Unit 1: Experimental study of C.G location

I. Orientation — Centre of Gravity and the Governing Principle

The centre of gravity (C.G.) of a body is the point through which its entire weight may be considered to act, regardless of the body’s orientation. Near Earth’s surface, where gravitational acceleration is effectively uniform over an ordinary object, the centre of gravity coincides with its centre of mass.

• Resultant weight: The gravitational forces acting on all particles of a body combine into one resultant force (W) acting vertically downward through the C.G.
• Moment principle: About the C.G., the algebraic sum of moments produced by the weights of all particles is zero.
• Suspension principle: When a freely suspended body reaches equilibrium, its C.G. lies vertically below the point of suspension.
  • If the C.G. were not below the support, the weight would create a turning moment.
  • The body rotates until the moment vanishes and the C.G. reaches the lowest possible position.
• Plumb-line convention: A plumb line indicates the true vertical because the plumb bob’s weight keeps the string aligned with gravity.
• Balancing principle: A body balances on a point or narrow edge when the vertical line through its C.G. passes through the support.
• Symmetry principle: If a homogeneous body possesses a line, plane, or axis of symmetry, its C.G. must lie on that line, plane, or axis.
• Intersection principle: Two or more experimentally obtained vertical lines through different suspension points intersect at the C.G.
• Standard assumptions:
  • The test object is rigid and does not bend during suspension.
  • Gravitational acceleration (g) is uniform.
  • The body is stationary when measurements are marked.
  • The suspension point and plumb line move freely with negligible friction.
  • A lamina has negligible thickness compared with its length and width.
• Units and notation: C.G. coordinates are measured in units of length, such as millimetres or centimetres, from chosen reference axes.

For particles or component masses, the theoretical coordinates are:

x̄ = Σ(mᵢxᵢ) / Σmᵢ
ȳ = Σ(mᵢyᵢ) / Σmᵢ

Here, (m_i) is the (i)-th mass, (x_i) and (y_i) are its coordinates, (\bar{x}) and (\bar{y}) are the C.G. coordinates, and (\Sigma) means “sum of.”

II. Symmetrical Shapes — Locating C.G. through Symmetry and Equilibrium

A. Purpose and Experimental Principle

The experiment determines whether the C.G. of a regular homogeneous lamina lies at the intersection of its geometric symmetry lines.

• Typical specimens: Uniform cardboard, sheet metal, or acrylic pieces shaped as a rectangle, square, circle, triangle, or other regular figure.
• Apparatus: The experiment commonly requires:
  • A symmetrical lamina
  • Retort stand or fixed horizontal pin
  • Plumb bob and light string
  • Pencil, ruler, and set square
  • Sharp balancing point or knife-edge
• Expected location: Symmetry predicts the C.G. before suspension measurements are made.
  • Rectangle or square: intersection of the diagonals
  • Circle or disc: geometric centre
  • Parallelogram: intersection of the diagonals
  • Homogeneous triangular lamina: intersection of the three medians, called the centroid
• Physical basis: Equal material distribution on opposite sides of a symmetry line produces equal and opposite moments about that line.
• Homogeneity condition: Geometric symmetry identifies the C.G. only if thickness and density are uniformly distributed.
  • A circular plate with an attached mass remains geometrically circular.
  • Its C.G., however, shifts toward the attached mass.

B. Experimental study of CG location of symmetrical shape

This study locates the C.G. by combining predicted symmetry lines with vertical lines obtained through suspension.

• Specimen preparation: Mark two or more geometrically known symmetry lines lightly on the lamina.
  • For a rectangle, draw both diagonals.
  • Their intersection gives the predicted C.G., (G_t).
• First suspension: Make a small hole near the boundary at point (P_1), suspend the lamina from the pin, and attach the plumb line to the same pin.
  • Allow all oscillations to stop.
  • Mark the vertical line along the string as (P_1V_1).
  • Because the body is in equilibrium, its C.G. lies somewhere on (P_1V_1).
• Second suspension: Suspend the lamina from a well-separated point (P_2) and mark the second vertical (P_2V_2).
  • The intersection of (P_1V_1) and (P_2V_2) is the experimental C.G., (G_e).
  • A third suspension line can confirm the result.
• Comparison: Compare (G_e) with the geometric point (G_t).
  • Close agreement supports both the suspension principle and the symmetry prediction.
  • A small cluster rather than a single intersection indicates experimental uncertainty.
• Balancing verification: Place the lamina horizontally on a pointed support positioned beneath (G_e).
  • If it remains level without turning, the vertical through the support passes approximately through its C.G.
  • Balancing is a check, not usually as precise as intersecting thin plumb lines.
• Moment interpretation: At equilibrium, the moment of total weight about the suspension point is zero because its line of action passes through that point.

τ = Wd = 0  when d = 0

Here, (\tau) is the moment of weight about the suspension point, (W) is the body’s weight, and (d) is the perpendicular distance from the support to the vertical line of action of (W).

