15.4
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…
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'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.
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Q1: What is elastic potential energy in a spring-mass system?
Elastic potential energy is the energy stored in a spring when it is compressed or stretched from equilibrium. According to Hooke's Law, the work done by the spring force during displacement equals the potential energy stored relative to that displacement. This energy is maximum when the block reaches maximum displacement and is directly related to how far the spring is stretched or compressed.
Q2: How does translational kinetic energy change during simple harmonic motion?
Translational kinetic energy is the energy of motion and is directly proportional to both the mass and the square of the velocity. During oscillation, kinetic energy is maximum at the equilibrium position where velocity is highest, and zero at maximum displacement where the block momentarily stops. As the block moves, kinetic energy continuously converts to and from potential energy.
Q3: Why does total energy remain constant in a simple harmonic oscillator?
Total energy remains constant because the system has no dissipative forces like friction. The sum of elastic potential energy and translational kinetic energy stays the same throughout oscillation. Energy continuously interconverts between these two forms—when potential energy decreases, kinetic energy increases proportionally, and vice versa, maintaining constant total energy.
Q4: What factors determine maximum velocity in simple harmonic motion?
Maximum velocity depends on three factors: amplitude, force constant, and mass. Maximum velocity is proportional to amplitude and the square root of the force constant, but inversely proportional to the square root of mass. Stiffer springs produce greater maximum velocities for the same amplitude, while heavier objects move more slowly at maximum velocity.
Q5: How are potential and kinetic energy distributed at different positions during oscillation?
At maximum displacement, elastic potential energy is maximum and kinetic energy is zero. At equilibrium position, kinetic energy is maximum and potential energy is zero. At intermediate positions, both energies have intermediate values that sum to the constant total energy. This energy distribution creates the sinusoidal waveforms characteristic of simple harmonic motion.
Q6: How is total energy related to amplitude in a simple harmonic oscillator?
Total energy in a simple harmonic oscillator is proportional to the square of the amplitude. This means doubling the amplitude quadruples the total energy. Since total energy equals the sum of potential and kinetic energy, and this sum remains constant throughout oscillation, the amplitude directly determines how much total energy the system possesses.
Q7: Why is kinetic energy maximum at the equilibrium position?
At equilibrium, the spring exerts no restoring force, so all the system's energy exists as kinetic energy. The block passes through equilibrium with maximum velocity because all elastic potential energy has been converted to motion energy. Beyond equilibrium, the spring begins pulling the block back, converting kinetic energy back into potential energy.