The period of oscillation of a simple pendulum is T = 2π√(l/g). The value of l is 20cm known to 1mm accuracy, and the time for 100 oscillations of the pendulum is found to be 90s using a wrist watch of 1s resolution. What is the accuracy in the determination of g?

Question

Asked on September 2, 2026Physics

This physics problem tests your understanding of fundamental physical laws and their applications. The step-by-step solution below breaks down the problem using relevant equations and physical reasoning.

Accepted Answer · 2026-09-02T05:39:10.449Z

Swali la 3: Tunapewa halijoto ya awali na ya mwisho ya kimiminika pamoja na uhakika wake. Tunahitaji kuamua ongezeko la halijoto. Step 1: Andika halijoto za awali na za mwisho. Halijoto ya awali, T_1 = (63.5 ± 0.5)^ C Halijoto ya mwisho, T_2 = (72.6 ± 0.4)^ C Step 2: Kokotoa ongezeko la halijoto kwa kutoa halijoto ya awali kutoka kwa halijoto ya mwisho. Ongezeko la halijoto, T = T_2 - T_1. Kwa thamani za wastani: T_wastani = 72.6^ C - 63.5^ C = 9.1^ C Step 3: Kokotoa uhakika katika ongezeko la halijoto. Wakati wa kutoa au kujumlisha vipimo, uhakika kamili huongezwa. ( T) = T_1 + T_2 = 0.5^ C + 0.4^ C = 0.9^ C Step 4: Andika ongezeko la halijoto na uhakika wake. Ongezeko la halijoto ni T = ( T_wastani ± ( T)). T = (9.1 ± 0.9)^ C Swali la 4: Tunapewa fomula ya kipindi cha bembea rahisi na vipimo vya urefu na muda. Tunahitaji kuamua usahihi katika uamuzi wa g. Step 1: Andika data uliyopewa na uhakika wake. Urefu, l = 20 cm. Usahihi ni 1 mm. Hivyo, l = 1 mm = 0.1 cm. l = (20.0 ± 0.1) cm Muda kwa mitetemo 100, t = 90 s. Azimio la saa ni 1 s. Hivyo, t = 1 s. t = (90 ± 1) s Step 2: Kokotoa kipindi T na uhakika wake. Kipindi T = (t)/(idadi ya mitetemo) = 90 s100 = 0.9 s. Uhakika katika kipindi T = ( t)/(idadi ya mitetemo) = 1 s100 = 0.01 s. Hivyo, T = (0.90 ± 0.01) s. Step 3: Panga upya fomula ya kipindi ili kupata g. Fomula ni T = 2 sqrt((l)/(g)). Piga mraba pande zote mbili: T^2 = (2)^2 (l)/(g) T^2 = 4^2 (l)/(g) Tenga g: g = (4^2 l)/(T^2) Step 4: Kokotoa uhakika wa asilimia katika g. Kwa fomula g = (4^2 l)/(T^2), uhakika wa asilimia (au uhakika wa sehemu) huhesabiwa kama: ( g)/(g) = ( l)/(l) + 2( T)/(T) Kokotoa uhakika wa sehemu kwa l: ( l)/(l) = 0.1 cm20.0 cm = 0.005 Kokotoa uhakika wa sehemu kwa T: ( T)/(T) = 0.01 s0.90 s = (1)/(90) ≈ 0.01111 Jumlisha uhakika wa sehemu: ( g)/(g) = 0.005 + 2 × (1)/(90) = 0.005 + (2)/(90) = 0.005 + (1)/(45) Ili kujumlisha, badilisha kuwa sehemu moja au desimali: ( g)/(g) = (1)/(200) + (1)/(45) = (9)/(1800) + (40)/(1800) = (49)/(1800) Badilisha kuwa asilimia: Usahihi wa asilimia katika g = (49)/(1800) × 100\% = (49)/(18)\% ≈ 2.722\% Kuzungusha hadi tarakimu muhimu mbili, tunapata 2.7\%. Usahihi katika uamuzi wa g = 2.72\% Tuma swali linalofuata 📸

