15.2 Energy in Simple Harmonic Motion
3 min read•june 24, 2024
is all about energy transfer in oscillating systems. As objects bounce back and forth, energy constantly shifts between potential and kinetic forms, following predictable patterns based on position and .
Understanding energy in these systems helps explain real-world oscillations, from playground swings to electrical circuits. We'll explore how to calculate energies, examine conservation principles, and see how forces relate to in equilibrium situations.
Energy in Simple Harmonic Motion
Energy calculations in harmonic oscillators
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- Potential energy () in a depends on the () and the from equilibrium () calculated using the formula
- Spring constant () measures the stiffness of the spring (N/m)
- () measures the distance from the (m)
- () in a simple harmonic oscillator depends on the mass () and the velocity () calculated using the formula
- Mass () is the amount of matter in the oscillating object (kg)
- Velocity () is the speed and direction of the oscillating object (m/s)
- Velocity in a simple harmonic oscillator can be calculated using the (), (), and displacement () with the formula
- () is the maximum displacement from equilibrium (m)
- () is the rate of (rad/s)
- () in a simple harmonic oscillator is the sum of potential and kinetic energy expressed as
- Total energy remains constant throughout the (J)
- This constant total energy is also known as the of the system
Conservation of energy in mass-spring systems
- In a simple harmonic oscillator, energy is continuously converted between potential and kinetic energy as the mass oscillates
- At the (), potential energy is zero and kinetic energy is at its maximum (all energy is in the form of motion)
- At the maximum displacement (), potential energy () is at its maximum and kinetic energy is zero (all energy is stored in the spring's compression or extension)
- Total energy remains constant throughout the oscillation, demonstrating the principle
- As the mass moves from equilibrium to maximum displacement, potential energy increases while kinetic energy decreases (energy is transferred from motion to spring)
- As the mass moves from maximum displacement to equilibrium, potential energy decreases while kinetic energy increases (energy is transferred from spring to motion)
- Conservation of energy in a simple harmonic oscillator can be expressed as
- Total energy () equals the maximum potential energy () and is always equal to the sum of instantaneous potential and kinetic energies
Force vs potential energy in equilibrium
- Force () in a simple harmonic oscillator is proportional to the displacement () and acts in the opposite direction, expressed as
- Negative sign indicates the acts opposite to the displacement
- Potential energy () is related to the force by
- Potential energy is the work done against the to displace the mass
- occurs when the potential energy is at a minimum
- A small displacement from the stable equilibrium position results in a restoring force that pushes the mass back towards equilibrium
- The potential energy curve around a is concave up (U-shaped)
- occurs when the potential energy is at a maximum
- A small displacement from the unstable equilibrium position results in a force that pushes the mass further away from equilibrium
- The potential energy curve around an unstable equilibrium point is concave down (inverted U-shaped)
Oscillations and Resonance
- An oscillation occurs when a system is displaced from its equilibrium position and experiences a restoring force
- The at which a system naturally oscillates is called its natural frequency
- occurs when an external force drives a system at its natural frequency, resulting in a large increase in the amplitude of oscillation
Key Terms to Review (43)
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 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.
Bar graphs of total energy: Bar graphs of total energy visually represent the distribution and conservation of energy in a system. Each bar corresponds to different forms of energy (e.g., kinetic, potential) at specific states or times.
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.
Displacement: Displacement is a vector quantity that refers to the change in position of an object. It is measured as the straight-line distance from the initial to the final position, along with the direction.
Displacement: Displacement is the change in position of an object relative to a reference point. It is a vector quantity, meaning it has both magnitude and direction, and is used to describe the movement of an object in physics.
Elastic potential energy: Elastic potential energy is the energy stored in elastic materials as a result of their stretching or compressing. It is quantified by the equation $U = \frac{1}{2} k x^2$, where $k$ is the spring constant and $x$ is the displacement from equilibrium.
Elastic Potential Energy: Elastic potential energy is the potential energy stored in an object due to its deformation or compression. It is the energy that is stored in an elastic material when it is stretched or compressed and has the ability to do work as the material returns to its original shape.
Equilibrium position: The equilibrium position is the point at which the net force acting on an oscillating system is zero. At this position, the system experiences no acceleration and remains at rest if undisturbed.
Equilibrium Position: The equilibrium position is the point at which a system is in a state of balance, where the net force or torque acting on the system is zero. This concept is fundamental in understanding the behavior of various physical systems, including those related to simple harmonic motion, circular motion, damped oscillations, and wave propagation.
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.
Hertz: Hertz (Hz) is the unit of frequency, which measures the number of cycles or oscillations that occur per second. It is a fundamental concept in physics, particularly in the study of wave phenomena, such as sound waves and electromagnetic waves.
