Question

The period \(T\), in seconds, of a pendulum of length \(l,\) in feet, may be approximated using the formula $$T=2 \pi \sqrt{\frac{l}{32}}$$ Express your answer both as a square root and as a decimal approximation. Find the period \(T\) of a pendulum whose length is 64 feet.

Short answer

\[T=2 \pi \sqrt{2} \approx 8.86\] seconds.

Step-by-step solution

- Formula Setup

Start with the given formula for the period of a pendulum: \[T=2 \pi \sqrt{\frac{l}{32}}.\]Identify the given values in the problem. The length of the pendulum, \(l\), is given as 64 feet.

- Substitute the Length

Substitute the given length \(l = 64\) feet into the formula:\[T=2 \pi \sqrt{\frac{64}{32}}.\]

- Simplify Inside the Square Root

Simplify the expression inside the square root:\[\frac{64}{32} = 2.\]So now, the formula becomes:\[T=2 \pi \sqrt{2}.\]

- Express as a Decimal

Find the decimal approximation for the expression. Use the value of \(\pi \approx 3.14\) and the square root of 2 approximately \(\sqrt{2} \approx 1.41\). Plug these into the formula:\[T \approx 2 \times 3.14 \times 1.41 \approx 8.86\] seconds.

Key concepts

pendulum formula

The pendulum formula is used to calculate the time period of a pendulum. It helps us understand how long it takes for the pendulum to complete one full swing back and forth. The basic formula is given by: \[T=2 \pi \sqrt{\frac{l}{32}}\]Here, **T** stands for the period (time in seconds), **l** is the length of the pendulum in feet, and \( \pi \) is a constant approximately equal to 3.14. This formula is derived from physics principles and can predict the motion of pendulums very accurately. In this exercise, we will use a pendulum with a length **l** of 64 feet, as given.

square root approximation

The square root approximation is an important step in the calculation. We need to simplify the term inside the square root. For our given equation:\[ T=2 \pi \sqrt{\frac{64}{32}} \]First, divide the values inside the square root. This gives us a simpler value to work with: \[\frac{64}{32} = 2\] Now, the formula reduces to: \[T=2 \pi \sqrt{2}\]The square root of 2 (\( \sqrt{2} \)) is approximately 1.41. This approximation helps us in further calculations. Remember, using approximations can slightly change the result but is usually enough for practical purposes.

substitution in equations

Substitution involves replacing the given variable (length of the pendulum) into the formula. In our case, we've been given a pendulum length of 64 feet. Substitution process is simple but crucial to get right values:\[T=2 \pi \sqrt{\frac{64}{32}}\]When we plug in the value 64 for **l**, it results in:\[T=2 \pi \sqrt{2}\]This step is essential for carrying out the following calculations correctly.

decimal approximation

To find the period **T** in decimal form, we need to approximate constants in the formula. Using **π ≈ 3.14** and **\( \sqrt{2} \) ≈ 1.41**, we can substitute these approximations into our simplified formula:\[T\approx 2 \times 3.14 \times 1.41\]Multiplying these values step by step:

• First, multiply **2** and **3.14** to get approximately **6.28**.
• Next, multiply **6.28** and **1.41** to get roughly **8.86**.

Thus, the decimal approximation for the period **T** of the pendulum is **8.86 seconds**.

simplifying expressions

Simplifying expressions involves breaking down complex mathematical statements into easier-to-manage parts. For the pendulum period formula, simplification is needed after substitution. Start from:\[T=2 \pi \sqrt{\frac{64}{32}}\]First simplify inside the square root:\[\frac{64}{32} = 2 \]Then, the formula becomes:\[T=2 \pi \sqrt{2}\]Next, recognize that we can use approximations like **\( \pi \approx 3.14 \)** and **\( \sqrt{2} \approx 1.41 \)**. Putting it all together, we get:\[T\approx 2 \times 3.14 \times 1.41 = 8.86\]Understanding each step in simplification helps to solve equations quickly and accurately.