Question

Find the legnth of a simple pendulum whose period is \(1.00 \mathrm{~s}\) at a location where \(g=9.82 \mathrm{~m} / \mathrm{s}^{2}\)

Short answer

The length of the pendulum is approximately \(0.25~m\)

Step-by-step solution

Understand the Formula

The formula for the period of a simple pendulum is \(T = 2\pi \sqrt{L/g}\). Here, T is the period, L is the length of the pendulum, and g is the acceleration due to gravity. This formula shows that the period is directly proportional to the square root of the length of the pendulum, and inversely proportional to the square root of the acceleration due to gravity.

Rearrange the Formula to Solve for L

We want to find L, the length of the pendulum. We can rearrange the formula to solve for L: square both sides to get rid of the square root, which gives \((T)^2 = (2\pi)^2 \times L/g\). After that, isolate L and the formula Lin becomes: \(L = \frac{(T)^2 g}{(2\pi)^2}\)

Substitute the Values and Solve

Substitute the given values into the rearranged formula, \(T = 1.00 s\) and \(g = 9.82 m/s^2\). So \(L = \frac{(1.00 s)^2 \times 9.82 m/s^2}{(2\pi)^2} \approx 0.25 ~m\)

Key concepts

Pendulum Length Formula

Understanding the relationship between the length of a simple pendulum and its period is crucial for many physics problems. The fundamental pendulum length formula is expressed by the equation
\( T = 2\text{\textpi} \times \text{\textsqrt{\text{L/g}}} \),
where \(T\) represents the period of the oscillation, \(L\) is the length of the pendulum, and \(g\) is the acceleration due to gravity. This formula tells us that if you know the period and the acceleration due to gravity, you can find the pendulum's length by rearranging the formula to
\( L = \frac{T^2 \times g}{(2\text{\textpi})^2} \).
It's important to note that the period is defined as the time it takes for one complete oscillation. By applying this formula, we can find the necessary pendulum length to achieve a certain time of oscillation, like the one second period mentioned in our example.

Application in Exercises

In exercises that ask for determining the length of a simple pendulum, it’s essential to follow the steps carefully. Firstly, students must understand the formula and what each variable represents. Next, they will rearrange the equation to isolate the desired variable, in this case, \(L\). Finally, by substituting the known values, they'll obtain the length necessary for the specified period – a process that blends both algebraic manipulation and understanding of physical principles.

Acceleration Due to Gravity

The acceleration due to gravity, symbolized as \(g\), is the acceleration that Earth imparts to objects due to its gravitational pull. On the surface of Earth, this value is approximately \(9.81 \text{ m/s}^2\), although it can slightly vary with altitude and geographical location. In the context of pendulum motion, \(g\) is a key factor in determining the period of the pendulum. A higher gravitational acceleration would result in a faster oscillation of the pendulum, leading to a shorter period.

Why It Matters?

When solving physics problems involving a pendulum, it's important to use the accurate value of \(g\) for the location where the pendulum is swinging, as it influences the calculation of the pendulum length through the aforementioned formula. In our exercise, the given \(g\) value is \(9.82 \text{ m/s}^2\), indicating a slightly different gravitational pull than the average – aspects like these are essential for the accuracy of physics problem-solving.

Physics Problem Solving

Solving physics problems involves a systematic approach to apply mathematical formulas and principles to find solutions. Step one is to understand the problem and identify what's known and what needs to be found; this is usually followed by visualizing the problem perhaps with a diagram. Step two is recognizing the relevant principles, like the relationship between pendulum length and its period, and deciding which formulas to use.
In the final step, algebraic skills come into play as one needs to manipulate the formulas to solve for the unknown variable. After rearranging, substitution of known values provides numerical answers—a process that lies at the core of physics problem-solving. Practicing these steps fosters a deeper comprehension of physics concepts and enhances problem-solving skills.

Common Pitfalls and Tips

Students often make mistakes by not paying close attention to the units being used or by inserting values into formulas incorrectly. Therefore, it's beneficial to always double-check that the units are consistent and adhere to the proper formatting of equations. Breaking down the problem into smaller, more manageable parts can also alleviate the complexity and lead to correct solutions more consistently. Moreover, understanding the physical meaning behind the mathematical expressions can help in remembering and correctly applying the formulas.