Question

What will be the time period of a simple pendulum if
a. the length of the pendulum is made 9 times the original length?
b. the length of the pendulum is made 1/9 of the original length?
c. the pendulum is taken to a place where the value of g is 4 times that on the surface of the earth?
d. the pendulum is taken to a place where the value of g is 1/9 times that on the surface of two earth?
e. the amplitude of the pendulum is doubled?
f. the mass of the bob is halved?​

Answer 1

Answer:

The formula to calculate the time period of a pendulum is given as:

$\boxed{ \bf{ T = 2\pi\sqrt{\dfrac{l}{g}}}}$

where, 'l' is the length of the pendulum and 'g' is the acceleration due to gravity.

Based on the formula we can say that:

• Time period of a pendulum is directly proportional to √l.

• Time period of a pendulum is inversely proportional to √g.

Coming to the question,

a) Length is made 9 times it's original length. (i.e. l' = 9l)

Therefore √l' = √9l = 3 √l (or) 3 times the original length.

Hence the Time period becomes 3 times the original Time period as it is directly proportional to length.

b) Length is made 1/9 times it's original length. (i.e. l' = l/9)

Therefore √l' = √l/9 = [(√l)/3] (or) 1/3 times the original length.

Hence the Time period becomes 1/3 times the original Time period as it is directly proportional to length.

c) Gravity is made 4 times the original gravity. (i.e. g' = 4g)

Therefore √g' = √4g = 2√g (or) 2 times the original gravity.

Hence the Time period becomes 1/2 times the original Time period as it is inversely proportional to gravity.

d) Gravity is made 1/9 times the original gravity. (i.e. g' = g/9)

Therefore √g' = √g/9 = [(√g)/3] (or) 1/3 times the original gravity.

Hence the Time period becomes 3 times the original Time period as it is inversely proportional to gravity.

Since Time period of pendulum is independent of mass of the bob and the amplitude of the pendulum, the time period remains constant for whichever values the mass and amplitude take.

Hence for (e) and (f) the time period is constant.