k₂ = 200 + 200 = Newtons 400/mita

Layi daya tsari maye gurbin maɓuɓɓugan ruwa (k)_p) da kuma bazara ta 3 (k₃) an shirya shi a jere. Madaidaitan maɓuɓɓugar ruwa mai maye gurbin shine:
1/k = 1/k_p + 1/k₃ = 1/400 + 1/200 = 1/400 + 2/400 = 3/400
k = 400/3 Newton/mita
Lissafa nauyin nauyin ta amfani da dabarar Dokar Hooke.
F = w = kx
w = (400/3)(0,01) = 4/3 Newtons
Nauyin nauyin shine 4/3 Newtons

2. Maɓuɓɓugan ruwa guda huɗu iri ɗaya kowannensu yana da madaidaicin ƙarfin N/m 500, an shirya su a jere kuma a layi ɗaya. A ƙayyade ƙaruwar tsawon tsarin maɓuɓɓugan ruwa idan aka ba da nauyin Newtons 20.
Tattaunawa
An san cewa:
Daidaiton kowace bazara (k)₁ = ku₂ = ku₃ = ku₄) = 500 N/m
Nauyin kayan (w) = Newtons 20
Tambaya: ƙaruwar tsawon tsarin bazara (x)
Amsa:
Da farko a ƙididdige haɗin ma'aunin bazara.
An shirya bazara ta 1, bazara ta 2 da bazara ta 3 a layi ɗaya. Daidaitaccen ma'aunin bazara shine:
k_p = ku₁ +k₂ +k₃ = 500 + 500 + 500 = 1500 Newtons/mita

Layi daya tsari maye gurbin maɓuɓɓugan ruwa (k)_p) da kuma bazara ta 4 (k₄) an shirya shi a jere. Madaidaitan maɓuɓɓugar ruwa mai maye gurbin shine:
1/k = 1/k_p + 1/k₃ = 1/1500 + 1/500 = 1/1500 + 3/1500 = 4/1500
k = 1500/4 = 375 Newton/mita
Yi amfani da dokar Hooke don tantance ƙaruwar tsawon tsarin bazara. Ƙarar tsawon tsarin bazara ita ce:
x = F / k = w / k
x = 20/375
x = mita 0,05
x = 5 cm

3. An shirya maɓuɓɓugan ruwa guda huɗu iri ɗaya a jere kuma a layi ɗaya kamar yadda aka nuna a cikin hoton da ke ƙasa. Idan aka ba da nauyin Newtons 20, tsarin maɓuɓɓugan ruwa yana ƙaruwa da tsayin santimita 4. A ƙayyade (a) haɗin ma'aunin tsarin. jerin maɓuɓɓugan layi ɗaya (b) daidaiton kowace bazara.
Tattaunawa
An san cewa:
Nauyin kayan (w) = Newtons 20
Ƙara tsawon tsarin bazara (x) = 4 cm = mita 0,04
Amsa:
(A) haɗin tsarin bazara mai ɗorewa
k = F / x = w / x
k = 20 / 0,04 = 500 Newtons/mita
(B) daidaito na kowane bazara
Maɓuɓɓugan guda huɗu iri ɗaya ne ta yadda dukkan maɓuɓɓugan guda huɗu suna da daidaito iri ɗaya. Idan aka maye gurbin maɓuɓɓugan 1, 2, da 3 da maɓuɓɓugar ruwa guda ɗaya, za a sami maɓuɓɓugan ruwa guda biyu, wato maɓuɓɓugan ruwa masu daidaitawa (k)_p) da kuma bazara ta 4 (k₄). Waɗannan maɓuɓɓugan ruwa guda biyu an shirya su a jere. Tsarin tantance daidaiton jerin shine:
1/k = 1/k_p + 1/k₄  
1/500 = 1/k_p + 1/k₄  —- lissafi na 1
k_p shine madadin madaidaitan maɓuɓɓuga don maɓuɓɓugan ruwa 1, 2, da 3 waɗanda aka shirya a layi ɗaya. Tunda maɓuɓɓugan ruwa guda uku iri ɗaya ne, madaidaitan kowane maɓuɓɓuga yana da girma iri ɗaya kuma ana iya wakilta shi da harafin k.
k_p = ku₁ +k₂ +k₃
k_p = k + k + k
k_p = 3k —- lissafi na 2
Sauya k_p a cikin lissafi 1 da k_p a cikin lissafi na 2. Haka kuma maye gurbin k₄ tare da k
1/500 = 1/3k + 1/k
1/500 = 1/3k + 3/3k
1/500 = 4/3k
3k = (4)(500)
3k = 2000
k = 2000 / 3
k = 667 N/m (sakamako mai zagaye)
Don haka daidaiton kowane bazara shine k₁ = ku₂ = ku₃ = ku₄ = 667 Newton/mita.

4. An shirya maɓuɓɓugai uku kamar yadda aka nuna a cikin hoton da ke ƙasa. Daidaiton kowace maɓuɓɓuga shine k.₁= 100 N/m, k₂ = 100 N/m, k₃ = 200 N/m. Ana rataye wani abu a ƙasan bazara don haka haɗuwar bazara ta sami ƙaruwa a tsawon 10 cm. Idan saurin gudu saboda nauyi ya kai 10 m/s² to nauyin kayan shine…
A. 1 kg
B. 2 kg
C. 3 kg
D. 4 kg
E. 5 kg

Tattaunawa:
An sani :
k₁= 100 N/m, k₂ = 100 N/m, k₃ = 200 N/m
x = 10 cm = mita 0,1
g = 10 m/s²
An tambaya :
Nauyin kaya (m)?
Jawab :
Tsarin Dokar Hooke :
F = kx
w = kx
mg = kx
Bayani: F = ƙarfi, w = nauyi, m = nauyin kaya, g = hanzari saboda nauyi, k = ma'aunin bazara, x = ƙaruwa a tsawon haɗuwar bazara
Lissafa haɗin ma'aunin bazara :
An shirya bazara ta 1 da bazara ta 2 a layi ɗaya. Daidaitaccen ma'aunin bazara shine:
k_p = ku₁ +k₂ = 100 + 100 = 200 N/m
An shirya madafun ruwa da bazara na 3 a jere. Madafun ruwa na maye gurbin shine:
1/k = 1/k_p + 1/k₃ = 1/200 + 1/200 = 2/200
k = 200/2 = 100 N/m
Lissafa nauyin nauyin :
mg = kx
m (10) = (100)(0,1)
m (10) = 10
m = 10/10
m = 1 kg
Amsar da ta dace ita ce A.