Name: simple harmonic motion: energy and speed
Uploaded: 2025-05-22T09:05:41.000Z
Duration: 1 min 23 s
Channel: JoVE
Description: Simple harmonic motion combines potential energy and kinetic energy. In a spring-like oscillator, Hooke’s law explains the potential energy stored when the string or spring is stretched or compressed. Because there are no dissipative forces, the object also has kinetic energy as it moves.

K12 Standards

Pennsylvania State Academic Standards

9-12

Physical Science

Energy

3.2.9-12.P

Energy in Simple Harmonic Motion

Consider a block of mass m attached to a spring with force constant k on a frictionless surface at equilibrium.
When the block is displaced, the work done by the spring force along the displacement from its initial to final position equals the amount of potential energy stored in the spring relative to its displacement. This is called elastic potential energy.
When released, the block undergoes simple harmonic motion. The energy required for its back-and-forth movement is called translational kinetic energy, and is directly proportional to the square of its velocity and its mass.
During oscillations, both energies continuously interchange and are represented by sinusoidal waveforms. The elastic potential energy is at maximum at the maximum displacement, while the translational kinetic energy is at maximum at the equilibrium position.
At other positions, the block has different kinetic and potential energy values, and their sum equals the system&#39;s total energy.
So, the total energy in the system remains constant and conserved, as it oscillates between translational kinetic energy and elastic potential energy.

energy in simple harmonic motion

To determine the energy of a simple harmonic oscillator, consider all the forms of energy it can have during its simple harmonic motion. According to Hooke's Law, the energy stored during the compression/stretching of a string in a simple harmonic oscillator is potential energy. As the simple harmonic oscillator has no dissipative forces, it also possesses kinetic energy. In the presence of conservative forces, both energies can interconvert during oscillation, but the total energy remains constant. The total energy for a simple harmonic oscillator is equal to the sum of the potential and kinetic energy and is proportional to the square of the amplitude. It can be expressed in the following form:
<img alt="Energy conservation equation; formula: 1/2mv&#178; + 1/2kx&#178; = 1/2kA&#178;; physics theory." src="/files/ftp_upload/12765/12765_Equation_1.svg" style="text-align: center; height: 35px;" data-altupdatedat="1754904545000" data-alt="Conservation of energy equation, physics formula." />
The magnitude of the velocity in a simple harmonic motion is obtained by rearranging and solving the equations of the total energy.
<img alt="Simple harmonic motion velocity equation, \(v = \pm \sqrt{\frac{k}{m}(A^2 - x^2)}\), formula." src="/files/ftp_upload/12765/12765_Equation_2.svg" style="text-align: center; height: 42px;" data-altupdatedat="1754904545000" data-alt="Harmonic motion equation: v=&#177;&#8730;(k/m)(A&#178;-x&#178;); mechanical oscillation concept." />
Manipulating this expression algebraically gives the following:
<img alt="Elliptical motion equation. Formula v=&#177;vmax&#8730;(1&#8722;x&#178;/A&#178;). Mathematical representation." src="/files/ftp_upload/12765/12765_Equation_3.svg" style="text-align: center; height: 50px;" data-altupdatedat="1754904545000" data-alt="Hyperbolic motion equation: v = &#177; vmax&#8730;(1-x&#178;/A&#178;); physics formula analysis." />
where
<img alt="Static equilibrium formula diagram, \( V_{max} = \sqrt{\frac{k}{m}}A \), physics concept." src="/files/ftp_upload/12765/12765_Equation_4.svg" style="text-align: center; height: 42px;" data-altupdatedat="1754904545000" data-alt="Vmax equation with square root, physics formula for maximum velocity, educational diagram." />
Notice that the maximum velocity depends on three factors and is proportional to the amplitude. If the displacement is maximal, the velocity will also be maximal. Additionally, the maximum velocity is greater for stiffer systems because they exert greater force for the same displacement. This observation can be seen in the expression for the maximum velocity. The maximum velocity is proportional to the square root of the force constant. Finally, the maximum velocity is smaller for objects with larger masses since the maximum velocity is inversely proportional to the square root of the mass.

To determine the energy of a simple harmonic oscillator, consider all the forms of energy it can have during its simple harmonic motion. According to ...

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KEY TERMS AND CONCEPTS

simple harmonic motion: energy and speed

Simple harmonic motion combines potential energy and kinetic energy. In a spring-like oscillator, Hooke’s law explains the potential energy stored when the string or spring is stretched or compressed. Because there are no dissipative forces, the object also has kinetic energy as it moves.

Simple Harmonic Motion: Energy and Speed

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Period and Frequency in Simple Harmonic Motion

Simple harmonic motion describes oscillatory motion when the net force follows Hooke's law. In this type of motion, the force acts toward the equilibrium position. If there is no damping from friction or other non-conservative forces, the oscillator moves with equal displacement on both sides of equilibrium. The period of a simple harmonic oscillator is found from the equation of motion. Period is the time for one complete oscillation. Frequency is the inverse of period, so the two values can...

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Simple harmonic motion depends on how displacement, force, and acceleration work together. In this motion, the acceleration of the system, and therefore the net force, is proportional to the displacement and points in the opposite direction. That restoring force pulls the object back toward equilibrium. The period and frequency of a simple harmonic oscillator do not depend on amplitude. A diving board shows this idea well. A stiff, thick diving board has a large force constant, so it has a...

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Simple harmonic motion combines potential energy and kinetic energy. In a spring-like oscillator, Hooke’s law explains the potential energy stored when the string or spring is stretched or compressed. Because there are no dissipative forces, the object also has kinetic energy as it moves.

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A spring-mass system shows simple harmonic motion, and its oscillation rate depends on the mass and the spring force constant. In this model, the angular frequency, period, and frequency are set by those two values. They do not depend on the motion’s amplitude or on the initial conditions. To find these relationships, the system can be analyzed with Newton’s second law. Imagine a block on a spring on a frictionless surface. Three forces act on the mass: weight, the normal force, and the spring...

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Simple Harmonic Motion from Circular Motion

Simple harmonic motion and uniform circular motion are closely linked. Simple harmonic motion is oscillatory motion in a system where the net force follows Hooke’s law. Uniform circular motion is motion along a circular path at constant speed. A simple way to create simple harmonic motion is to use uniform circular motion. One example is a ball attached to a vertical turntable that spins at a steady rate. If you watch the ball’s shadow on the floor, the shadow, or projection, moves in simple...

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key terms and concepts

Learn how simple harmonic motion stores potential and kinetic energy, why total energy stays constant, and what controls maximum velocity.

Simple Harmonic Motion: Energy and Speed | JoVE K12

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3.2.9-12.Q

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3.2.9-12.R

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Simple Harmonic Motion: Energy and Speed

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