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Question:

In an experiment to determine the period of a simple pendulum of length 1m, it is attached to different spherical bobs of radii r1 and r2. The two spherical bobs have uniform mass distribution. If the relative difference in the periods is found to be 5×10⁻⁸s, the difference in radii, |r1−r2| is best given by: 1cm, 0.01cm, 0.1cm, 0.5cm

0.5cm

0.01cm

0.1cm

1cm

Solution:

T1=√2(l+r1)/g
T2=√2(l+r2)/g
(T1−T2)/T1=√2(l+r1)−√2(l+r2)/√2(l+r1)
Upon simplification,
5×10⁻⁸=√2(l+r1)−√2(l+r2)/√2(l+r1)
√l+r1−√l+r2=(5×10⁻⁸)×√l+r1
√l+r1−√l+r2=(5×10⁻⁸)(√l+r1)
(l+r1)/(l+r2)=(1+5×10⁻⁸)²(√l+r1)
l+r1/l+r2=1+10×10⁻⁸(l+r1)/l+r2=1+10⁻⁷
l+r1/l+r2=1.0000001
(l+r1)/(l+r2)=1.0000001
(l+r1)=1.0000001(l+r2)
r2−r1=10⁻⁷l
r2−r1=10⁻⁷(1m)
r2−r1=10⁻⁷m
|r2−r1|=0.0001m
|r2−r1|=0.1cm