The period of a pendulum varies directly with the square root of the length of the pendulum.
a.) Express this relationship by writing an equation.
Let T and ℓ be the period of length of the pendulum respectively. So, T=k√ℓ
b.) To double the period, how would we have to change the length ℓ
Let T1 and ℓ1 be the information of the original pendulum. On the other hand, T2 and ℓ2 is the information of the new pendulum.
T1=k√ℓ1k=T1√ℓ1⟸Equation 1T2=k√ℓ2Recall that T2=2T12T1=k√ℓ2k=2T1√ℓ2⟸Equation 2
Use equations 1 and 2 to solve for the requested length.
T1√ℓ1=2T1√ℓ2Cancel out T11√ℓ1=2√ℓ2Apply cross multiplication√ℓ2=2√ℓ1Square both sidesℓ2=4ℓ1
It shows that, in order to double the period, the length must be four times the original length.
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