mechanics_functions

shortfx.fxNumeric.mechanics_functions

Moments of inertia, centroids, and section properties.

Classical formulas for mass properties and centroids of common geometric shapes from Murray R. Spiegel's Mathematical Handbook of Formulas and Tables.

Functions

centroid_circular_sector(r: float, alpha: float) -> Tuple[float, float]

Returns centroid distance from center of a circular sector.

For a sector of half-angle α symmetric about x-axis, centroid is at (2r sin(α)/(3α), 0).

Parameters:

Name	Type	Description	Default
r	float	Radius.	required
alpha	float	Half-angle of the sector in radians (0 < α <= π).	required

Returns:

Type	Description
Tuple[float, float]	Centroid as (x_c, 0).

Example

x, _ = centroid_circular_sector(3, math.pi / 2) round(x, 6) 1.273240

Complexity: O(1)

Source code in shortfx/fxNumeric/mechanics_functions.py

def centroid_circular_sector(r: float, alpha: float) -> Tuple[float, float]:
    """Returns centroid distance from center of a circular sector.

    For a sector of half-angle α symmetric about x-axis, centroid is at
    (2r sin(α)/(3α), 0).

    Args:
        r: Radius.
        alpha: Half-angle of the sector in radians (0 < α <= π).

    Returns:
        Centroid as (x_c, 0).

    Example:
        >>> x, _ = centroid_circular_sector(3, math.pi / 2)
        >>> round(x, 6)
        1.273240

    Complexity: O(1)
    """
    _check_positive(r, "r")
    _check_positive(alpha, "alpha")
    return (2.0 * r * math.sin(alpha) / (3.0 * alpha), 0.0)

centroid_cone(h: float) -> float

Returns the height of the centroid of a solid cone above its base.

The centroid is at h/4 from the base.

Parameters:

Name	Type	Description	Default
h	float	Height of the cone.	required

Returns:

Type	Description
float	Distance from base to centroid.

Example

centroid_cone(12) 3.0

Complexity: O(1)

Source code in shortfx/fxNumeric/mechanics_functions.py

def centroid_cone(h: float) -> float:
    """Returns the height of the centroid of a solid cone above its base.

    The centroid is at h/4 from the base.

    Args:
        h: Height of the cone.

    Returns:
        Distance from base to centroid.

    Example:
        >>> centroid_cone(12)
        3.0

    Complexity: O(1)
    """
    _check_positive(h, "h")
    return h / 4.0

centroid_hemisphere(r: float) -> float

Returns the height of the centroid of a solid hemisphere above the flat face.

The centroid is at 3r/8 from the base.

Parameters:

Name	Type	Description	Default
r	float	Radius.	required

Returns:

Type	Description
float	Distance from flat face to centroid.

Example

centroid_hemisphere(4) 1.5

Complexity: O(1)

Source code in shortfx/fxNumeric/mechanics_functions.py

def centroid_hemisphere(r: float) -> float:
    """Returns the height of the centroid of a solid hemisphere above the flat face.

    The centroid is at 3r/8 from the base.

    Args:
        r: Radius.

    Returns:
        Distance from flat face to centroid.

    Example:
        >>> centroid_hemisphere(4)
        1.5

    Complexity: O(1)
    """
    _check_positive(r, "r")
    return 3.0 * r / 8.0

centroid_polygon(vertices: List[Tuple[float, float]]) -> Tuple[float, float]

Computes the centroid of a simple polygon using the Shoelace-derived formula.

Parameters:

Name	Type	Description	Default
vertices	List[Tuple[float, float]]	Ordered list of (x, y) vertices (not repeated at end).	required

Returns:

Type	Description
Tuple[float, float]	Centroid as (x, y).

Raises:

Type	Description
ValueError	If polygon has fewer than 3 vertices or has zero area.

Example

centroid_polygon([(0, 0), (4, 0), (4, 4), (0, 4)]) (2.0, 2.0)

Complexity: O(n)

Source code in shortfx/fxNumeric/mechanics_functions.py

def centroid_polygon(vertices: List[Tuple[float, float]]) -> Tuple[float, float]:
    """Computes the centroid of a simple polygon using the Shoelace-derived formula.

    Args:
        vertices: Ordered list of (x, y) vertices (not repeated at end).

    Returns:
        Centroid as (x, y).

    Raises:
        ValueError: If polygon has fewer than 3 vertices or has zero area.