Hooke's Law: Hooke's law is a fundamental principle in physics that describes the linear relationship between the force applied to an elastic object and the resulting deformation or displacement of that object. It is a crucial concept that underpins the understanding of various physical phenomena, including work, conservative and non-conservative forces, potential energy diagrams and stability, stress, strain, and elasticity, as well as simple harmonic motion.
Joule: A joule is the SI unit of work or energy, equivalent to one newton-meter. It represents the amount of work done when a force of one newton displaces an object by one meter in the direction of the force.
Joule: The joule (J) is the standard unit of energy in the International System of Units (SI). It represents the amount of work done or energy expended when a force of one newton acts through a distance of one meter.
Kinetic energy: Kinetic energy is the energy possessed by an object due to its motion. It depends on the mass and velocity of the object.
Lennard-Jones 6-12 potential: The Lennard-Jones 6-12 potential is a mathematical model that approximates the interaction between a pair of neutral atoms or molecules based on their distance. It is characterized by an attractive term proportional to $r^{-6}$ and a repulsive term proportional to $r^{-12}$, where $r$ is the interatomic distance.
Mass-Spring System: A mass-spring system is a physical model that consists of a mass attached to a spring, which represents a simple harmonic oscillator. This system is commonly used to study the principles of vibration, energy, and stability in various fields of physics.
Mechanical energy: Mechanical energy is the sum of kinetic energy and potential energy in a system. It is the energy associated with the motion and position of an object.
Mechanical Energy: Mechanical energy is the sum of the kinetic energy and potential energy possessed by an object due to its motion and position within a physical system. It represents the total energy available to do work or cause change in the system.
Newton's Second Law: Newton's Second Law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. It describes the relationship between an object's motion and the forces acting upon it, providing a quantitative framework for understanding the dynamics of physical systems.
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.
Oscillation: Oscillation is the repetitive variation, typically in time, of some measure about a central value or between two or more different states. It is commonly seen in mechanical systems like pendulums and springs.
Oscillation: Oscillation refers to the repetitive motion of an object or system back and forth between two or more positions or states. This periodic movement is a fundamental concept that underlies various physical phenomena, including the behavior of potential energy diagrams, the energy dynamics of simple harmonic motion, and the propagation of waves.
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.
Potential Energy: Potential energy is the stored energy possessed by an object due to its position or state, which can be converted into kinetic energy or other forms of energy when the object is moved or transformed. This term is central to understanding various physical phenomena and the conservation of energy.
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 force: A restoring force is a force that gives rise to an equilibrium in a physical system. It acts in the direction opposite to the displacement of the object, aiming to bring it back to its equilibrium position.
Restoring Force: The restoring force is the force that acts to return an object or system to its equilibrium or original state after it has been displaced or disturbed from that state. This force plays a crucial role in understanding various physical phenomena, including the behavior of oscillating systems, the stability of structures, and the energy changes associated with different types of motion.
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 harmonic oscillator: A simple harmonic oscillator is a system where the force acting on an object is directly proportional to its displacement from equilibrium and acts in the opposite direction. This results in periodic motion, such as that of a mass on a spring or a pendulum.
Spring Constant: The spring constant, often denoted as 'k', is a measure of the stiffness of a spring. It quantifies the force required to stretch or compress a spring by a unit distance, and it is a fundamental property of a spring that is crucial in understanding its behavior in various physical contexts.
Stability: Stability in oscillatory systems refers to the ability of a system to return to its equilibrium position after being disturbed. A stable system will exhibit bounded oscillations without diverging.
Stable Equilibrium: Stable equilibrium is a state of balance in a system where any small disturbance or displacement from the equilibrium position will result in a restoring force that pushes the system back towards its original state. This concept is crucial in understanding the behavior of physical systems and their tendency to maintain a state of stability.
Stable equilibrium point: A stable equilibrium point is a position where an object, when slightly displaced, experiences a net force or torque directed towards that position, causing it to return to equilibrium. In simple harmonic motion, this corresponds to the lowest potential energy configuration.
Total Energy: Total energy is the sum of all forms of energy possessed by an object or system, including kinetic energy, potential energy, and any other forms of energy that may be present. It represents the complete energy state of the system and is a fundamental concept in physics.
Turning points: Turning points are specific positions in the motion of an oscillating system where the object momentarily comes to rest before reversing its direction. These points correspond to the maximum and minimum displacements in simple harmonic motion.
Unstable Equilibrium: Unstable equilibrium refers to a state of balance where the slightest disturbance or perturbation can cause the system to move away from its initial position, leading to a significant change in the system's behavior. This term is particularly relevant in the context of potential energy diagrams, static equilibrium, and simple harmonic motion.
Velocity: Velocity is a vector quantity that describes the rate of change of an object's position with respect to time. It includes both the speed and the direction of an object's motion, making it a more complete description of an object's movement compared to just speed alone.