The formula that relates period, T, to length, L is
#T = 2*pi*sqrt(L/g)#
Let the original period and length be #T_1 and L_1# respectively. Therefore in the original configuration,
#T_1 = 2*pi*sqrt(L_1/g)" "# Eq(1)
After the length is increased, the formula looks like this
#1.5*T_1 = 2*pi*sqrt((L_1+0.6 m)/g)#
Rearranging the formula
#sqrt((L_1+0.6 m)/g) = (1.5*T_1)/(2*pi) = 0.239 * T_1#
Squaring both sides
#(L_1+0.6 m)/g = 0.057*T_1^2#
Working to get L_1 by itself
#L_1/g + 0.6 m/g = 0.057*T_1^2#
#L_1/g = 0.057*T_1^2 - 0.6 m/g#
#L_1 = (0.057*T_1^2 - 0.6 m/g)*g#
#L_1 = 0.057*g*T_1^2 - 0.6 m " "#Eq(2)
Now we will substitute the expression for #T_1# from Eq(1) in for the #T_1# from Eq(2).
#L_1 = 0.057*g*(2*pi*sqrt(L_1/g))^2 - 0.6 m#
Working to simplify that
#L_1 = 0.057*cancel(g)*(2*pi)^2*(L_1/cancel(g)) - 0.6 m#
#L_1 = 0.057*(2*pi)^2*(L_1) - 0.6 m#
#L_1 = 2.25*L_1 - 0.6 m#
Getting close now.
#1.25L_1 = 0.6 m#
#L_1 = (0.6 m)/1.25 = 0.48 m#
I hope this helps,
Steve
