Unit 1: Experimental study of C.G location - Subjective Questions

The centre of gravity of a body is the point through which its entire weight may be considered to act, irrespective of the body's orientation.

It can be identified experimentally by the suspension method:

• Suspend the body freely from a point near its edge.
• Hang a plumb line from the same suspension point.
• Mark the vertical line indicated by the plumb line.
• Repeat the procedure using another suspension point.
• The intersection of the marked vertical lines gives the position of the C.G.

This works because, at equilibrium, the C.G. lies vertically below the point of suspension.

A uniform symmetrical lamina has its mass distributed equally on opposite sides of each axis of symmetry.

• The gravitational effects of equal mass elements on opposite sides balance one another.
• Therefore, the C.G. must lie on every axis of symmetry.
• The point at which the axes of symmetry intersect is the geometrical centre.

Hence, for a uniform symmetrical lamina, the centre of gravity coincides with the geometrical centre. For example, the C.G. of a rectangle lies at the intersection of its diagonals.

Apparatus: Rectangular cardboard lamina, retort stand, pin, plumb line, pencil and ruler.

Procedure:

1. Make a small hole near one corner of the lamina.
2. Suspend the lamina freely from the hole using a pin.
3. Suspend a plumb line from the same pin.
4. Allow both the lamina and plumb line to come to rest.
5. Draw a line along the plumb line on the lamina.
6. Repeat the process from a second hole near another corner.
7. Mark the intersection of the two vertical lines as the experimental C.G.

For verification, draw the two diagonals of the rectangle. Their intersection should coincide approximately with the experimentally determined C.G.

Experimental determination:

• Make a small hole near one vertex of the triangular lamina.
• Suspend the lamina from that hole and hang a plumb line from the suspension point.
• Mark the vertical line shown by the plumb line.
• Repeat from a second vertex.
• The intersection of the two lines is the experimental C.G.

Geometrical verification:

• Find the midpoint of each side.
• Join each vertex to the midpoint of the opposite side to construct the medians.
• The medians intersect at the centroid.
• The centroid lies at a distance of of a median from the corresponding vertex.

For a uniform triangular lamina, the centroid and C.G. coincide.

The suspension method is based on the principle of rotational equilibrium.

When a body is freely suspended and comes to rest:

• Its C.G. lies vertically below the suspension point.
• The line of action of its weight passes through the suspension point.
• The perpendicular distance between the suspension point and the line of action of weight becomes zero.
• Therefore, the moment of weight about the suspension point is zero.

Mathematically,

where is the weight and is the perpendicular distance from the suspension point to the line of action of .

Let a lamina of weight be suspended from a point . