Dimensional Analysis of a Pendulum Period

If a pendulum is taken to Mars, how would its period change?

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Mars

Q) If a pendulum has a period of 4s on the earth, what would its period be if it were placed on Mars? (Use g_M/g_E~ 1/3.) Use only dimensional analysis.

A) First, we must determine the relationship between the pendulum's period and gravity.

The period (P) of a pendulum of length l is a time, so

=T

if P α l^ag^b

T=[l]^a[g]^b

T=L^a(L/T²)^b

Thus, a=1/2 and b=-1/2

P α √(l/g)

Now, if we move to Mars, we don't change the length of the pendulum - Thus:

T_M/T_E=√(g_E/g_M)

T_M=T_E√(g_E/g_M)

T_M=(4s)√3

T_M~7s