16.4 The Simple Pendulum
3 min read•june 18, 2024
A is a fascinating example of harmonic motion. It consists of a suspended by a string, swinging back and forth due to gravity. The pendulum's depends on its length and local gravity, not on the bob's mass or swing .
Understanding pendulums helps us grasp key concepts in oscillatory motion. We can calculate a pendulum's using a simple formula, and explore how changing its length affects its swing time. This knowledge has practical applications in timekeeping and beyond.
Simple Pendulum Mechanics
Pendulum as harmonic oscillator
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- consists of a bob (point mass) suspended by a massless string from a fixed point
- When displaced from equilibrium position, pendulum experiences a due to gravity
- For small angles of displacement (less than 15°), is approximately proportional to displacement
- Linear relationship between force and displacement is characteristic of a
- Pendulum oscillates back and forth with a constant period, as long as angle remains small
- Period is independent of bob mass and oscillation amplitude (maximum displacement from equilibrium)
- Motion of pendulum described by equation:
- is angular displacement
- is pendulum length
- is acceleration due to gravity (9.81 m/s^2 on Earth)
Calculation of pendulum period
- Period () of simple pendulum is time taken to complete one full oscillation
- Period depends on pendulum length () and acceleration due to gravity ()
- Formula for period of simple pendulum:
- is mathematical constant pi (3.14159)
- To calculate period:
- Measure length () of pendulum from to center of bob
- Determine acceleration due to gravity () at pendulum location
- Substitute and values into formula and calculate period ()
- Example: pendulum with m on Earth ( m/s^2) has period s
- () of the pendulum is related to the period:
Length vs period in pendulums
- Period of simple pendulum is directly proportional to square root of its length
- Increasing pendulum length increases period
- Decreasing pendulum length decreases period
- Relationship between period () and length () expressed as:
- If pendulum length is doubled, period increases by factor of (1.41)
- Example: 1 m pendulum with 2 s period, 2 m pendulum has 2.83 s period ()
- If pendulum length is halved, period decreases by factor of (0.71)
- Example: 1 m pendulum with 2 s period, 0.5 m pendulum has 1.41 s period ()
- Pendulum length can be adjusted to achieve desired period for applications like clocks
Energy and forces in a pendulum
- As the pendulum swings, energy is continuously converted between potential and kinetic forms
- At the highest points of swing, the pendulum has maximum and minimum
- At the lowest point (equilibrium position), the pendulum has maximum kinetic energy and minimum potential energy
- The in the string varies throughout the swing, being greatest at the bottom of the swing
- In real-world pendulums, effects cause a gradual decrease in amplitude over time due to air resistance and friction
Key Terms to Review (27)
$ ext{pi}$: $ ext{pi}$ is a mathematical constant that represents the ratio of a circle's circumference to its diameter. It is an irrational number, meaning its decimal representation never repeats or terminates, and it is commonly approximated as 3.14159. $ ext{pi}$ is a fundamental constant in mathematics, physics, and various other scientific disciplines.
$\frac{d^2\theta}{dt^2} + \frac{g}{L}\theta = 0$: This equation, known as the differential equation of motion for a simple pendulum, describes the angular acceleration of a simple pendulum as a function of the angular displacement. It is a fundamental equation in the study of simple pendulum dynamics and is derived from the application of Newton's second law of motion to the pendulum system.
$\theta$: $\theta$ is a symbol commonly used to represent an angle in various physical contexts, especially in rotational dynamics and oscillatory motion. In the context of rotational kinetic energy, $\theta$ helps describe the orientation of an object and its angular displacement. In relation to a simple pendulum, $\theta$ is crucial for understanding the angular position of the pendulum relative to its resting vertical position, influencing calculations of potential and kinetic energy throughout its swing.
$g$: $g$ is the acceleration due to gravity, a fundamental constant that describes the acceleration experienced by an object near the Earth's surface due to the force of gravity. This term is crucial in understanding various physical phenomena, including fluid dynamics and the motion of pendulums.
$L$: $L$ is a fundamental quantity that describes the inductance of an electrical circuit or component. Inductance is a measure of the magnetic field produced by an electric current, and it is a crucial concept in understanding the behavior of circuits involving inductors, transformers, and other electromagnetic devices.
$T \propto \sqrt{L}$: $T \propto \sqrt{L}$ is a mathematical relationship that describes the period of a simple pendulum. The period, or the time it takes for the pendulum to complete one full oscillation, is proportional to the square root of the length of the pendulum. This relationship is a fundamental principle in the study of simple pendulum motion.
$T = 2 ext{pi} ext{sqrt}{ rac{L}{g}}$: $T = 2 ext{pi} ext{sqrt}{ rac{L}{g}}$ is a fundamental equation that describes the period of a simple pendulum. The period, denoted by $T$, is the time it takes for the pendulum to complete one full oscillation or cycle. This equation relates the period to the length of the pendulum, $L$, and the acceleration due to gravity, $g$, which are the two key parameters that determine the behavior of a simple pendulum.
