Question

The period T of a pendulum (the time it takes to make one complete oscillation) varies directly as the square root of its length L. A pendulum 7 ft long has a period of 2.9 seconds. find the period (in s) of a pendulum that is 10 ft long. round to the nearest tenth of a second. _____ swhat is the length (in ft) of a pendulum that beats seconds (that is, has a 2 second period)? round to the nearest tenth of a ft. ______ ft

Answer 1

The period T of a pendulum varies directly as the square root of its length L. To find the period of a 10 ft pendulum, use the direct variation formula and the known period of a 7 ft pendulum. Similarly, to calculate the length of a pendulum with a 2-second period, use the period formula and solve for L.

Step-by-step explanation:

The period T of a pendulum is determined by the formula T = 2π√(L/g), where L is the length of the pendulum and g is the acceleration due to gravity. Given that a 7 ft pendulum has a period of 2.9 seconds, we can establish a ratio and calculate the period for other lengths, assuming g remains constant.

For a 7 ft long pendulum:
T1 = 2π√(L1/g) = 2.9 seconds

For a 10 ft long pendulum, using the direct variation formula T2 / T1 = √(L2/L1), we can find:
T2 = T1 √(L2/L1)

Similarly, to find the length of a pendulum with a 2-second period, use the formula L = (T/2π)2 g, and solve for L.

Answer 2

The problem is about a directly proportional variation between T and the square root of L.

`T=k\cdot\sqrt[]{L}`

Where k is the constant of proportionality. Let's use L = 7ft and T = 2.9 sec to find k.

`2.9=k\cdot\sqrt[]{7}\to k=\frac{2.9}{\sqrt[]{7}}\approx1.10`

The constant of proportionality is 1.10.

Then, use the constant to find the period of a pendulum that is 10 feet long.

`T=1.10\sqrt[]{10}\approx3.5\sec`

The period of a pendulum that is 10 ft long is 3.5 seconds.

Then, use T = 2 sec to find the length.

`\begin{gathered} 2=1.10\sqrt[]{L} \\ L=((2)/(1.10))^2 \\ L\approx3.3ft \end{gathered}`

The length of a pendulum that beats seconds is 3.3 feet.