Adadin kimiyyar lissafi a cikin motsi na zagaye

Adadin kimiyyar lissafi a cikin motsi na zagaye ya haɗa da canjin kusurwa, saurin kusurwa, da haɓaka kusurwa.

1. Matsar da kusurwa (θ)

Ana kiran sauyawa a cikin motsi na zagaye. Saboda haka, sauyawar kusurwa tare da adadin vector yana da girma da alkibla. Alkiblar sauyawar kusurwa yawanci ana bayyana ta a alkiblar agogo (agogo ko akasin agogo).

Akwai raka'a uku na matsar kusurwa. Na farko, digiri (o). Da'ira ɗaya ta da'irar daidai take da 360.^oNa biyu, juyin juya hali. Da'ira ɗaya ta da'irar daidai take da juyin juya hali ɗaya. Na uku, radians. Lura da hoton da ke ƙasa. Idan wani abu yana motsawa a cikin da'ira to r = radius na da'irar, x = tsawon hanyar da'ira da abin ya wuce = da'irar da'irar.

Da'ira ɗaya ta da'irar tana daidai da 2π radians.

Misalin matsala ta 1:

Juyin juya hali 1 = 360 ^o . ½ juyin juya hali = …. Rad?

Magani:

1 juyin juya hali = 360 ^o = 2 π rad = 2 (3.14) rad = 6.28 rad

Juyin juya hali 1/2 = 180 ^o = 1/2 (6.28 rad) = 3.14 rad

Misalin matsala ta 2:

1 rad = …… ^o ? 1 ^o =… rad?

Magani:

180 ^o = π rad = 3.14 rad

2. Gudun Kusurwa (ω)

a. Matsakaicin Gudun Kusurwa

Misalin matsala ta 3:

Tayar tana juyawa a hannun agogo, tana juyawa kusurwar 180 ^o na tsawon daƙiƙa 2 da kuma 90 ^o na tsawon daƙiƙa 1. Menene girma da alkiblar matsakaicin saurin kusurwa?

Magani:

Alkiblar matsakaicin saurin kusurwa = alkiblar agogo.

9 ^o /s = … rad/s?

Misalin matsala ta 4:

Tayar tana juyawa a hannun agogo, tana juyawa a kusurwar 360 ^o na tsawon daƙiƙa 4. Sannan tayar tana juyawa a hannun agogo, tana juyawa a kusurwar 180 ^o na tsawon daƙiƙa 2. Menene matsakaicin saurin kusurwa?

Magani:

Alkiblar matsakaicin saurin kusurwa = alkiblar agogo.

3 ^o /s = … rad/s?

b. Saurin kusurwa nan take

Sau da yawa ana kiran saurin kusurwar da ke tafe da saurin kusurwa. Idan aka ambaci saurin kusurwar kawai, to abin da ake nufi shine saurin kusurwar da ke tafe. Girman saurin kusurwar da ke tafe = girman saurin kusurwar a cikin ɗan gajeren lokaci. A lissafi:

Idan a cikin motsi na layi, za mu iya maye gurbin girman saurin da sauri, ta yadda a cikin motsi na zagaye za mu iya maye gurbin girman saurin kusurwa da saurin kusurwa.

3. Hanzarta kusurwa (α)

a. Matsakaicin hanzarin kusurwa

Misalin matsala ta 5:

Da farko wani injin niƙa mai ƙarfi ya huta, iska ta hura shi, don haka ya juya a hannun agogo. Bayan daƙiƙa 2, saurin kusurwar ya zama 90 ^o / s. Menene matsakaicin saurin kusurwa?

Magani:

A matsakaici, saurin kusurwar injin niƙa iska yana canzawa da 45 o/daki a kowace dakika 1 = … rad / s a ​​kowace dakika 1?

b. Hanzarin kusurwa nan take

Sau da yawa ana taƙaita saurin kusurwoyi nan take a matsayin saurin kusurwoyi. Saurin kusurwoyi nan take canji ne a cikin saurin kusurwoyi a cikin ɗan gajeren lokaci. A lissafi:

Girman layi a cikin motsi na zagaye

Adadin layi adadi ne a cikin motsi na layi, kamar motsi (nisa), gudu (gudu), da hanzari. A gefe guda kuma, ana iya kiran adadin motsi na zagaye adadi mai kusurwa.

1. Kaura

Ka lura da wani abu da ke juyawa, kamar ƙafa ko fanka ko injin niƙa ko agogo, da sauransu. Idan abu kamar fanka ya juya, dukkan sassan fanka suna juyawa tare. Idan fanka ya ɗauki juyawa ɗaya (360 ^o ), to dukkan sassan fanka, waɗanda ke gefen da kuma kusa da axis suma suna ɗaukar juyin juya hali ɗaya ko 360 ^o . Juyin juya hali ɗaya ko 360 ^o girma ne na canjin kusurwa da dukkan sassan fanka suka yi, duka a gefen da kuma kusa da axis.

Idan fanka ta ɗauki juyi ɗaya, ɓangaren fanka da ke gefen da ɓangaren fanka kusa da axis na juyawa yana motsawa har zuwa da'ira ɗaya (2). Idan radius na fanka ya kai cm 20, nisan da ke tsakanin gefen fanka da axis na juyawa shine cm 20. Misali, nisan da ke tsakanin axis da wani ɓangare na fanka da ke kusa da axis = cm 1. Lokacin da fanka ta yi juyi ɗaya, gefen fanka yana motsawa da'ira har zuwa (2)(3.14)(20 cm) = 125.6 cm,

yayin da wurin da ke kusa da axis yana motsawa da'ira har zuwa (2)(3.14)(1 cm) = 6.28 cm. 125.6 cm shine girman motsi da gefen fanka ya yi,

yayin da 6.28 cm girma ne na matsugunin da aka yi ta hanyar wani wuri da ke kusa da axis na juyawa. Ƙaramin r, ƙaramin matsugunin. Alaƙar da ke tsakanin matsugunin (d) da matsugunin kusurwa (θ) ana bayyana ta ta hanyar lissafi:

θ = d / r

d = r θ

d = matsar (mita), r = radius ko nisa daga axis na juyawa (mita), θ = matsar kusurwa (radians)

Misalin matsala ta 6:

CD mai tsawon santimita 5 a radius yana juyawa ta kusurwar digiri 90. Menene ma'aunin da ke tsakanin maki 2 cm da axis na juyawa?