• Worked observation—rectangular lamina: For a uniform rectangle measuring (20\,\text{cm} \times 12\,\text{cm}), the geometric C.G. is (10\,\text{cm}) from either vertical side and (6\,\text{cm}) from either horizontal side. If three plumb lines meet near this point, the experimental result confirms the theoretical location.
• Recording the result: Measure (\bar{x}) and (\bar{y}) from chosen reference edges and report them with suitable precision, such as (10.0\,\text{cm}) and (6.0\,\text{cm}).

C. Reliability, Applications, and Limitations

The value of the method depends on careful alignment, uniform material, and sufficiently distinct suspension lines.

• Accuracy precautions:
  • Use small suspension holes so the support point is well defined.
  • Wait until the plumb bob and lamina are completely stationary.
  • Draw fine lines beside the string without pushing it.
  • Choose suspension points far apart so the lines cross at a clear angle.
  • Keep the lamina in a vertical plane to reduce parallax.
• Likely errors:
  • Thick pencil lines produce a broad intersection region.
  • Friction at the pin can prevent free rotation.
  • Air currents can displace the plumb line.
  • Non-uniform cardboard, paint, tape, or enlarged holes can shift the actual C.G.
• Application: The method checks the balance point of model-aircraft components, signs, plates, and structural templates.
• Limitation: Symmetry alone cannot locate the C.G. when the material density or thickness varies, even if the outline is perfectly regular.

III. Unsymmetrical Shapes — Locating C.G. by Successive Suspension

A. Purpose and Experimental Principle

The experiment locates the C.G. of an irregular lamina whose outline provides no dependable geometric centre or intersecting symmetry lines.

• Typical specimens: An L-shaped plate, an irregular cardboard cut-out, or a model component with a non-regular outline.
• Essential method: Suspend the object from at least two points and trace the vertical through each support.
• Reason for multiple suspensions: One suspension identifies only a line containing the C.G.; a second non-parallel line identifies its position.
• Possible external location: The C.G. need not lie within the material.
  • For a ring, it lies at the empty central point.
  • For some curved or strongly concave shapes, it can lie outside the boundary.
• No symmetry assumption: The result depends on equilibrium under gravity rather than visual estimation of the shape’s centre.

B. Experimental study of CG location of unsymmetrical shape

This study uses successive suspension lines to determine the unique point through which the weight of an irregular lamina acts.

• Preparation: Cut or select a rigid irregular lamina and mark three suspension points, (P_1), (P_2), and (P_3), near different parts of its boundary.
• First observation: Suspend the lamina freely at (P_1), hang the plumb bob from the same support, and mark vertical line (L_1).
  • The unknown C.G. lies on (L_1).
• Second observation: Repeat from (P_2) and mark vertical line (L_2).
  • The intersection (L_1 \cap L_2) gives the experimental C.G., (G).
• Third observation: Suspend from (P_3) and mark (L_3).
  • Ideally, (L_3) passes through (G).
  • In practice, the three lines may form a small triangle; its central region is taken as the best estimate.
• Coordinate measurement: Select perpendicular reference axes along convenient edges or on background graph paper and measure (\bar{x}) and (\bar{y}).
• Worked observation—irregular lamina: Suppose two suspension lines intersect (8.4\,\text{cm}) from the chosen vertical reference and (5.7\,\text{cm}) above the horizontal reference. The experimental C.G. is recorded as:

G = (x̄, ȳ) = (8.4 cm, 5.7 cm)

Here, (G) is the centre of gravity, while (\bar{x}) and (\bar{y}) are its measured horizontal and vertical coordinates.

• Balancing check: Support the lamina beneath the marked point (G).
  • Stable balance indicates that the upward reaction and downward weight act along the same vertical line.
  • If it consistently tips, recheck the suspension lines or inspect the lamina for added mass.
• Interpretation: Changing the suspension point changes the body’s orientation but not the C.G.’s fixed position relative to the body.

C. Applications and Experimental Limitations

The suspension method is especially useful for irregular components, although its accuracy is restricted by practical measurement errors.

• Applications:
  • Locating the balance point of an irregular aircraft model or wing profile
  • Positioning supports for hanging signs and decorative panels
  • Studying stability before mounting mechanical components
  • Checking whether added material has shifted a component’s C.G.
• Importance in flight: An aircraft’s overall C.G. affects trim, stability, and controllability.
  • A forward shift generally increases the balancing force required from the tail.
  • An excessively rearward position can reduce longitudinal stability.
  • A two-dimensional lamina experiment demonstrates the same underlying moment principle, though real aircraft require three-dimensional mass calculations.
• Limitations:
  • The simple method directly locates only the in-plane C.G. of a thin lamina.
  • Thick three-dimensional bodies require suspension in different planes or calculation from component masses.
  • Flexible objects may change shape and redistribute mass during testing.
  • A C.G. outside a concave object cannot be checked by placing a support directly beneath material at that point.
• Experimental quality: Repeated suspensions, fine markings, and an average intersection region make the final location more reliable than a single visual estimate.