Question

A clock has a pendulum that performs one full swing every \(1.0 \mathrm{s}(\) back and forth). The object at the end of the pendulum weighs $10.0 \mathrm{N}$. What is the length of the pendulum?

Short answer

Answer: The length of the pendulum is approximately 0.248 m.

Step-by-step solution

Identify the given values

We are given the period of the pendulum \(T = 1.0 \mathrm{s}\) and the weight of the object at the end \(W = 10.0 \mathrm{N}\).

Recall the formula for the period of a simple pendulum

The formula for the period of a simple pendulum is: \(T = 2\pi\sqrt{\frac{l}{g}}\), where \(T\) is the period, \(l\) is the length, and \(g\) is the acceleration due to gravity.

Solve the formula for the length

We need to find the length of the pendulum, so we need to rearrange the formula to solve for \(l\). By squaring both sides of the equation and isolating \(l\), we get: \(l = \frac{gT^2}{4\pi^2}\).

Calculate the acceleration due to gravity

Knowing that the weight of the object is \(10.0 \mathrm{N}\), we can use Newton's second law to find the mass of the object: \(W = mg\) where \(W\) is the weight, \(m\) is the mass, and \(g\) is the acceleration due to gravity. In this case, \(g\) is approximately \(9.81 \mathrm{m/s^2}\).

Solve for the length

Now we can plug the values of \(T\) and \(g\) into the formula for the length and find the length of the pendulum: \(l = \frac{(9.81 \mathrm{m/s^2})(1.0 \mathrm{s})^2}{4\pi^2} \approx 0.248 \mathrm{m}\).

Provide the answer

The length of the pendulum is approximately \(0.248 \mathrm{m}\).