Magani:

r = nisan daga axis na juyawa = 2 cm = 0.02 m

θ = 90 ^o = 1.57 rad (dole ne a faɗi a cikin radians)

d = (0.02 m)(1.57 rad) = 0.03 m.

Matsar da kusurwa ba ta da naúrar tsarin ƙasa da ƙasa kuma ba ta da girma (girmanta shine 1), saboda haka a cikin lissafin kamar yadda aka ambata a sama, kawai cire naúrar radians daga sakamakon lissafi.

2. Gudu

Idan aka juya, fanka ko wani abu mai juyawa, ba shakka, yana buƙatar takamaiman tazara. Idan fanka ta juya a hannun agogo kuma ta ɗauki juyawa ɗaya (360 ^o ) na daƙiƙa 1,

to, saurin kusurwar dukkan sassan fanka shine 1 rev/s = 360 ^o /s = 6.28 rad/s kuma alkiblar saurin kusurwa iri ɗaya ce da alkiblar allurar awa.

Idan radius na fanka ya kai santimita 20, to gefen fanka yana motsawa da'ira tare da gudun 2(3.14)(20 cm) / daƙiƙa 1 = 125.6 cm/s = 1.256 m/s. Ma'aunin da yake 1 cm (0.01 m) daga axis na juyawa tare da gudun 2 (3.14) (1 cm)/daƙiƙa 1 = 6.28 cm/s = 0.0628 m/s. Ƙaramin r, ƙaramin gudun. A cikin motsi na zagaye, saurin yakan zama ana kiransa saurin tangent.

Alaƙar da ke tsakanin saurin gudu (v) da saurin kusurwa (ω) ana bayyana ta ta hanyar lissafi:

v = gudu (m/s), r = radius ko nisa daga axis na juyawa (m), ω = saurin kusurwa (rad/s)

Misalin matsala ta 7:

Gudun allurar ta biyu shine 6.28 rad/minti = 6.28 rad / daƙiƙa 60 = 0.1 rad/s. Menene saurin wurin da yake 2 cm (0.02 m) daga axis na juyawa?

Magani:

ω = 0.1 rad/s, r = 0.02 m

v = r

ω = (0.02 m) (0.1 rad/s) = 0.002 m/s

Cire radians daga sakamakon lissafi.

3. Hanzartawa

Ma'aunin motsi na zagaye yana ƙaruwa idan girma da alkiblar saurin ya canza. Saboda haka, akwai nau'ikan hanzari guda biyu a cikin motsi na zagaye. Na farko, hanzarin tsakiya (a ₎ ko kuma wanda ake kira hanzarin radial (a _r ). Haɓaka centripetal yana faruwa ne saboda canje-canje a alkiblar gudu. Alkiblar hanzarin tsakiya koyaushe yana zuwa tsakiyar da'irar.

Girman hanzarin centripetal shine:

a _c = hanzarin centripetal (m/s ² ), r = radius ko nisa daga axis na juyawa (m), v = gudun (m/s), ω = saurin kusurwa (rad/s)

Na biyu, saurin gudu (a _t ). Hawan gudu yana faruwa ne saboda canje-canje a girman saurin. Bita wani wuri a gefen fankar da ke juyawa. Idan, da farko, fankar tana hutawa, to wurin da ke gefen fankar shi ma yana hutawa (v = 0). Idan daƙiƙa ɗaya bayan haka, fankar tana juyawa da saurin kusurwa na 1 rev/s = 360 ^o /s = 6.28 rad/s don haka wurin da ke gefen fankar yana motsawa da'ira tare da saurin 1.2 m/s

sannan wurin da ke gefen iskar fanka yana fuskantar saurin gudu na 1.2 m/s a kowace daƙiƙa = 1.2 m/s ².

Alaƙar da ke tsakanin hanzarin tangential (a ₎ da hanzarin kusurwa ( α ) an bayyana ta ta hanyar lissafi:

a _t = hanzarin tangential (m/s ² ), r = radius ko nisa daga axis na juyawa (m), α = hanzarin kusurwa (rad/s ² )

Alaƙa tsakanin lokaci (T) da mita (f) da gudu da kuma saurin kusurwa

Wannan lokacin yana nuna tazara tsakanin abu don yin juyin juya hali ɗaya, yayin da mitar ke nuna adadin juyin juya hali na daƙiƙa ɗaya. Alaƙar da ke tsakanin lokaci da mita ana bayyana ta ta hanyar lissafi: f = 1/T

Sashen Tsarin Ƙasashen Duniya na wannan lokacin shine na biyu, Sashen Tsarin Ƙasashen Duniya don mita shine 1/daƙiƙa (= hertz). Ana iya bayyana saurin (v) da saurin kusurwa (ω) na ƙwayoyin da ke motsawa a zagaye a cikin periods ko mitoci.

Ana bayyana saurin (v) na ƙwayar da ke motsi ta hanyar lissafi:

Girman saurin kusurwa (ω) na ƙwayoyin da ke motsi da'ira ana bayyana su ta hanyar lissafi: