The energy of an oscillating system Revision Notes for AQA A-Level Physics

The energy of an oscillating system

Introduction to energy in oscillating systems

Periodic motion occurs in various systems, including pendulums swinging, masses oscillating on springs, and atoms vibrating within solids. In all these examples, energy continuously transfers between potential energy and kinetic energy. Understanding this energy interchange is essential for analyzing oscillating systems.

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Energy Transfer in a Pendulum

When a pendulum bob is displaced from its equilibrium position, it gains height and therefore acquires gravitational potential energy. Upon release, this potential energy converts to kinetic energy as the bob accelerates and the pendulum swings downward. At the equilibrium position, the bob reaches its maximum speed, meaning all the energy is now kinetic. The bob's momentum carries it past equilibrium, causing it to rise on the opposite side. As it does so, kinetic energy converts back to gravitational potential energy.

Although no resultant force acts on the bob at equilibrium, its momentum ensures it continues moving, maintaining the oscillation. This continuous energy transformation is the hallmark of all oscillating systems.

Free oscillation

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Definition: Free Oscillation

A free oscillation is an idealized oscillation in which the amplitude of the oscillating object remains constant because no energy enters the system and no energy is removed by friction or other dissipative forces.

In a free oscillation, the total energy equals the sum of potential energy and kinetic energy, and this total remains constant throughout the oscillation. This constancy arises from the continuous interchange between the two forms of energy.

Energy variation with displacement

The potential energy $E_p$ and kinetic energy $E_k$ of an oscillating system vary with displacement from the equilibrium position. The graph of energy against displacement shows how these quantities change:

At maximum displacement (amplitude $A$ or $-A$ ), all energy is potential energy, and kinetic energy is zero because the object momentarily stops before reversing direction
At equilibrium (displacement = 0), all energy is kinetic energy, and potential energy is at its minimum because the object passes through this point at maximum speed
At intermediate positions, the system possesses both potential and kinetic energy
The total energy line remains horizontal, indicating constant total energy

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The curves for potential and kinetic energy are smooth and continuous, demonstrating the gradual interchange between the two forms throughout the oscillation cycle. This smooth transfer of energy ensures the oscillation continues indefinitely in an ideal (friction-free) system.

Energy in simple harmonic motion

For an object oscillating with Simple Harmonic Motion (SHM), the kinetic energy varies in proportion to $\sin^2(\omega t)$ , where $\omega$ represents the angular frequency and $t$ represents time. The potential energy varies in proportion to $\cos^2(\omega t)$ .

These relationships assume that at time $t = 0$ , the displacement (and therefore the potential energy) is at its maximum value. This convention is useful for mathematical analysis of SHM.

The graph of energy against time for one complete cycle shows:

Kinetic energy and potential energy curves oscillate sinusoidally
Both energy forms complete two full cycles for each oscillation cycle of the object
When kinetic energy is maximum, potential energy is minimum, and vice versa
The sum of the two energies at any instant equals the constant total energy

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Key Energy Relationships in SHM:

Kinetic energy $\propto \sin^2(\omega t)$
Potential energy $\propto \cos^2(\omega t)$
At any instant: $E_{\text{total}} = E_k + E_p = \text{constant}$

These relationships show that both energy forms vary with twice the frequency of the displacement oscillation.

Components of potential energy in vertical oscillations

When a mass oscillates vertically on a spring, the potential energy comprises two components:

Elastic strain energy stored in the spring, which is maximum at the extreme positions where the spring is most stretched or compressed
Gravitational potential energy, which is maximum at the highest point of the oscillation

The elastic strain energy in a spring is given by:

$E = \frac{1}{2}k(\Delta l)^2$

where:

$E$ is the elastic potential energy (J)
$k$ is the spring constant (N m $^{-1}$ )
$\Delta l$ is the extension or compression from the spring's natural length (m)

The change in gravitational potential energy is:

$\Delta E = mg\Delta h$

where:

$\Delta E$ is the change in gravitational potential energy (J)
$m$ is the mass (kg)
$g$ is gravitational field strength (N kg $^{-1}$ )
$\Delta h$ is the change in height (m)

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At the bottom extreme of the oscillation, elastic strain energy is at its maximum, while at the top extreme, gravitational potential energy is at its maximum. The interplay between these two forms, along with kinetic energy, governs the motion of the vertical oscillating system.

Worked example: energy calculations in a mass-spring system

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Worked Example: Energy Transformations in a Vertical Spring System

A mass of 500 g is attached to a spring with spring constant 81.8 N m $^{-1}$ . The spring is initially stretched by 6.0 cm. The mass is then pulled down a further 4.0 cm and released.