    Example:
        >>> centroid_polygon([(0, 0), (4, 0), (4, 4), (0, 4)])
        (2.0, 2.0)

    Complexity: O(n)
    """
    n = len(vertices)

    if n < 3:
        raise ValueError("Polygon must have at least 3 vertices.")

    area = 0.0
    cx = 0.0
    cy = 0.0

    for i in range(n):
        j = (i + 1) % n
        cross = vertices[i][0] * vertices[j][1] - vertices[j][0] * vertices[i][1]
        area += cross
        cx += (vertices[i][0] + vertices[j][0]) * cross
        cy += (vertices[i][1] + vertices[j][1]) * cross

    area *= 0.5

    if abs(area) < 1e-15:
        raise ValueError("Polygon has zero area.")

    factor = 1.0 / (6.0 * area)
    return (cx * factor, cy * factor)

centroid_semicircle(r: float) -> Tuple[float, float]

Returns the centroid of a semicircle of radius r lying above the x-axis.

The centroid is at (0, 4r/(3π)) above the diameter.

Parameters:

Name	Type	Description	Default
r	float	Radius.	required

Returns:

Type	Description
Tuple[float, float]	Centroid as (0, y_c).

Example

_, y = centroid_semicircle(3) round(y, 6) 1.273240

Complexity: O(1)

Source code in shortfx/fxNumeric/mechanics_functions.py

def centroid_semicircle(r: float) -> Tuple[float, float]:
    """Returns the centroid of a semicircle of radius r lying above the x-axis.

    The centroid is at (0, 4r/(3π)) above the diameter.

    Args:
        r: Radius.

    Returns:
        Centroid as (0, y_c).

    Example:
        >>> _, y = centroid_semicircle(3)
        >>> round(y, 6)
        1.273240

    Complexity: O(1)
    """
    _check_positive(r, "r")
    return (0.0, 4.0 * r / (3.0 * math.pi))

centroid_triangle(p1: Tuple[float, float], p2: Tuple[float, float], p3: Tuple[float, float]) -> Tuple[float, float]

Computes the centroid of a triangle given its vertices.

Parameters:

Name	Type	Description	Default
p1	Tuple[float, float]	First vertex (x, y).	required
p2	Tuple[float, float]	Second vertex (x, y).	required
p3	Tuple[float, float]	Third vertex (x, y).	required

Returns:

Type	Description
Tuple[float, float]	Centroid as (x, y).

Example

centroid_triangle((0, 0), (6, 0), (0, 6)) (2.0, 2.0)

Complexity: O(1)

Source code in shortfx/fxNumeric/mechanics_functions.py

def centroid_triangle(
    p1: Tuple[float, float],
    p2: Tuple[float, float],
    p3: Tuple[float, float]
) -> Tuple[float, float]:
    """Computes the centroid of a triangle given its vertices.

    Args:
        p1: First vertex (x, y).
        p2: Second vertex (x, y).
        p3: Third vertex (x, y).

    Returns:
        Centroid as (x, y).

    Example:
        >>> centroid_triangle((0, 0), (6, 0), (0, 6))
        (2.0, 2.0)

    Complexity: O(1)
    """
    return ((p1[0] + p2[0] + p3[0]) / 3.0, (p1[1] + p2[1] + p3[1]) / 3.0)

mass_moment_cone(m: float, r: float) -> float

Mass moment of inertia of a solid cone about its axis.

I = (3/10)mr².

Parameters:

Name	Type	Description	Default
m	float	Mass.	required
r	float	Base radius.	required

Returns:

Type	Description
float	Moment of inertia I.

Example

mass_moment_cone(10, 2) 12.0

Complexity: O(1)

Source code in shortfx/fxNumeric/mechanics_functions.py

def mass_moment_cone(m: float, r: float) -> float:
    """Mass moment of inertia of a solid cone about its axis.

    I = (3/10)mr².

    Args:
        m: Mass.
        r: Base radius.

    Returns:
        Moment of inertia I.

    Example:
        >>> mass_moment_cone(10, 2)
        12.0

    Complexity: O(1)
    """
    _check_positive(m, "m")
    _check_positive(r, "r")
    return 3.0 * m * r * r / 10.0

mass_moment_cylinder(m: float, r: float) -> float

Mass moment of inertia of a solid cylinder about its axis.

I = mr²/2.

Parameters:

Name	Type	Description	Default
m	float	Mass.	required
r	float	Radius.	required

Returns:

Type	Description
float	Moment of inertia I.

