Problem 13 - Classical Dynamics of Particles and Systems

Question

A simple pendulum of length \(b\) and bob with mass \(m\) is attached to a massless support moving horizontally with constant acceleration \(a\). Determine (a) the equations of motion and (b) the period for small oscillations.

Step-by-Step Solution

Answer

The equation of motion for the pendulum with small oscillations is given by: \(\theta = \frac{b\ddot{\theta} - a}{b\ddot{\theta}\theta + \frac{mg\theta}{T}}\) The period of oscillations for small angles is: \(T = \frac{2\pi}{\sqrt{\frac{g}{b}}}\)

Step 1: Draw a Free Body Diagram (FBD) of the pendulum

Draw a Free Body Diagram (FBD) of the pendulum, showing all forces acting on the bob. There are three forces acting on the bob: 1. Gravitational force (\(mg\)), acting vertically downwards 2. Tension force (\(T\)), acting along the length of the pendulum 3. Horizontal inertial force (\(ma\)), acting horizontally in the opposite direction of the horizontal acceleration

Step 2: Define the angles and coordinates

Define the angle between the vertical axis and the pendulum as \(\theta\). Let (x, y) be the coordinates of the bob, with the origin at the center of the support.

Step 3: Write the equations for the coordinates

Write the equations for the coordinates x and y in terms of the angle \(\theta\) and the length of the pendulum \(b\): \(x = b \sin(\theta) - at\) \(y = b - b\cos(\theta)\)

Step 4: Differentiate the coordinate equations with respect to time

Differentiate the x and y coordinate equations with respect to time to find the velocity components: \(\dot{x} = b\dot{\theta}\cos(\theta) - a\) \(\dot{y} = b\dot{\theta}\sin(\theta)\)

Step 5: Apply Newton's second law of motion to the bob

Apply Newton's second law of motion (\(F = ma\)) on the two components (horizontal and vertical) of the bob to get two equations: Horizontal component: \(T\sin(\theta) = m(b\ddot{\theta}\cos(\theta) - a) \) Vertical component: \(T\cos(\theta)-mg= mb\ddot{\theta}\sin(\theta)\)

Step 6: Divide the two equations

Divide the horizontal component equation by the vertical component equation to eliminate the tension force and obtain a differential equation for \(\theta\): \(\frac{\sin(\theta)}{\cos(\theta)-\frac{mg}{T}} = \frac{b\ddot{\theta}\cos(\theta) - a}{b\ddot{\theta}\sin(\theta)}\)

Step 7: Simplify the differential equation

Simplify the equation from step 6: \(\tan(\theta) = \frac{(b\ddot{\theta}\cos(\theta) - a)}{b\ddot{\theta}\sin(\theta) + \frac{mg\sin(\theta)}{T}}\) For small oscillations, we can approximate \(\tan(\theta) \approx \theta\) and \(\cos(\theta) \approx 1\): \(\theta = \frac{b\ddot{\theta} - a}{b\ddot{\theta}\theta + \frac{mg\theta}{T}}\) This equation is the equation of motion for the pendulum, which is the solution to part (a) of the exercise.

Step 8: Find the period of oscillations

To find the period of oscillations (part b), linearize the differential equation in Step 7 by assuming small angles (\(\theta \approx 0\)) and neglecting higher-order terms in \(\theta\): \(\ddot{\theta} + \frac{g}{b}\theta + \frac{a}{b^2}t = 0\) This equation is a linear, second-order, non-homogeneous ordinary differential equation. The homogeneous part of the equation is: \(\ddot{\theta} + \frac{g}{b}\theta = 0\) The homogeneous equation has a characteristic equation: \(p^2 + \frac{g}{b} = 0\) The roots of the characteristic equation are: \(p = \pm \sqrt{\frac{g}{b}}\) The general solution to the homogeneous equation is: \(\Theta(t) = C_1\cos(\sqrt{\frac{g}{b}}t) + C_2\sin(\sqrt{\frac{g}{b}}t)\) To find the period, notice that the coefficient of the sine term is: \(\sqrt{\frac{g}{b}}\) So the period T of the pendulum for small oscillations is: \(T = \frac{2\pi}{\sqrt{\frac{g}{b}}}\) This is the solution to part (b) of the exercise.

Key Concepts

Equations of Motion

In the context of pendulum motion, the equation of motion describes how the pendulum swings over time. For a pendulum attached to a massless support, we consider the forces acting on the bob and how these influences relate to its motion. Using a set of equations, we can predict the path and behavior of the pendulum.
The key equations involve the position coordinates of the bob, defined by the length of the pendulum and the angle of swing. Starting with the expressions:

• Position in the x-direction: \[ x = b \sin(\theta) - at \]
• Position in the y-direction: \[ y = b - b\cos(\theta) \]

Deriving the velocities by differentiating these with respect to time gives insights into the pendulum's speed in both horizontal and vertical components. Analyzing these, we relate each to the forces in action using Newton's laws to formulate a differential equation expressing the pendulum’s motion.

Small Oscillations

Small oscillations refer to the pendulum's motion when the angle of displacement from the vertical position is minimal. At such small angles, some simplifying approximations can be employed, allowing us to treat the pendulum's behavior in a more straightforward manner.
In small-angle approximations, we make use of the mathematical properties:

• \(\tan(\theta) \approx \theta\)
• \(\sin(\theta) \approx \theta\)
• \(\cos(\theta) \approx 1\)

By applying these approximations to the pendulum's motion equations, the resulting expressions become linear, making them easier to solve and analyze. This simplification allows us to derive the period of the pendulum where the dynamics are largely consistent and predictable over time.

Free Body Diagram

A Free Body Diagram (FBD) is an essential tool in physics for visualizing the various forces acting on an object or system, such as a pendulum.In the case of a pendulum, the FBD includes several forces:

• The gravitational force \( mg \) which acts downwards
• The tension force \( T \) in the string or rod, pointing along the length of the pendulum
• The inertial force \( ma \), reflecting the horizontal acceleration, acts horizontally opposite to the direction of the support's motion

Each force contributes to the equation of motion as their interactions define how the pendulum swings. By correctly identifying and representing these forces on the FBD, we can configure and solve the motion equations accurately, offering insight into how the pendulum will behave under varying conditions.

Newton's Second Law

Newton's Second Law is a cornerstone of dynamics, providing the framework to understand motion in terms of forces. It states that the force applied to an object is equal to the mass of the object multiplied by its acceleration ( \( F = ma \)).
For the pendulum, Newton's second law helps us set up the equations governing the swing motion, collaborated closely with the identified forces and their components on the FBD:

• Horizontal component: \[ T\sin(\theta) = m(b\ddot{\theta}\cos(\theta) - a) \]
• Vertical component: \[ T\cos(\theta) - mg = mb\ddot{\theta}\sin(\theta) \]

Using these equations, we eliminate the tension to get a differential equation in terms of \(\theta\), allowing us to explore solution pathways leading to insights on motion behavior and pendulum characteristics, particularly for small oscillations.