Step 1: Calculate initial elastic energy stored (at 6.0 cm extension)

Using $E = \frac{1}{2}k(\Delta l)^2$ :

$E = \frac{1}{2} \times 81.8 \times (0.06)^2$

$E = \frac{1}{2} \times 81.8 \times 0.0036$

$E = 0.147 \text{ J}$

Step 2: Calculate elastic energy after pulling down further 4.0 cm (total extension 10.0 cm)

$E = \frac{1}{2}k(\Delta l)^2$

$E = \frac{1}{2} \times 81.8 \times (0.10)^2$

$E = \frac{1}{2} \times 81.8 \times 0.01$

$E = 0.409 \text{ J}$

Step 3: Calculate energy transformations during motion

When released from the 10.0 cm position, the mass moves upward. As it passes through the equilibrium position (6.0 cm extension), the elastic energy decreases from 0.409 J back to 0.147 J.

During this motion, the gravitational potential energy increases by:

$\Delta E = mg\Delta h$

$\Delta E = 0.5 \times 9.81 \times 0.04$

$\Delta E = 0.196 \text{ J}$

Step 4: Apply conservation of energy

As the mass passes through equilibrium:

Kinetic energy = Loss in elastic energy - Gain in gravitational potential energy

$E_k = 0.409 - 0.147 - 0.196$

$E_k = 0.066 \text{ J}$

Conclusion: This calculation demonstrates how conservation of energy allows us to track energy transformations throughout the oscillation. The elastic potential energy decreases, some converts to gravitational potential energy as the mass rises, and the remainder becomes kinetic energy.

Damping in oscillating systems

In practice, most real oscillations experience resistive forces that transfer energy away from the oscillating system. These resistive forces include friction and air resistance, which always act in the opposite direction to the motion of the oscillating object.

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Definition: Damping

Damping is the removal of energy from an oscillating system. The extent of damping determines how quickly oscillations die away.

A displacement-time graph for a damped oscillation shows the amplitude decreasing over successive cycles until the oscillation eventually stops.

Types of damping

The degree of damping significantly affects the behavior of an oscillating system:

Light damping (under-damped): If damping is light, oscillations die away gradually over many cycles. The amplitude decreases slowly, but the oscillation persists for a considerable time. In some cases, under-damping may be problematic because unnecessary vibrations continue, potentially causing damage to mechanical systems.

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Heavy Damping

When resistive forces are very high, the system experiences heavy damping. The displaced object moves slowly back to its equilibrium position without oscillating. The return to equilibrium takes a long time because the strong resistive forces significantly impede motion.

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Critical Damping - The Ideal State

A system is critically damped when the resistive forces are precisely sufficient to prevent oscillation while allowing the object to return to equilibrium in the minimum possible time. This represents the ideal state of damping for many mechanical systems where the goal is to prevent vibration damage while ensuring quick settling.

Critical damping is commonly used in applications such as:

Vehicle suspension systems
Door closers
Measuring instruments

The effects of different damping levels can be visualized on displacement-time graphs:

Critically damped: smooth exponential decay to equilibrium, reaching it in minimum time
Heavily damped: slow exponential decay, taking longer to reach equilibrium
Under-damped: oscillations with decreasing amplitude, potentially persisting unnecessarily

Applications of damping

An example of a critically damped system is the front suspension on a mountain bike. The combination of the suspension spring and a shock absorber work together to ensure a smoother ride. The shock absorber provides the damping force necessary to prevent the wheel from bouncing excessively after encountering bumps, while the spring absorbs impact energy. Together, these components allow the wheel to return quickly to its working position without oscillating, improving control and comfort.

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Key Points to Remember:

In a free oscillation, total energy (potential + kinetic) remains constant throughout the motion
Energy continuously interchanges between potential and kinetic forms during oscillation
At maximum displacement, energy is entirely potential; at equilibrium, energy is entirely kinetic
For SHM, kinetic energy varies as $\sin^2(\omega t)$ and potential energy as $\cos^2(\omega t)$
In vertical spring systems, potential energy includes both elastic strain energy ( $E = \frac{1}{2}k(\Delta l)^2$ ) and gravitational potential energy ( $\Delta E = mg\Delta h$ )
Damping removes energy from oscillating systems through resistive forces
Critical damping returns a system to equilibrium in minimum time without oscillation, making it ideal for many practical applications

The energy of an oscillating system (AQA A-Level Physics): Revision Notes