Physics - Classical Mechanics - Exercises around Pendulums

almost 3 years ago

Introduction

Hey it's a me again @drifter1!

Today we continue with Physics, and more specifically the branch of "Classical Mechanics" in order to get into Exercises around Pendulums.

So, without further ado, let's get straight into it!

Example 1: Simple Pendulum

Consider a simple pendulum of length L = 4.9 m and maximum angular displacement θ₀ = 0.5 rad, as shown in the figure below.

Let's:

calculate the angular frequency (ω), period (T) and frequency (f) of the periodic motion
write the angular displacement (θ) as a function of time. The initial phase is φ = 0 (maximum positive displacement - as in the figure)
calculate the maximum "tangent" acceleration (a_max)

Note: Approximate simple pendulum motion as SHM by applying the small angle approximation formulas. The acceleration of gravity is g = 9.8 m / s ².

Solution

Applying small angle approximation, the angular frequency equals:

And so, the oscillation period and frequency are:

The maximum angular displacement is known, and thus writing the angular displacement as a function of time is a simple substitution of the known values:

The only force acting upon the motion is gravity. Thus, taking the component of gravity which is tangent to the motion, the maximum acceleration equals:

The maximum acceleration can also be calculated by taking the square of the angular frequency and multiplying it with the amplitude. Of course, the amplitude in that case is equal to L sinθ₀, which is the maximum horizontal displacement, giving us:

With small angle approximation (sinθ ≈ θ) the maximum acceleration would be 4.9 m / s ² in both cases, which introduces a little bit of error, and so sin(0.5) ≈ 0.48 was used instead!

Continuing On...

In a similar notion, the maximum velocity equals:

And writing the velocity and acceleration as functions of time is as simple as:

Example 2: Physical Pendulum

Let's next consider a uniform rod of length d and mass m, that is pivoted at one end, as shown in the figure below. This physical pendulum starts oscillating with a maximum angular displacement of θ₀.

Express the angular frequency (ω) and period of oscillation (T) in terms of d and g only. How does the period change when the length of the rod is doubled?

Note: The moment of inertia at the end points of a rod is given by I_p = 1 / 3 md ². Approximate the physical pendulum oscillation as SHM.

Solution

The center of mass (and gravity) of a uniform rod is exactly at it's center, and so L = d / 2.

Thus, the angular frequency is given by:

and the period by:

For d' = 2d, the corresponding L' equals d, which gives a period of:

which, in turn, relates to the original period as follows:

Continuing On...

Let's suppose that m = 1 Kg, g = 9.8 m / s ² and d = 2 m. We can now estimate the moment of inertia by calculating the period (T) and solving the period equation for I. Let's afterwards compare that value to the expected one!

For the given values, the period equals:

and so the moment of inertia is:

The corresponding expected value is:

The results are very close and so the error can be considered infinitesimal!

RESOURCES:

References

Previous parts...

Images

https://commons.wikimedia.org/wiki/File:Pendulum-no-text.gif

Mathematical equations used in this article, where made using quicklatex.

Visualizations were made using draw.io.

Previous articles of the series

Rectlinear motion

Velocity and acceleration in a rectlinear motion -> velocity, acceleration and averages of those
Rectlinear motion with constant acceleration and free falling -> const acceleration motion and free fall
Rectlinear motion with variable acceleration and velocity relativity -> integrations to calculate pos and velocity, relative velocity
Rectlinear motion exercises -> examples and tasks in rectlinear motion

Plane motion

Position, velocity and acceleration vectors in a plane motion -> position, velocity and acceleration in plane motion
Projectile motion as a plane motion -> missile/bullet motion as a plane motion
Smooth Circular motion -> smooth circular motion theory
Plane motion exercises -> examples and tasks in plane motions

Newton's laws and Applications

Force and Newton's first law -> force, 1st law
Mass and Newton's second law -> mass, 2nd law
Newton's 3rd law and mass vs weight -> mass vs weight, 3rd law, friction
Applying Newton's Laws -> free-body diagram, point equilibrium and 2nd law applications
Contact forces and friction -> contact force, friction
Dynamics of Circular motion -> circular motion dynamics, applications
Object equilibrium and 2nd law application examples -> examples of object equilibrium and 2nd law applications
Contact force and friction examples -> exercises in force and friction
Circular dynamic and vertical circle motion examples -> exercises in circular dynamics
Advanced Newton law examples -> advanced (more difficult) exercises