Accepted Answer · 2026-09-02T05:39:10.449Z

Swali la 3: Tunapewa halijoto ya awali na ya mwisho ya kimiminika pamoja na uhakika wake. Tunahitaji kuamua ongezeko la halijoto. Step 1: Andika halijoto za awali na za mwisho. Halijoto ya awali, T_1 = (63.5 ± 0.5)^ C Halijoto ya mwisho, T_2 = (72.6 ± 0.4)^ C Step 2: Kokotoa ongezeko la halijoto kwa kutoa halijoto ya awali kutoka kwa halijoto ya mwisho. Ongezeko la halijoto, T = T_2 - T_1. Kwa thamani za wastani: T_wastani = 72.6^ C - 63.5^ C = 9.1^ C Step 3: Kokotoa uhakika katika ongezeko la halijoto. Wakati wa kutoa au kujumlisha vipimo, uhakika kamili huongezwa. ( T) = T_1 + T_2 = 0.5^ C + 0.4^ C = 0.9^ C Step 4: Andika ongezeko la halijoto na uhakika wake. Ongezeko la halijoto ni T = ( T_wastani ± ( T)). T = (9.1 ± 0.9)^ C Swali la 4: Tunapewa fomula ya kipindi cha bembea rahisi na vipimo vya urefu na muda. Tunahitaji kuamua usahihi katika uamuzi wa g. Step 1: Andika data uliyopewa na uhakika wake. Urefu, l = 20 cm. Usahihi ni 1 mm. Hivyo, l = 1 mm = 0.1 cm. l = (20.0 ± 0.1) cm Muda kwa mitetemo 100, t = 90 s. Azimio la saa ni 1 s. Hivyo, t = 1 s. t = (90 ± 1) s Step 2: Kokotoa kipindi T na uhakika wake. Kipindi T = (t)/(idadi ya mitetemo) = 90 s100 = 0.9 s. Uhakika katika kipindi T = ( t)/(idadi ya mitetemo) = 1 s100 = 0.01 s. Hivyo, T = (0.90 ± 0.01) s. Step 3: Panga upya fomula ya kipindi ili kupata g. Fomula ni T = 2 sqrt((l)/(g)). Piga mraba pande zote mbili: T^2 = (2)^2 (l)/(g) T^2 = 4^2 (l)/(g) Tenga g: g = (4^2 l)/(T^2) Step 4: Kokotoa uhakika wa asilimia katika g. Kwa fomula g = (4^2 l)/(T^2), uhakika wa asilimia (au uhakika wa sehemu) huhesabiwa kama: ( g)/(g) = ( l)/(l) + 2( T)/(T) Kokotoa uhakika wa sehemu kwa l: ( l)/(l) = 0.1 cm20.0 cm = 0.005 Kokotoa uhakika wa sehemu kwa T: ( T)/(T) = 0.01 s0.90 s = (1)/(90) ≈ 0.01111 Jumlisha uhakika wa sehemu: ( g)/(g) = 0.005 + 2 × (1)/(90) = 0.005 + (2)/(90) = 0.005 + (1)/(45) Ili kujumlisha, badilisha kuwa sehemu moja au desimali: ( g)/(g) = (1)/(200) + (1)/(45) = (9)/(1800) + (40)/(1800) = (49)/(1800) Badilisha kuwa asilimia: Usahihi wa asilimia katika g = (49)/(1800) × 100\% = (49)/(18)\% ≈ 2.722\% Kuzungusha hadi tarakimu muhimu mbili, tunapata 2.7\%. Usahihi katika uamuzi wa g = 2.72\% Tuma swali linalofuata 📸

The period of oscillation of a simple pendulum is T = 2π√(l/g). The value of l is 20cm known to 1mm accuracy, and the time for 100 oscillations of the pendulum is found to be 90s using a wrist watch of 1s resolution. What is the accuracy in the determination of g?

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The period of oscillation of a simple pendulum is T = 2π√(l/g). The value of l is 20cm known to 1mm accuracy, and the time for 100 oscillations of the pendulum is found to be 90s using a wrist watch of 1s resolution. What is the accuracy in the determination of g?

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