15.4 Pendulums
3 min read•june 24, 2024
Simple pendulums are fascinating oscillators that demonstrate key principles of physics. They consist of a mass suspended by a string, swinging back and forth due to gravity and tension forces. Their motion is predictable and can be described using simple equations.
Physical and torsional pendulums build on these concepts, introducing more complex factors. Physical pendulums involve extended objects, while torsional pendulums twist around an axis. Both types showcase how mass distribution and material properties affect oscillation periods.
Simple Pendulums
Forces on simple pendulums
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- acts along the string towards the point of suspension, equal in magnitude and opposite in direction to the pendulum's weight ()
- Weight () acts vertically downward, depends on the pendulum's mass and gravitational acceleration (, )
- These forces create a that causes the pendulum to oscillate back and forth ()
- The restoring torque is proportional to the from equilibrium
Calculations for pendulum motion
- () calculated using , where is gravitational acceleration (Earth's gravity) and is the length of the pendulum (string length)
- () represents the number of oscillations per unit time (seconds), calculated using
- () is the time required for one complete oscillation (back and forth), calculated using
- Increasing the length of the pendulum increases the period and decreases the frequency (slower oscillations)
- Increasing gravitational acceleration (different planets) decreases the period and increases the frequency (faster oscillations)
Energy and Motion Characteristics
- The pendulum's motion involves the continuous conversion between potential and kinetic energy
- The maximum of the swing corresponds to the highest point of potential energy
- As the pendulum swings, energy is gradually lost due to effects (air resistance, friction)
- can occur when an external force is applied at the pendulum's natural frequency, leading to increased
Physical and Torsional Pendulums
Simple vs physical pendulums
- consists of an extended object oscillating about a fixed axis (), while a is a point mass suspended by a massless string
- Period of a depends on the object's () and the distance between the and the (), calculated using
- period depends only on the length of the string () and gravitational acceleration (), calculated using
- Physical pendulums have a more complex motion due to the distribution of mass (non-uniform density)
Factors in torsional pendulum periods
- () depends on the material properties (stiffness) and geometry (thickness, length) of the
- Higher results in a shorter period (stiffer wire, faster oscillations)
- Lower results in a longer period (more flexible wire, slower oscillations)
- () depends on the mass distribution of the oscillating object (shape, size)
- Higher results in a longer period (more mass farther from the axis of rotation)
- Lower results in a shorter period (less mass or mass concentrated closer to the axis)
- Period of a calculated using , combining the effects of torsional constant and moment of inertia
Key Terms to Review (34)
Amplitude: Amplitude is the maximum displacement of a point on a wave from its equilibrium position. It is a measure of the energy carried by the wave.
Amplitude: Amplitude is the maximum displacement or extent of a periodic motion, such as a wave or an oscillation, from its equilibrium position. It represents the magnitude or size of the motion and is a fundamental characteristic of various physical phenomena described in the topics of 1.7 Solving Problems in Physics, 8.4 Potential Energy Diagrams and Stability, 15.1 Simple Harmonic Motion, and beyond.
Angular Displacement: Angular displacement is a measure of the change in the angular position of an object or a system. It describes the rotation or the change in the orientation of an object around a fixed axis or point. This concept is fundamental in understanding rotational motion and its relationship with linear motion in various physics topics.
Angular frequency: Angular frequency, denoted by $\omega$, is the rate of change of angular displacement with time. It is commonly measured in radians per second (rad/s).
Angular Frequency: Angular frequency, often represented by the Greek letter $\omega$ (omega), is a fundamental concept that describes the rate of change of the angular position of an object undergoing rotational or oscillatory motion. It is a crucial parameter in understanding various physical phenomena, including simple harmonic motion, wave propagation, and the behavior of oscillating systems.
Center of mass: The center of mass is the point in an object or system where all its mass can be considered to be concentrated for the purpose of analyzing translational motion. It is the weighted average position of all the mass in the system.
Center of Mass: The center of mass is the point at which an object's entire mass can be considered to be concentrated. It is the average position of the mass of an object, and it is the point around which the object's rotation and motion can be analyzed.
Conservation of Energy: The conservation of energy principle states that energy cannot be created or destroyed, only transformed from one form to another. This fundamental concept links various phenomena, illustrating how mechanical, kinetic, and potential energies interconvert while keeping the total energy constant in a closed system.
Damping: Damping refers to the process of reducing or controlling the amplitude or oscillation of a system over time. It is a phenomenon that occurs in various physical systems, including mechanical, electrical, and electronic systems, where it serves to dissipate energy and prevent excessive vibrations or oscillations.
Earth's gravity: Earth's gravity is the natural force that attracts objects with mass towards the center of the planet, causing them to experience weight. This gravitational pull is responsible for various phenomena such as free fall, projectile motion, and the oscillatory motion of pendulums. It plays a crucial role in determining how objects move and interact within Earth's environment.
Frequency: Frequency is a fundamental concept in physics that describes the number of occurrences of a repeating event or phenomenon per unit of time. It is a crucial parameter in various areas of physics, including wave behavior, oscillations, and sound propagation.
