Question

Two pendulums with identical lengths of \(1.000 \mathrm{~m}\) are suspended from the ceiling and begin swinging at the same time. One is at Manila, in the Philippines, where \(g=9.784 \mathrm{~m} / \mathrm{s}^{2},\) and the other is at Oslo, Norway, where \(g=9.819 \mathrm{~m} / \mathrm{s}^{2}\). After how many oscillations of the Manila pendulum will the two pendulums be in phase again? How long will it take for them to be in phase again?

Short answer

In this problem, two pendulums with identical lengths swing in different locations with different gravity values. We want to determine how many oscillations of the Manila pendulum it takes for both pendulums to be in phase again and the time it takes for this to occur. The steps to solve this problem involve calculating the periods of each pendulum, finding the least common multiple of the periods, and calculating the time taken for both pendulums to be in phase again.

Step-by-step solution

Calculate the periods of each pendulum

We'll first determine the periods of oscillation for each pendulum using the formula for the period of a simple pendulum: $$ T = 2 \pi \sqrt{\frac{l}{g}} $$ where T is the period of oscillation, l is the length of the pendulum, and g is the gravitational acceleration. For the Manila pendulum: $$ T_{Manila} = 2 \pi \sqrt{\frac{1.000}{9.784}} $$ And for the Oslo pendulum: $$ T_{Oslo} = 2 \pi \sqrt{\frac{1.000}{9.819}} $$ Calculate the periods for each pendulum.

Find the least common multiple of the periods

Since we want to find the number of oscillations after which both pendulums are in phase again, we need to find the least common multiple of their periods. We can do this by first finding the ratio between the periods and then determining the smallest integers that are multiples of each period. Let's denote the ratio between the periods as R: $$ R = \frac{T_{Manila}}{T_{Oslo}} $$ Calculate R and round it to the nearest integer. Now that we have the ratio of the periods, we can find the least common multiple of the periods by multiplying them by the smallest integers that ensure their equality: $$ n_{Manila} \times T_{Manila} = n_{Oslo} \times T_{Oslo} $$ where \(n_{Manila}\) and \(n_{Oslo}\) are the smallest integers to make both sides equal. Solve for \(n_{Manila}\) and \(n_{Oslo}\).

Calculate the time taken for both pendulums to be in phase again

Now that we have the smallest integers for both pendulums to be in phase again, we can determine how long this will take. The time taken for both pendulums to be in phase again is equal to the product of the number of oscillations and the period of the Manila pendulum: $$ t_{phase} = n_{Manila} \times T_{Manila} $$ Calculate the time taken for both pendulums to be in phase again.

Key concepts

Simple Pendulum Period

Understanding the period of a simple pendulum is key to grasping pendular motion. A simple pendulum consists of a mass, also known as the 'bob,' that hangs from a fixed point by means of a string or rod that has negligible mass.

The period of a simple pendulum, denoted as T, is the time it takes for one complete cycle of oscillation, which includes a swing in one direction and a return swing back to the starting position. The formula given by

\[T = 2 \pi \sqrt{\frac{l}{g}}\]
is crucial as it relates the period (T) to the length of the pendulum (l) and the acceleration due to gravity (g). A peculiar characteristic of the simple pendulum is that its period is independent of the mass of the bob and the amplitude of the swing, as long as the amplitude is not too large.

When applying this formula to solve problems, such as the textbook exercise where different values of g cause variations in periods, remember that a longer pendulum or a location with less gravitational acceleration will increase the period of oscillation.

Gravitational Acceleration

Gravitational acceleration, often expressed as g, is the acceleration that the Earth imparts to objects due to the force of gravity. Contrary to common belief, g is not constant everywhere on Earth. It varies slightly due to factors like altitude, latitude, and local geological structures.

For instance, the question posed in the textbook exercise highlights two different values of g: one in Manila, Philippines, and one in Oslo, Norway. The differences in g impact the period of oscillation for pendulums located in these cities, as seen from the formula for the simple pendulum period.

Typically, g is approximately 9.81 m/s². However, as showcased by the exercise, g can be slightly less or more depending on location. Being mindful of these variations is essential when performing precise physical calculations or experiments that rely on gravitational force.

Oscillatory Motion

Oscillatory motion refers to the repetitive back and forth movement of a system about an equilibrium position. Examples of such motion include the swings of a pendulum, the vibrations of a guitar string, and the alternating current in electrical circuits.

In the context of pendulum oscillations, as presented in the textbook problem, the concept of 'being in phase' comes into play. Two pendulums are said to be in phase when they simultaneously reach the same position in their oscillatory cycles. If their periods are slightly different, due to varying gravitational accelerations, they will gradually become out of phase.

The step-by-step solution details a method to calculate after how many oscillations pendulums with different periods will be in phase again. This requires understanding of the least common multiple of their periods and amplifies the importance of oscillatory motion concepts in analyzing the dynamic behavior of simple pendulum systems.