Simple harmonic energy is the energy that an oscillator possesses when it performs simple harmonic motion (SMH). During simple harmonic motion, energy is being continuously exchanged between kinetic and potential energies.
Simple harmonic motion is the periodic oscillation where the acceleration of an oscillator is proportional to the displacement but acts in the opposite direction.
Kinetic energy is the energy that is acquired when a mass m is in motion, while potential energy is the energy stored in the oscillator when it has been displaced from its equilibrium position. The potential kinetic energy can be in the form of:
- Gravitational potential energy, as in the case of a mass on a pendulum that is in a different position than the equilibrium.
- Elastic potential energy, as in the case of a mass on a spring.
What is the energy in a simple harmonic system?
The net energy in a simple harmonic system is constant and is equal to the sum of kinetic and potential energy.
This can be interpreted as follows:
- As the two forms of energy are interchanging continuously, when one form increases, the other decreases. Hence, when one form of energy reaches its maximum point, the other form reaches its minimum value (zero).
- If the energy is constant, the system is oscillating.
Let’s assume we have a mass on a pendulum starting to oscillate from its equilibrium position. The sequence of motion is shown in figure 1 below, where the pendulum moves from the initial position to the maximum position on the right of the equilibrium point, which is considered to be the maximum positive position.
The pendulum will return to its equilibrium point but with the opposite direction of acceleration. The pendulum will then move to the maximum position on the left of the equilibrium point, which is considered to be the maximum negative position.
Figure 1.Simple Harmonic Motion,Georgia Panagi, StudySmarter.
The form of energy in the pendulum depends on its position. The pendulum initially gains kinetic energy as soon as it starts to oscillate, while potential energy is at its minimum (0). When the pendulum reaches maximum amplitude, it momentarily stops moving. The kinetic energy decreases to 0, while potential energy reaches its maximum.
When the pendulum continues to move in the opposite direction, the kinetic energy starts to increase while potential energy decreases. When the pendulum reaches its equilibrium point, completing one periodic cycle, the kinetic energy reaches its maximum, and the potential energy again reaches its minimum.
Mechanical energy conservation in simple harmonic motion
In an ideal oscillator, the mechanical energy is conserved in simple harmonic motion. This means that external forces, such as friction, air resistance, etc., are ignored. If energy is not being lost due to external forces, it is conserved in the system. Mechanical energy is the sum of kinetic and potential energy and is constant. Energy is continuously interchanging between the two forms in simple harmonic motion, as mentioned earlier.
Therefore, it can be concluded that:
- KE = KEmax at equilibrium or x = 0
- PE = PEmax at maximum amplitudes
- When KE = KEmax, PE = 0
- When PE = PEmax, KE = 0
Energy vs time graph in simple harmonic motion
The conservation of mechanical energy is illustrated in the energy vs time graph in simple harmonic motion in figure 2, where the following properties can be derived:
- When the potential energy is 0, the kinetic energy is at its maximum point and vice versa.
- Both kinetic and potential energies are represented by periodic functions (sine or cosine).
- The two periodic functions vary in opposite directions.
- Energy is always positive.
- The total energy is represented by a horizontal straight line at the maximum value of both kinetic and potential energy.
- During one period of oscillation, kinetic and potential energy go through two complete cycles, as one period reaches the maximum amplitude point twice (negative and positive).
Figure 2. Energy vs time graph. Source: Georgia Panagi, StudySmarter.
The average energy in an oscillator performing simple harmonic motion is the total energy of the oscillator in one time period, which is the time it takes for the oscillator to return to its initial equilibrium position after it has reached both of the amplitude points once.
Energy vs displacement graph in simple harmonic motion
Another graph that can be conducted from the principle of mechanical conservation of energy is the energy vs displacement graph in figure 3, where the total energy is shown as well as the energy at the maximum amplitude points. An interchanging pattern can be seen in the graph.
Figure 3. Energy vs displacement graph. Source: Georgia Panagi, StudySmarter.
- As displacement is a vector quantity, the graph has both positive and negative displacement values.
- The potential energy is at its maximum at the maximum amplitude position, where x = ±Xmax, and at 0 at the equilibrium position, where x = 0.
- This is represented by a U-shaped curve.
- The total energy is represented by a horizontal straight line above the curves, and it is constant.
- The kinetic energy is at its maximum at the equilibrium position, where x = 0, and at 0 at the amplitude position x = ±Xmax.
- This is represented by an upside-down U-shaped curve.
Simple harmonic motion energy equation
The simple harmonic motion energy equation provides the numerical magnitude of energy of an oscillator. This kinetic energy equation can be derived by starting with the equation of energy in translational motion (see below), where KE is the kinetic energy, m is the mass, and V is the speed.
\[KE = 0.5 \cdot m \cdot v^2\]
By substituting the velocity equation of simple harmonic motion at any given time, we get the following, where Xmax is the maximum amplitude, while x is the current position of an object at a given time.
If the position of the object is the equilibrium, the kinetic energy is at its maximum and proportional to the maximum amplitude.
\[v = \sqrt{\pm \omega (X_{max}^2 - x^2)} \qquad v^2 = \omega^2(X_{max}^2 - x^2) \qquad KE = \frac{1}{2} \cdot m \cdot \omega^2 (X^2_{max} - x^2) \qquad KE_{max} = \frac{1}{2} \cdot m \cdot \omega^2 \cdot X_{max}^2\]
Where \(\omega = 2\pi f\)
For the potential energy, we also use the kinetic energy but substitute the spring constant. The potential energy equation is shown below, where PE is the potential energy, k is the spring constant, and ω and t are the angular frequency and time, respectively.
If the position of the object is the equilibrium, the potential energy is at its maximum and proportional to the maximum amplitude.
\[k = m \omega^2 PE = \frac{1}{2} \cdot k \cdot (X_{max}^2 - x^2) \qquad PE_{max} = \frac{1}{2} \cdot k \cdot X_{max}^2\]
The total energy can be determined by the sum of the kinetic and potential energy, which is shown below, where k is the spring constant measured in N/m, and x is the displacement measured in m.
\[E_{total}[J] = KE + PE\]
A mass of 5 kg is performing SHM. Its position [m] is given by the equation \(x(t) = 10 \sin(2t)\). Determine the maximum kinetic energy.
As the object seems to have no phase, we begin by using the maximum kinetic energy equation and substitute the angular frequency, mass, and amplitude values from the given equation.
\(KE = 0.5 \cdot m \cdot X_{max}^2 \cdot \omega^2\)
We use the equation given, and by comparing it to the formula for displacement at a given time, we can derive that maximum amplitude is equal to 10, while ω is equal to 2.
\(x(t) = 10 \sin(2t) \qquad X_{max} = 10 m \text{ and } \omega = 2s^{-1}\)
Lastly, we substitute the maximum amplitude and angular velocity found in the energy equation above and get:
\(KE_{max} = 0.5 \cdot 5 kg \cdot (10 m)^2 \cdot (2s^{-1})^2 = 1000J\)
Energy in Simple Harmonic Motion - Key takeaways
- Simple harmonic energy is the energy that an oscillator possesses when it performs simple harmonic motion.
- During simple harmonic motion, energy is continuously being exchanged between kinetic and potential energies.
- As mechanical energy is conserved in simple harmonic motion, the total energy is constant.
- The kinetic and potential energies cannot be negative.
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