3.1.1.2. Amplitude

Table of Contents

Meaning of Amplitude

In simple harmonic motion, amplitude tells us how far the object moves from its equilibrium position at the greatest extent of its motion. The equilibrium position is the central position where the net restoring effect balances so that the object would stay there if it were not moving.

If the displacement from equilibrium is called $x$, then the amplitude is usually written as $A$. During the motion, the displacement changes between $+A$ and $-A$.

For example, if a mass on a spring moves 5 cm to the right of equilibrium at its farthest point, and 5 cm to the left at the other farthest point, then the amplitude is 5 cm, not 10 cm. The 10 cm value is the total distance between the two extreme positions.

Amplitude is the maximum displacement from equilibrium.
If $x(t)$ is the displacement, then always
$$
-A \le x(t) \le A
$$
and the amplitude is
$$
A = |x|_{\max}
$$

Amplitude in the SHM Equation

A common mathematical form for simple harmonic motion is

$$
x(t) = A \cos(\omega t + \phi)
$$

$$
x(t) = A \sin(\omega t + \phi)
$$

In these equations, $A$ is the amplitude. It is the coefficient in front of the sine or cosine function, and it sets the size of the oscillation. A larger amplitude means a larger swing away from equilibrium.

If $A$ doubles, the object moves twice as far from the center at its extreme positions.

Physical Interpretation

Amplitude measures the strength or size of the oscillation. It does not tell us how fast the object oscillates, and it does not directly tell us how long one cycle takes. Those ideas belong to frequency, period, and angular frequency.

What amplitude does tell us is how large the motion is. A small amplitude means gentle oscillation. A large amplitude means wider oscillation.

Some everyday examples help make this clear.

System	Meaning of amplitude
Mass on a spring	Maximum stretch or compression from equilibrium
Pendulum, for small oscillations	Maximum displacement from the center position
Vibrating string	Maximum sideways displacement of a point on the string
Sound wave, in a simple sense	Size of the pressure or displacement variation

Amplitude and Extreme Positions

At the two turning points of the motion, the object reaches its largest positive and negative displacements. These are the extreme positions.

If the equilibrium position is chosen as $x = 0$, then the turning points are

$$
x = +A \quad \text{and} \quad x = -A
$$

At these points, the object reverses direction. The amplitude is therefore closely connected to the boundaries of the motion.

Rendering scientific sketch…

Amplitude in simple harmonic motion

Amplitude and Energy

In simple harmonic motion, amplitude is strongly related to the total mechanical energy of the oscillator. A larger amplitude means more total energy.

For many SHM systems, the total energy is proportional to $A^2$. In a spring mass system, for example,

$$
E = \frac{1}{2}kA^2
$$

where $k$ is the spring constant.

This means that if the amplitude becomes twice as large, the total energy becomes four times as large.

For a spring mass oscillator,
$$
E = \frac{1}{2}kA^2
$$
So energy is proportional to the square of the amplitude:
$$
E \propto A^2
$$

Amplitude Is Always Positive

The displacement $x$ can be positive or negative, depending on which side of equilibrium the object is on. But the amplitude itself is taken as a positive quantity, because it is a magnitude.

For instance, saying the amplitude is $-4 \text{ cm}$ is not correct. The correct statement is that the object may reach a displacement of $-4 \text{ cm}$, while the amplitude is $4 \text{ cm}$.

Amplitude Versus Total Distance of Motion

A common beginner mistake is to confuse amplitude with the full width of the motion.

Suppose an object moves between $x = -8 \text{ cm}$ and $x = +8 \text{ cm}$. Then

$$
A = 8 \text{ cm}
$$

but the total distance from one extreme to the other is

$$
2A = 16 \text{ cm}
$$

This distinction is very important.

Quantity	Value if motion is from $-8$ cm to $+8$ cm
Amplitude	$8$ cm
Total span	$16$ cm

Reading Amplitude from a Graph

On a displacement time graph, amplitude is the largest vertical distance from the equilibrium line to a peak or trough. If the equilibrium line is at $x=0$, then the highest point is $+A$ and the lowest point is $-A$.

Rendering scientific sketch…

Displacement-time graph showing amplitude

Final Idea

Amplitude is one of the simplest and most important quantities in simple harmonic motion. It tells us the maximum displacement from equilibrium, sets the size of the oscillation, and helps determine the energy of the system.

Key facts about amplitude:
$$
A = |x|_{\max}
$$
The extreme positions are
$$
x = \pm A
$$
The total distance between extremes is
$$
2A
$$
Amplitude is positive and represents the size of the oscillation.

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