Problem 42 - Physics

Question

The length of a simple pendulum is \(0.79 \mathrm{~m}\) and the mass of the particle (the "bob") at the end of the cable is \(0.24 \mathrm{~kg}\). The pendulum is pulled away from its equilibrium position by an angle of \(8.50^{\circ}\) and released from rest. Assume that friction can be neglected and that the resulting oscillatory motion is simple harmonic motion. (a) What is the angular frequency of the motion? (b) Using the position of the bob at its lowest point as the reference level, determine the total mechanical energy of the pendulum as it swings back and forth. (c) What is the bob's speed as it passes through the lowest point of the swing?

Step-by-Step Solution

Answer

(a) \(3.52\text{ rad/s}\), (b) \(0.0078\text{ J}\), (c) \(0.255\text{ m/s}\)

Step 1: Identify given parameters and known formulas

We are given the following information about the pendulum: length \( L = 0.79 \text{ m} \), mass \( m = 0.24 \text{ kg} \), initial angle \( \theta_0 = 8.50^{\circ} \), which when converted to radians is \( \theta_0 = \frac{8.50 \pi}{180} \). Simplified formulas for a simple pendulum include: \( \omega = \sqrt{\frac{g}{L}} \) for angular frequency, where \( g \) is the acceleration due to gravity (approximately \( 9.81 \text{ m/s}^2 \)).

Step 2: Calculate the angular frequency (a)

To find the angular frequency \( \omega \) of the system, we use the formula: \( \omega = \sqrt{\frac{g}{L}} = \sqrt{\frac{9.81}{0.79}} \). Solving this gives \( \omega \approx 3.52 \text{ rad/s} \).

Step 3: Determine the total mechanical energy (b)

The total mechanical energy \( E \) is conserved and can be calculated using potential energy at the maximum amplitude. \( E = mgh \), where \( h = L(1 - \cos(\theta_0)) \). Calculating \( h \) gives \( h = 0.79(1 - \cos(8.50^{\circ})) \), \( h \approx 0.0033 \text{ m} \). Thus, \( E = 0.24 \times 9.81 \times 0.0033 \approx 0.0078 \text{ J} \).

Step 4: Calculate the bob's speed at the lowest point (c)

At the lowest point, all energy is kinetic and can be given by \( E = \frac{1}{2} mv^2 \). Setting this equal to the total mechanical energy calculated previously gives: \( \frac{1}{2} \times 0.24 \times v^2 = 0.0078 \). Solving for \( v \), we find \( v = \sqrt{\frac{0.0078 \times 2}{0.24}} \approx 0.255 \text{ m/s} \).

Key Concepts

Pendulum Mechanics

A pendulum is a simple yet fascinating mechanical system that exhibits periodic oscillatory motion. In its simplest form, a pendulum consists of a mass, known as the 'bob,' connected by a string or rod to a fixed point. When the bob is displaced from its equilibrium position and released, it swings back and forth under the influence of gravity. This to-and-fro motion is an example of simple harmonic motion, particularly when the displacement angle is small.

• Length: The length of the pendulum, in this case, is the distance from the pivot point to the center of mass of the bob, which affects the pendulum's period.
• Mass: The mass contributes to the gravitational force acting on the pendulum but does not affect the period in a simple pendulum scenario.
• Angle of Displacement: The angle at which the pendulum is initially displaced affects the amplitude but, when small, does not significantly change the period of oscillation.

Pendulum mechanics form the basis for various applications, including timekeeping in clocks and studying motion dynamics in physics.

Angular Frequency

Angular frequency is a key concept in analyzing harmonic motion, representing how many oscillations occur in a given time period. In the context of a simple pendulum, angular frequency is determined by the formula:\[ \omega = \sqrt{\frac{g}{L}} \]where \( g \) is the acceleration due to gravity (approximately 9.81 m/s²), and \( L \) is the length of the pendulum.
For our specific pendulum problem, we calculate the angular frequency as:\[ \omega = \sqrt{\frac{9.81}{0.79}} \approx 3.52 \text{ rad/s} \]The higher the angular frequency, the quicker the pendulum oscillates. This inverse relationship reveals that longer pendulums oscillate more slowly, while shorter ones oscillate faster. Hence, the angular frequency gives us insight into the temporal dynamics of the pendulum's motion.

Mechanical Energy Conservation

The principle of mechanical energy conservation states that, in the absence of friction and air resistance, the total mechanical energy in a system remains constant throughout the motion. In the case of the pendulum, there are two main forms of energy involved:

• Potential Energy: Maximum at the highest points of the swing due to height.
• Kinetic Energy: Maximum at the lowest point of the swing when the speed is highest.

For the pendulum considered here, the total mechanical energy can be calculated at any point using:\[ E = mgh \]where \( h = L(1 - \cos(\theta_0)) \). Substituting the given values:\[ h \approx 0.0033 \text{ m} \]\[ E \approx 0.0078 \text{ J} \]This calculation shows how gravitational potential energy at the start is fully converted into kinetic energy at the lowest point, illustrating energy conservation in oscillatory systems.

Oscillatory Motion

Oscillatory motion refers to any motion that repeats itself over regular intervals. A classic example is the motion of a simple pendulum, which swings back and forth in a consistent, rhythmical manner. Such motion is characterized by several key features:

• Amplitude: The maximum extent of displacement from the equilibrium position, which corresponds to the initial displacement angle in our pendulum problem.
• Period: The time it takes to complete one full cycle of motion. For a pendulum, this period is independent of amplitude when the angles are small.
• Frequency: The number of oscillations per unit time, related to the period by being its inverse.

In this pendulum situation, when the bob passes through its lowest point, its velocity is at a maximum, calculated as:\[ v \approx 0.255 \text{ m/s} \]This demonstrates how the velocity changes throughout the motion but cycles back to its original state, embodying the essence of oscillatory motion.