Example

mass_moment_cylinder(10, 2) 20.0

Complexity: O(1)

Source code in shortfx/fxNumeric/mechanics_functions.py

def mass_moment_cylinder(m: float, r: float) -> float:
    """Mass moment of inertia of a solid cylinder about its axis.

    I = mr²/2.

    Args:
        m: Mass.
        r: Radius.

    Returns:
        Moment of inertia I.

    Example:
        >>> mass_moment_cylinder(10, 2)
        20.0

    Complexity: O(1)
    """
    _check_positive(m, "m")
    _check_positive(r, "r")
    return m * r * r / 2.0

mass_moment_cylinder_transverse(m: float, r: float, h: float) -> float

Mass moment of inertia of a solid cylinder about a transverse centroidal axis.

I = m(3r² + h²)/12.

Parameters:

Name	Type	Description	Default
m	float	Mass.	required
r	float	Radius.	required
h	float	Height.	required

Returns:

Type	Description
float	Moment of inertia I.

Example

round(mass_moment_cylinder_transverse(10, 2, 6), 6) 40.0

Complexity: O(1)

Source code in shortfx/fxNumeric/mechanics_functions.py

def mass_moment_cylinder_transverse(m: float, r: float, h: float) -> float:
    """Mass moment of inertia of a solid cylinder about a transverse centroidal axis.

    I = m(3r² + h²)/12.

    Args:
        m: Mass.
        r: Radius.
        h: Height.

    Returns:
        Moment of inertia I.

    Example:
        >>> round(mass_moment_cylinder_transverse(10, 2, 6), 6)
        40.0

    Complexity: O(1)
    """
    _check_positive(m, "m")
    _check_positive(r, "r")
    _check_positive(h, "h")
    return m * (3.0 * r * r + h * h) / 12.0

mass_moment_disk(m: float, r: float) -> float

Mass moment of inertia of a thin uniform disk about its axis.

I = mr²/2 (same as solid cylinder about axis).

Parameters:

Name	Type	Description	Default
m	float	Mass.	required
r	float	Radius.	required

Returns:

Type	Description
float	Moment of inertia I.

Example

mass_moment_disk(10, 3) 45.0

Complexity: O(1)

Source code in shortfx/fxNumeric/mechanics_functions.py

def mass_moment_disk(m: float, r: float) -> float:
    """Mass moment of inertia of a thin uniform disk about its axis.

    I = mr²/2 (same as solid cylinder about axis).

    Args:
        m: Mass.
        r: Radius.

    Returns:
        Moment of inertia I.

    Example:
        >>> mass_moment_disk(10, 3)
        45.0

    Complexity: O(1)
    """
    _check_positive(m, "m")
    _check_positive(r, "r")
    return m * r * r / 2.0

mass_moment_hollow_sphere(m: float, r: float) -> float

Mass moment of inertia of a thin hollow sphere about any diameter.

I = (2/3)mr².

Parameters:

Name	Type	Description	Default
m	float	Mass.	required
r	float	Radius.	required

Returns:

Type	Description
float	Moment of inertia I.

Example

round(mass_moment_hollow_sphere(10, 2), 6) 26.666667

Complexity: O(1)

Source code in shortfx/fxNumeric/mechanics_functions.py

def mass_moment_hollow_sphere(m: float, r: float) -> float:
    """Mass moment of inertia of a thin hollow sphere about any diameter.

    I = (2/3)mr².

    Args:
        m: Mass.
        r: Radius.

    Returns:
        Moment of inertia I.

    Example:
        >>> round(mass_moment_hollow_sphere(10, 2), 6)
        26.666667

    Complexity: O(1)
    """
    _check_positive(m, "m")
    _check_positive(r, "r")
    return 2.0 * m * r * r / 3.0

mass_moment_rod(m: float, length: float) -> float

Mass moment of inertia of a thin rod about its center (perpendicular axis).

I = mL²/12.

Parameters:

Name	Type	Description	Default
m	float	Mass.	required
length	float	Length of the rod.	required

Returns:

Type	Description
float	Moment of inertia I.

Example

round(mass_moment_rod(6, 4), 6) 8.0

Complexity: O(1)

Source code in shortfx/fxNumeric/mechanics_functions.py

def mass_moment_rod(m: float, length: float) -> float:
    """Mass moment of inertia of a thin rod about its center (perpendicular axis).

    I = mL²/12.

    Args:
        m: Mass.
        length: Length of the rod.

    Returns:
        Moment of inertia I.

