The period p of a pendulum, or the time it takes for the pendulum to make one complete swing, varies directly as the square root of the length L of the pendulum. If the period of a pendulum is 1.3 s when the length is 2 ft, find the period when the length is 7 ft. Round to the nearest hundredth.
p = k√L
That is,
p/√L = k is constant. So, you want to find p where
1.3/√2 = p/√7
To solve this problem, we can use the formula for direct variation:
p = k√L
where p is the period, L is the length of the pendulum, and k is a constant of variation.
First, let's find the value of k using the given information. We know that when L is 2 ft, the period is 1.3 s. We can substitute these values into the equation:
1.3 = k√2
To solve for k, divide both sides of the equation by the square root of 2:
k = 1.3 / √2
Now we can use this value of k to find the period when the length of the pendulum is 7 ft. Again, substitute the values into the equation:
p = (1.3 / √2)√7
Now let's calculate the period:
p ≈ (1.3 / √2)√7 = 2.249
Rounding to the nearest hundredth, the period of the pendulum when the length is 7 ft is approximately 2.25 s.
To solve this problem, we need to use the direct variation equation. The equation can be written as:
p = k * √L
where p is the period, L is the length, and k is the constant of variation.
We are given that when the length is 2 ft, the period is 1.3 seconds. We can use this information to find the value of k.
1.3 = k * √2
To isolate k, we divide both sides of the equation by √2:
k = 1.3 / √2
Using a calculator, we find that k is approximately 0.919.
Now we can use this value of k to find the period when the length is 7 ft using the same direct variation equation:
p = 0.919 * √7
Calculating this expression, we find that the period is approximately 2.61 seconds when the length is 7 ft (rounded to the nearest hundredth).