The equation shows that the period of a simple pendulum is proportional to the square root of the length of the pendulum and inversely proportional to the acceleration due to gravity. This relationship is crucial in understanding the dynamics of a simple pendulum and its applications in various fields, such as timekeeping, physics experiments, and the study of oscillatory motion.
Amplitude: Amplitude refers to the maximum extent of a vibration or oscillation, measured from the position of equilibrium. It plays a crucial role in understanding how energy is transferred in oscillatory systems, impacting the characteristics of waves and sounds.
Angular frequency: Angular frequency is a measure of how quickly an object oscillates or rotates, typically expressed in radians per second. It relates to the periodic motion of systems, connecting the time taken to complete a full cycle (the period) with how many cycles occur per unit of time (the frequency). This concept plays a crucial role in understanding various physical phenomena involving oscillations and circular motion.
Bob: Bob is a term used in the context of the simple pendulum, which is a fundamental concept in classical mechanics. A simple pendulum is a weight suspended from a fixed point by a lightweight, inextensible string or rod, and its motion is governed by the force of gravity and the tension in the string.
Critical damping: Critical damping occurs when a damping force is applied to an oscillating system, bringing it to rest in the shortest possible time without oscillation. It represents the threshold between overdamping and underdamping.
Damping: Damping refers to the process of reducing or controlling the amplitude of an oscillating or vibrating system over time. It involves the dissipation of energy, which causes the system to gradually come to rest or a steady-state condition.
Elastic potential energy: Elastic potential energy is the energy stored in an object when it is deformed elastically, such as when a spring is stretched or compressed. It can be calculated using the formula $U = \frac{1}{2} k x^2$, where $k$ is the spring constant and $x$ is the displacement from equilibrium.
Harmonic Oscillator: A harmonic oscillator is a system that exhibits oscillations, or repetitive motion, around an equilibrium position. It is a fundamental concept in physics that describes the behavior of various physical systems, including mechanical, electrical, and quantum-mechanical systems.
Internal kinetic energy: Internal kinetic energy is the sum of the kinetic energies of all particles within a system. It plays a crucial role in understanding how energy is distributed and conserved during elastic collisions.
Kinetic Energy: Kinetic energy is the energy of motion possessed by an object. It is the energy an object has by virtue of being in motion and is directly proportional to the mass of the object and the square of its velocity. Kinetic energy is a crucial concept in physics, as it relates to the work done on an object, the conservation of energy, and various other physical phenomena.
Period: The period is the time it takes for one complete cycle of an oscillation or wave to occur. It is typically measured in seconds.
Period: The period of an oscillation or wave is the time taken for one complete cycle to occur. It represents the time interval between successive repetitions of a particular state or event in a periodic motion or wave. This term is crucial in understanding various concepts related to oscillations, simple harmonic motion, pendulums, and waves.
Potential Energy: Potential energy is the stored energy an object possesses due to its position or state, which can be converted into kinetic energy or other forms of energy. This term is central to understanding various physical phenomena and energy transformations in the context of introductory college physics.
Restoring force: A restoring force is a force that acts to bring a system back to its equilibrium position. It is directly proportional to the displacement and acts in the opposite direction.
Restoring Force: The restoring force is a force that acts to return a system to its equilibrium or resting state after it has been displaced or disturbed. This force arises from the inherent properties of the system and acts to counteract the external forces that caused the displacement, thereby restoring the system to its original position or configuration.
Simple harmonic oscillator: A simple harmonic oscillator is a system where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement. It exhibits periodic motion characterized by sinusoidal oscillations.
Simple pendulum: A simple pendulum consists of a mass (called the bob) attached to the end of a lightweight string or rod that swings freely under the influence of gravity. The motion is periodic and can be described by harmonic motion for small angles.
Simple Pendulum: A simple pendulum is a weight (known as a pendulum bob) suspended from a fixed point by a light, inextensible string or rod. It is a classic example of an oscillating system, where the weight swings back and forth due to the force of gravity and the tension in the string.
Small-Angle Approximation: The small-angle approximation is a mathematical simplification used in physics when the angle of a quantity is very small. It allows for the simplification of trigonometric functions, making calculations more manageable without significantly affecting the accuracy of the results.
Suspension Point: The suspension point refers to the point at which a pendulum is attached to its support, allowing it to swing freely. This point is crucial in the study of the simple pendulum, as it determines the dynamics and behavior of the pendulum's motion.
Tension: Tension is the force transmitted through a string, rope, cable, or similar object when it is pulled tight by forces acting from opposite ends. This concept is crucial in understanding how forces interact in various systems, as it provides insights into how objects transmit forces and maintain equilibrium.