    Example:
        >>> round(mass_moment_rod(6, 4), 6)
        8.0

    Complexity: O(1)
    """
    _check_positive(m, "m")
    _check_positive(length, "length")
    return m * length * length / 12.0

mass_moment_sphere(m: float, r: float) -> float

Mass moment of inertia of a solid sphere about any diameter.

I = (2/5)mr².

Parameters:

Name	Type	Description	Default
m	float	Mass.	required
r	float	Radius.	required

Returns:

Type	Description
float	Moment of inertia I.

Example

mass_moment_sphere(10, 2) 16.0

Complexity: O(1)

Source code in shortfx/fxNumeric/mechanics_functions.py

def mass_moment_sphere(m: float, r: float) -> float:
    """Mass moment of inertia of a solid sphere about any diameter.

    I = (2/5)mr².

    Args:
        m: Mass.
        r: Radius.

    Returns:
        Moment of inertia I.

    Example:
        >>> mass_moment_sphere(10, 2)
        16.0

    Complexity: O(1)
    """
    _check_positive(m, "m")
    _check_positive(r, "r")
    return 2.0 * m * r * r / 5.0

moment_of_inertia_annulus(r_outer: float, r_inner: float) -> float

Computes the area moment of inertia of an annulus (hollow circle) about a diameter.

I = π(R⁴ - r⁴)/4.

Parameters:

Name	Type	Description	Default
r_outer	float	Outer radius.	required
r_inner	float	Inner radius (< r_outer).	required

Returns:

Type	Description
float	Moment of inertia I.

Raises:

Type	Description
ValueError	If r_inner >= r_outer.

Example

round(moment_of_inertia_annulus(3, 2), 6) 50.267544

Complexity: O(1)

Source code in shortfx/fxNumeric/mechanics_functions.py

def moment_of_inertia_annulus(r_outer: float, r_inner: float) -> float:
    """Computes the area moment of inertia of an annulus (hollow circle) about a diameter.

    I = π(R⁴ - r⁴)/4.

    Args:
        r_outer: Outer radius.
        r_inner: Inner radius (< r_outer).

    Returns:
        Moment of inertia I.

    Raises:
        ValueError: If r_inner >= r_outer.

    Example:
        >>> round(moment_of_inertia_annulus(3, 2), 6)
        50.267544

    Complexity: O(1)
    """
    _check_positive(r_outer, "r_outer")
    _check_positive(r_inner, "r_inner")

    if r_inner >= r_outer:
        raise ValueError("r_inner must be less than r_outer.")

    return math.pi * (r_outer ** 4 - r_inner ** 4) / 4.0

moment_of_inertia_circle(r: float) -> float

Computes the area moment of inertia of a circle about a diameter.

I = πr⁴/4.

Parameters:

Name	Type	Description	Default
r	float	Radius.	required

Returns:

Type	Description
float	Moment of inertia I.

Example

round(moment_of_inertia_circle(2), 6) 12.566371

Complexity: O(1)

Source code in shortfx/fxNumeric/mechanics_functions.py

def moment_of_inertia_circle(r: float) -> float:
    """Computes the area moment of inertia of a circle about a diameter.

    I = πr⁴/4.

    Args:
        r: Radius.

    Returns:
        Moment of inertia I.

    Example:
        >>> round(moment_of_inertia_circle(2), 6)
        12.566371

    Complexity: O(1)
    """
    _check_positive(r, "r")
    return math.pi * r ** 4 / 4.0

moment_of_inertia_ellipse(a: float, b: float) -> Tuple[float, float]

Computes the area moments of inertia of an ellipse about its centroidal axes.

I_x = πab³/4 (about major axis), I_y = πa³b/4 (about minor axis).

Parameters:

Name	Type	Description	Default
a	float	Semi-axis along x.	required
b	float	Semi-axis along y.	required

Returns:

Type	Description
Tuple[float, float]	Tuple (I_x, I_y).

Example

ix, iy = moment_of_inertia_ellipse(3, 2) (round(ix, 6), round(iy, 6)) (18.849556, 42.411501)

Complexity: O(1)

Source code in shortfx/fxNumeric/mechanics_functions.py

def moment_of_inertia_ellipse(a: float, b: float) -> Tuple[float, float]:
    """Computes the area moments of inertia of an ellipse about its centroidal axes.

    I_x = πab³/4 (about major axis), I_y = πa³b/4 (about minor axis).

    Args:
        a: Semi-axis along x.
        b: Semi-axis along y.

    Returns:
        Tuple (I_x, I_y).

