Question

Pendulum A has a bob of mass \(m\) hung from a string of length \(I_{i}\) pendulum \(B\) is identical to \(A\) except its bob has mass \(2 m\). Compare the frequencies of small oscillations of the two pendulums.

Short answer

Question: Compare the frequencies of small oscillations of two pendulums: Pendulum A with a mass m and length l, and Pendulum B with a mass of 2m and the same length l. Answer: The frequencies of small oscillations of pendulums A and B are equal.

Step-by-step solution

Formula for frequency of small oscillations

The formula for the frequency of small oscillations of a simple pendulum is given by: \[f = \frac{1}{2\pi}\sqrt{\frac{g}{l}}\] where f is the frequency, g is the acceleration due to gravity, and l is the length of the pendulum.

Frequency of Pendulum A

For pendulum A, we have a mass m and length \(l_i\). Using the formula from step 1, the frequency of pendulum A will be: \[f_A = \frac{1}{2\pi}\sqrt{\frac{g}{l_i}}\]

Frequency of Pendulum B

Since Pendulum B is identical to Pendulum A except for having a mass of 2m, its length is also \(l_i\), and the frequency of Pendulum B will be given by the same formula: \[f_B = \frac{1}{2\pi}\sqrt{\frac{g}{l_i}}\]

Compare the frequencies

Now, we can compare the frequencies of both pendulums. Since both pendulum A and pendulum B have the same length, their frequencies will be the same (since the mass does not affect the frequency in this case) : \[f_A = f_B\] In conclusion, the frequencies of small oscillations of pendulums A and B are equal, despite the difference in mass of their bobs.

Key concepts

Simple Pendulum

A simple pendulum is a device that consists of a weight, known as a bob, connected to a fixed point by a string or rod. The concept revolves around how the pendulum swings back and forth due to the gravitational force acting on the bob. The main characteristics of a simple pendulum that affect its oscillation include the length of the string and the acceleration due to gravity. The motion it exhibits is simple harmonic motion when the angles involved are small. In these small oscillations, the restoring force is approximately proportional to the displacement from the equilibrium position, which leads to periodic motion.

Understanding the motion of a simple pendulum is crucial in physics due to its practical applications, such as clocks and seismology equipment. Additionally, the pendulum model helps students better grasp concepts of harmonic motion and energy conservation.

Oscillation

Oscillation refers to the repeated back and forth movement of a pendulum through its equilibrium position. This movement is periodic and can be characterized by its frequency, which is the number of oscillations per unit time. For a simple pendulum, the frequency is determined by the length of the pendulum and the acceleration due to gravity, as seen in the formula:

\[f = \frac{1}{2\pi}\sqrt{\frac{g}{l}}\]

In the context of the problem given, both pendulums A and B undergo small oscillations, and their frequencies can be calculated using the same formula, as their lengths are equal. Therefore, the mass of the bob does not influence the frequency in the case of small oscillations. It's interesting to note how the pendulum's period (time for one complete cycle) remains constant for a given length in a uniform gravitational field, a principle utilized in timekeeping devices.

Acceleration Due to Gravity

The acceleration due to gravity, denoted as \(g\), is a constant which affects how objects fall toward the Earth. It plays a crucial role in determining the frequency of a simple pendulum. The standard value of \(g\) on Earth's surface is approximately \(9.81 \, \text{m/s}^2\). In the frequency formula for a simple pendulum, \(g\) is a crucial component:

\[f = \frac{1}{2\pi}\sqrt{\frac{g}{l}}\]

This formula shows that the frequency of small oscillations is directly proportional to the square root of the gravitational acceleration. Hence, on a planet with stronger gravity, a pendulum would have a higher frequency, leading to faster oscillations. Conversely, in spaces with weaker gravity, oscillations slow down. This interconnection illustrates how fundamental constants like gravitational acceleration influence everyday mechanics.

Mass of the Bob

The mass of the bob in a simple pendulum setup, surprisingly, does not affect the frequency of its oscillations, provided the oscillations are small. This is because the restoring force that acts to return the pendulum to its equilibrium position is dependent on the displacement and not the mass of the bob. This principle is clearly evidenced by the formula:

\[f = \frac{1}{2\pi}\sqrt{\frac{g}{l}}\]

which does not contain mass \(m\) as a factor. Consequently, whether the bob is heavy or light, the frequency remains constant as long as the other conditions are identical. This surprising result is rooted in how gravitational force and inertia proportionally affect the movement of objects, allowing for the mass to cancel out in the pendulum's equation of motion. This principle reaffirms that a simple pendulum's frequency is governed primarily by its length and the gravitational pull it experiences.