Gravitational Force: Gravitational force is the attractive force that exists between any two objects with mass. It is the force that causes objects to be pulled towards each other, and is the fundamental force responsible for the motion of celestial bodies and the behavior of objects on Earth.
Moment of inertia: Moment of inertia is a measure of an object's resistance to changes in its rotational motion about a fixed axis. It depends on the mass distribution relative to the axis of rotation.
Moment of Inertia: The moment of inertia is a measure of an object's resistance to rotational acceleration. It is a scalar quantity that depends on the mass and distribution of an object's mass about a given axis of rotation. The moment of inertia is a crucial concept in the study of rotational dynamics, as it determines how an object will respond to applied torques.
Newton's Third Law: Newton's Third Law, also known as the law of action and reaction, states that for every action, there is an equal and opposite reaction. This fundamental principle of physics describes the relationship between forces acting on interacting objects.
Orbital period: The orbital period is the time taken for a satellite or celestial body to complete one full orbit around another object. It is typically measured in seconds, minutes, hours, or years.
Period: The period of a periodic phenomenon is the time taken for one complete cycle or repetition of the event. This concept is fundamental in understanding various physics topics, including uniform circular motion, simple harmonic motion, and wave phenomena.
Physical pendulum: A physical pendulum is a rigid body that oscillates about a pivot point, where the mass distribution of the body affects its oscillatory motion. Unlike an ideal simple pendulum, its period depends on the shape and mass distribution.
Physical Pendulum: A physical pendulum is a rigid body that oscillates about a fixed axis under the influence of gravity. Unlike an ideal mathematical pendulum, which is a point mass attached to a massless string, a physical pendulum has a finite mass and size, and the axis of rotation is not part of the pendulum itself.
Pivot Point: A pivot point is the fixed point around which an object rotates or swings. In the context of pendulums, it is crucial for understanding how the pendulum moves back and forth as it swings under the influence of gravity. The pivot point remains stationary while the pendulum swings, allowing it to convert potential energy at its highest point to kinetic energy at its lowest point.
Resonance: Resonance occurs when a system is driven at its natural frequency, leading to a significant increase in amplitude. It is a crucial concept in oscillations and wave phenomena.
Resonance: Resonance is a phenomenon that occurs when a system is driven by a force that matches the system's natural frequency of oscillation, leading to a significant increase in the amplitude of the system's response. This concept is fundamental across various fields in physics, including mechanics, acoustics, and electromagnetism.
Restoring Torque: Restoring torque is the torque that acts on an object to return it to its equilibrium or stable position. It is a crucial concept in the study of pendulums and other oscillating systems, as it is the force that drives the object's motion back towards its equilibrium state.
Rigid Body: A rigid body is an idealized object that maintains its shape and size regardless of the forces acting upon it. It is a fundamental concept in classical mechanics that simplifies the analysis of the motion and behavior of objects.
Simple Harmonic Motion: Simple harmonic motion is a type of periodic motion where the restoring force acting on an object is directly proportional to its displacement from the equilibrium position. This motion is characterized by a sinusoidal pattern and is found in various physical systems, including pendulums, mass-spring systems, and vibrating molecules.
Simple harmonic motion (SHM): Simple Harmonic Motion (SHM) is a type of periodic motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement. It is characterized by its sinusoidal oscillations in time.
Simple pendulum: A simple pendulum consists of a mass (called the bob) attached to the end of a string or rod of fixed length, which swings freely in a vertical plane under the influence of gravity. It exhibits periodic motion with its restoring force proportional to the sine of its displacement angle.
Simple Pendulum: A simple pendulum is a weight suspended from a fixed point by a light, inextensible string or rod, and allowed to swing back and forth under the influence of gravity. It is a classic example of a system that exhibits simple harmonic motion.
Small-Angle Approximation: The small-angle approximation is a mathematical simplification used in various physics contexts, including the analysis of pendulum motion. It allows for the replacement of trigonometric functions with simpler linear expressions when the angle of interest is small, simplifying calculations and providing accurate approximations.
Suspension Wire: A suspension wire is a type of wire or cable used to support the pendulum of a clock or other oscillating device. It provides a flexible connection that allows the pendulum to swing freely while maintaining its position within the clock's mechanism.
Tension Force: Tension force is a type of contact force that acts between two objects that are connected by a string, rope, cable, or some other medium. It is the force that pulls on an object, keeping it from moving away from the point of attachment.
Torsional Constant: The torsional constant, also known as the torsion constant, is a measure of a material's resistance to twisting or torsional deformation. It is a crucial parameter in the analysis and design of structures, mechanical systems, and devices that experience torsional loads or moments.
Torsional pendulum: A torsional pendulum is an oscillatory system where an object suspended by a wire or rod twists back and forth around its axis due to torsional restoring forces. The angular displacement and period of oscillation are key characteristics of this system.
Torsional Pendulum: A torsional pendulum is a type of pendulum that oscillates by twisting about its vertical axis. This motion occurs when a restoring torque, caused by a spring-like behavior of the twisted medium, acts to return the pendulum to its equilibrium position. Torsional pendulums are essential in studying rotational dynamics and can illustrate concepts like torque, moment of inertia, and angular frequency.