    Example:
        >>> ix, iy = moment_of_inertia_ellipse(3, 2)
        >>> (round(ix, 6), round(iy, 6))
        (18.849556, 42.411501)

    Complexity: O(1)
    """
    _check_positive(a, "a")
    _check_positive(b, "b")
    return (math.pi * a * b ** 3 / 4.0, math.pi * a ** 3 * b / 4.0)

moment_of_inertia_rectangle(b: float, h: float) -> Tuple[float, float]

Computes the area moments of inertia of a rectangle about centroidal axes.

I_x = bh³/12 (about horizontal axis), I_y = hb³/12 (about vertical axis).

Parameters:

Name	Type	Description	Default
b	float	Width (base).	required
h	float	Height.	required

Returns:

Type	Description
Tuple[float, float]	Tuple (I_x, I_y).

Example

ix, iy = moment_of_inertia_rectangle(4, 6) (ix, iy) (72.0, 32.0)

Complexity: O(1)

Source code in shortfx/fxNumeric/mechanics_functions.py

def moment_of_inertia_rectangle(b: float, h: float) -> Tuple[float, float]:
    """Computes the area moments of inertia of a rectangle about centroidal axes.

    I_x = bh³/12 (about horizontal axis), I_y = hb³/12 (about vertical axis).

    Args:
        b: Width (base).
        h: Height.

    Returns:
        Tuple (I_x, I_y).

    Example:
        >>> ix, iy = moment_of_inertia_rectangle(4, 6)
        >>> (ix, iy)
        (72.0, 32.0)

    Complexity: O(1)
    """
    _check_positive(b, "b")
    _check_positive(h, "h")
    return (b * h ** 3 / 12.0, h * b ** 3 / 12.0)

moment_of_inertia_semicircle(r: float) -> float

Computes the area moment of inertia of a semicircle about its diameter.

I = πr⁴/8.

Parameters:

Name	Type	Description	Default
r	float	Radius.	required

Returns:

Type	Description
float	Moment of inertia I about the diameter.

Example

round(moment_of_inertia_semicircle(2), 6) 6.283185

Complexity: O(1)

Source code in shortfx/fxNumeric/mechanics_functions.py

def moment_of_inertia_semicircle(r: float) -> float:
    """Computes the area moment of inertia of a semicircle about its diameter.

    I = πr⁴/8.

    Args:
        r: Radius.

    Returns:
        Moment of inertia I about the diameter.

    Example:
        >>> round(moment_of_inertia_semicircle(2), 6)
        6.283185

    Complexity: O(1)
    """
    _check_positive(r, "r")
    return math.pi * r ** 4 / 8.0

moment_of_inertia_triangle(b: float, h: float) -> float

Computes the area moment of inertia of a triangle about its base.

I = bh³/12.

Parameters:

Name	Type	Description	Default
b	float	Base length.	required
h	float	Height.	required

Returns:

Type	Description
float	Moment of inertia I about the base.

Example

moment_of_inertia_triangle(6, 4) 32.0

Complexity: O(1)

Source code in shortfx/fxNumeric/mechanics_functions.py

def moment_of_inertia_triangle(b: float, h: float) -> float:
    """Computes the area moment of inertia of a triangle about its base.

    I = bh³/12.

    Args:
        b: Base length.
        h: Height.

    Returns:
        Moment of inertia I about the base.

    Example:
        >>> moment_of_inertia_triangle(6, 4)
        32.0

    Complexity: O(1)
    """
    _check_positive(b, "b")
    _check_positive(h, "h")
    return b * h ** 3 / 12.0

moment_of_inertia_triangle_centroidal(b: float, h: float) -> float

Computes the area moment of inertia of a triangle about its centroidal axis.

I = bh³/36.

Parameters:

Name	Type	Description	Default
b	float	Base length.	required
h	float	Height.	required

Returns:

Type	Description
float	Moment of inertia I about centroidal axis parallel to base.

Example

round(moment_of_inertia_triangle_centroidal(6, 4), 6) 10.666667

Complexity: O(1)

Source code in shortfx/fxNumeric/mechanics_functions.py

def moment_of_inertia_triangle_centroidal(b: float, h: float) -> float:
    """Computes the area moment of inertia of a triangle about its centroidal axis.

    I = bh³/36.

    Args:
        b: Base length.
        h: Height.

    Returns:
        Moment of inertia I about centroidal axis parallel to base.

    Example:
        >>> round(moment_of_inertia_triangle_centroidal(6, 4), 6)
        10.666667

    Complexity: O(1)
    """
    _check_positive(b, "b")
    _check_positive(h, "h")
    return b * h ** 3 / 36.0

parallel_axis_theorem(i_cm: float, m: float, d: float) -> float

Applies the parallel axis theorem: I = I_cm + md².

