SOLUTION: the period of a simple pendulum is directly proportional to the square root of its length. If a pendulum has a length of 6 feet and a period of 2 seconds, to what length should it

Algebra ->  Rational-functions -> SOLUTION: the period of a simple pendulum is directly proportional to the square root of its length. If a pendulum has a length of 6 feet and a period of 2 seconds, to what length should it      Log On


   

Question 46407: the period of a simple pendulum is directly proportional to the square root of its length. If a pendulum has a length of 6 feet and a period of 2 seconds, to what length should it be shortened to achieve a 1 second period?

Answer by Earlsdon(6294) About Me  (Show Source):
You can put this solution on YOUR website!
You can start by writing the equation of proportionality:
P+=+k%2Asqrt%28L%29 Where: P = period, L = Length, and k is the constant of proportionality.
We need to find the value of k. We can do this by substituting the given values of L (6 ft) and P (2 secs) into the above formula and solving for k.
2+=+k%2Asqrt%286%29 Divide both sides by sqrt%286%29
k+=+2%2Fsqrt%286%29
Now the formula for the period is:
P+=+%282%2Fsqrt%286%29%29sqrt%28L%29
To answer your probem however, we need to solve this equation for L.
P+=+%282%2Fsqrt%286%29%29sqrt%28L%29 Multiply both sides by sqrt%286%29 and divide by 2.
sqrt%286%29P%2F2+=+sqrt%28L%29 Now square both sides.
%286%2F4%29P%5E2+=+L Simplify.
L+=+%283%2F2%29P%5E2
Now we can answer your problem. What should L be if P is to be 1 second.
Just substitute P = 1 into the above equation and solve for L.
L+=+%283%2F2%29%281%5E2%29
L+=+3%2F2
L+=+1.5ft.