Problem 39 - Physics

Question

If the period of a simple pendulum is to be \(2.0 \mathrm{~s},\) what should be its length?

Step-by-Step Solution

Answer

The length of the pendulum should be approximately 1.006 meters.

Step 1: Understanding the Problem

We need to find the length of a simple pendulum given its period is 2.0 seconds. The formula for the period of a simple pendulum is given by \(T = 2\pi\sqrt{\frac{L}{g}}\), where \(T\) is the period, \(L\) is the length, and \(g\) is the acceleration due to gravity \((9.81 \mathrm{~m/s^2})\).

Step 2: Rearranging the Formula

To find the length \(L\), we need to rearrange the equation \(T = 2\pi\sqrt{\frac{L}{g}}\) to solve for \(L\). Start by dividing both sides by \(2\pi\): \(\frac{T}{2\pi} = \sqrt{\frac{L}{g}}\).

Step 3: Squaring Both Sides

Next, square both sides of the equation to eliminate the square root: \(\left(\frac{T}{2\pi}\right)^2 = \frac{L}{g}\).

Step 4: Isolating L

Now, isolate \(L\) by multiplying both sides by \(g\): \(L = g \left(\frac{T}{2\pi}\right)^2\).

Step 5: Substituting the Values

Substitute \(T = 2.0\,s\) and \(g = 9.81\, \mathrm{m/s^2}\) into the equation: \(L = 9.81 \left(\frac{2.0}{2\pi}\right)^2\).

Step 6: Calculating L

Calculate \(L\) using the substituted values: \(L = 9.81 \left(\frac{2.0}{6.2832}\right)^2 = 9.81 \times 0.1013^2 = 9.81 \times 0.01026 \approx 1.006 \mathrm{~m}\).

Key Concepts

Pendulum Period Formula

The pendulum period formula is crucial for understanding how pendulums operate in a physics context. This formula helps determine the time it takes for a pendulum to complete one full swing or cycle. The formula is:
\[ T = 2\pi\sqrt{\frac{L}{g}} \]where:

• \(T\) is the period of the pendulum (the time for one full cycle).
• \(L\) is the length of the pendulum.
• \(g\) is the acceleration due to gravity.

This formula tells us several important things:

• The period \(T\) is independent of the mass of the pendulum.
• It depends only on the length and the acceleration due to gravity. Longer pendulums or those in stronger gravitational fields will have different periods.

Understanding this formula is the first step in solving physics problems involving pendulums.

Acceleration due to Gravity

The acceleration due to gravity \(g\) is a fundamental concept in physics. It is the rate at which an object accelerates when it is in free fall. On Earth, this is typically \(9.81 \, \mathrm{m/s^2}\). This value can vary slightly depending on your location, such as altitude or geographical position, but \(9.81\) is a good average for most calculations.
This constant is essential for solving pendulum problems because it influences the period of the pendulum. In the period formula \(T = 2\pi\sqrt{\frac{L}{g}}\), \(g\) is used to determine how quickly the pendulum returns to its starting position. Higher values of \(g\) would mean faster pendulum swings.
By knowing \(g\), you can understand how a pendulum will behave in different environments and accurately calculate its period when combined with its length.

Pendulum Length Calculation

Calculating the length of a pendulum involves rearranging and solving the period formula. When given the period, you can find the length with this modified formula:
\[ L = g \left(\frac{T}{2\pi}\right)^2 \]To perform this calculation, you:

• Set up the equation based on the known period \(T\).
• Square the result of \(\frac{T}{2\pi}\), which converts the period term appropriately within the formula.
• Multiply by the gravitational constant \(g\) to find the pendulum length \(L\).

For example, if you have a period of \(2.0 \, s\), the calculation using \(g = 9.81 \, \mathrm{m/s^2}\) results in a length of approximately \(1.006 \, m\). This process shows how closely the pendulum's length is tied to the period and gravitational force.

Physics Problem Solving

Physics problem solving often involves applying known formulas to uncover unknown values. It requires identifying the known quantities and systematically using mathematical operations to find the unknowns.
With pendulums, knowing the period formula \(T = 2\pi\sqrt{\frac{L}{g}}\) is your starting point. Problem-solving steps typically include:

• Read the problem carefully to understand what is known and what is being asked.
• Identify the formula that connects these quantities.
• Rearrange the formula if necessary to isolate the unknown variable.
• Substitute the known values into the formula.
• Perform the calculations step by step, ensuring accuracy at each stage.

Remember to double-check your units and ensure they are consistent throughout the calculation. Practice with various problems strengthens your ability to approach similar tasks with confidence and improves your overall problem-solving skills in physics.