Parameters:

Name	Type	Description	Default
i_cm	float	Moment of inertia about the centroidal axis.	required
m	float	Mass (or area for second moment of area).	required
d	float	Distance between the two parallel axes.	required

Returns:

Type	Description
float	Moment of inertia about the new parallel axis.

Example

parallel_axis_theorem(8.0, 6, 3) 62.0

Complexity: O(1)

Source code in shortfx/fxNumeric/mechanics_functions.py

def parallel_axis_theorem(
    i_cm: float, m: float, d: float
) -> float:
    """Applies the parallel axis theorem: I = I_cm + md².

    Args:
        i_cm: Moment of inertia about the centroidal axis.
        m: Mass (or area for second moment of area).
        d: Distance between the two parallel axes.

    Returns:
        Moment of inertia about the new parallel axis.

    Example:
        >>> parallel_axis_theorem(8.0, 6, 3)
        62.0

    Complexity: O(1)
    """
    if not isinstance(i_cm, (int, float)):
        raise TypeError("i_cm must be numeric.")

    if not isinstance(m, (int, float)):
        raise TypeError("m must be numeric.")

    if not isinstance(d, (int, float)):
        raise TypeError("d must be numeric.")

    return i_cm + m * d * d

polar_moment_of_inertia(ix: float, iy: float) -> float

Computes the polar moment of inertia J = I_x + I_y.

Parameters:

Name	Type	Description	Default
ix	float	Moment of inertia about x-axis.	required
iy	float	Moment of inertia about y-axis.	required

Returns:

Type	Description
float	Polar moment of inertia J.

Example

polar_moment_of_inertia(72, 32) 104

Complexity: O(1)

Source code in shortfx/fxNumeric/mechanics_functions.py

def polar_moment_of_inertia(ix: float, iy: float) -> float:
    """Computes the polar moment of inertia J = I_x + I_y.

    Args:
        ix: Moment of inertia about x-axis.
        iy: Moment of inertia about y-axis.

    Returns:
        Polar moment of inertia J.

    Example:
        >>> polar_moment_of_inertia(72, 32)
        104

    Complexity: O(1)
    """
    if not isinstance(ix, (int, float)) or not isinstance(iy, (int, float)):
        raise TypeError("ix and iy must be numeric.")

    return ix + iy

radius_of_gyration(i: float, m: float) -> float

Computes the radius of gyration k = sqrt(I/m).

Parameters:

Name	Type	Description	Default
i	float	Moment of inertia.	required
m	float	Mass (or area).	required

Returns:

Type	Description
float	Radius of gyration.

Example

round(radius_of_gyration(48, 12), 6) 2.0

Complexity: O(1)

Source code in shortfx/fxNumeric/mechanics_functions.py

def radius_of_gyration(i: float, m: float) -> float:
    """Computes the radius of gyration k = sqrt(I/m).

    Args:
        i: Moment of inertia.
        m: Mass (or area).

    Returns:
        Radius of gyration.

    Example:
        >>> round(radius_of_gyration(48, 12), 6)
        2.0

    Complexity: O(1)
    """
    if not isinstance(i, (int, float)) or not isinstance(m, (int, float)):
        raise TypeError("i and m must be numeric.")

    if m <= 0:
        raise ValueError("m must be positive.")

    if i < 0:
        raise ValueError("i must be non-negative.")

    return math.sqrt(i / m)

section_modulus(i: float, c: float) -> float

Computes the section modulus S = I/c.

Parameters:

Name	Type	Description	Default
i	float	Moment of inertia about the neutral axis.	required
c	float	Distance from neutral axis to extreme fiber.	required

Returns:

Type	Description
float	Section modulus S.

Example

section_modulus(72, 3) 24.0

Complexity: O(1)

Source code in shortfx/fxNumeric/mechanics_functions.py

def section_modulus(i: float, c: float) -> float:
    """Computes the section modulus S = I/c.

    Args:
        i: Moment of inertia about the neutral axis.
        c: Distance from neutral axis to extreme fiber.

    Returns:
        Section modulus S.

    Example:
        >>> section_modulus(72, 3)
        24.0

    Complexity: O(1)
    """
    if not isinstance(i, (int, float)) or not isinstance(c, (int, float)):
        raise TypeError("i and c must be numeric.")

    if c <= 0:
        raise ValueError("c must be positive.")

    return i / c