Work and Energy

Work and Kinetic Energy -> Definition of Work, Work by a constant and variable Force, Work and Kinetic Energy, Power, Exercises
Conservative and Non-Conservative Forces -> Conservation of Energy, Conservative and Non-Conservative Forces and Fields, Calculations and Exercises
Potential and Mechanical Energy -> Gravitational and Elastic Potential Energy, Conservation of Mechanical Energy, Problem Solving Strategy & Tips
Force and Potential Energy -> Force as Energy Derivative (1-dim) and Gradient (3-dim)
Potential Energy Diagrams -> Energy Diagram Interpretation, Steps and Example
Internal Energy and Work -> Internal Energy, Internal Work

Momentum and Impulse

Conservation of Momentum -> Momentum, Conservation of Momentum
Elastic and Inelastic Collisions -> Collision, Elastic Collision, Inelastic Collision
Collision Examples -> Various Elastic and Inelastic Collision Examples
Impulse -> Impulse with Example
Motion of the Center of Mass -> Center of Mass, Motion analysis with examples
Explaining the Physics behind Rocket Propulsion -> Required Background, Rocket Propulsion Analysis

Angular Motion

Angular motion basics -> Angular position, velocity and acceleration
Rotation with constant angular acceleration -> Constant angular acceleration, Example
Rotational Kinetic Energy & Moment of Inertia -> Rotational kinetic energy, Moment of Inertia
Parallel Axis Theorem -> Parallel axis theorem with example
Torque and Angular Acceleration -> Torque, Relation to Angular Acceleration, Example
Rotation about a moving axis (Rolling motion) -> Fixed and moving axis rotation
Work and Power in Angular Motion -> Work, Work-Energy Theorem, Power
Angular Momentum -> Angular Momentum and its conservation
Explaining the Physics behind Mechanical Gyroscopes -> What they are, History, How they work (Precession, Mathematical Analysis) Difference to Accelerometers
Exercises around Angular motion -> Angular motion examples

Equilibrium and Elasticity

Rigid Body Equilibrium -> Equilibrium Conditions of Rigid Bodies, Center of Gravity, Solving Equilibrium Problems
Force Couple System -> Force Couple System, Example
Tensile Stress and Strain -> Tensile Stress, Tensile Strain, Young's Modulus, Poisson's Ratio
Volumetric Stress and Strain -> Volumetric Stress, Volumetric Strain, Bulk's Modulus of Elasticity, Compressibility
Cross-Sectional Stress and Strain -> Shear Stress, Shear Strain, Shear Modulus
Elasticity and Plasticity of Common Materials -> Elasticity, Plasticity, Stress-Strain Diagram, Fracture, Common Materials
Rigid Body Equilibrium Exercises -> Center of Gravity Calculation, Equilibrium Problems
Exercises on Elasticity and Plasticity -> Young Modulus, Bulk Modulus and Shear Modulus Examples

Gravity

Newton's Law of Gravitation -> Newton's Law of Gravity, Gravitational Constant G
Weight: The Force of Gravity -> Weight, Gravitational Acceleration, Gravity on Earth and Planets of the Solar System
Gravitational Fields -> Gravitational Field Mathematics and Visualization
Gravitational Potential Energy -> Gravitational Potential Energy, Potential and Escape Velocity
Exercises around Newtonian Gravity (part 1) -> Examples on the Universal Law of Gravitation
Exercises around Newtonian Gravity (part2) -> Examples on Gravitational Fields and Potential Energy
Explaining the Physics behind Satellite Motion -> The Circular Motion of Satellites
Kepler's Laws of Planetary Motion -> Kepler's Story, Elliptical Orbits, Kepler's Laws
Spherical Mass Distributions -> Spherical Mass Distribution, Gravity Outside and Within a Spherical Shell, Simple Examples
Earth's Rotation and its Effect on Gravity -> Gravity on Earth, Apparent Weight
Black Holes and Schwarzschild Radius -> Black Holes (Creation, Types, How To "See" Them), Schwarzschild Radius

Periodic Motion

Periodic Motion Fundamentals -> Fundamentals (Period, Frequency, Angular Frequency, Return Force, Acceleration, Velocity, Amplitude), Simple Harmonic Motion, Example
Energy in Simple Harmonic Motion -> Forms of Energy in SHM (Potential, Kinetic, Total and Maximum Energy, Maximum Velocity), Simple Example
Simple Harmonic Motion Equations -> SHM Equations (Displacement, Velocity, Acceleration, Phase Angle, Amplitude)
Simple Harmonic Motion and Reference Circle -> SHM and Smooth Circular Motion, Reference Circle
Simple Harmonic Motion Exercises -> 2 Complete Examples on Simple Harmonic Motion
Simple Pendulum -> Simple Pendulum (Return Force, Small Angle Approximations, More Accurate Period, Gravity Approximation)
Physical Pendulum -> Physical Pendulum (Return Torque, Small Angle Approximations, Estimating Moment of Inertia)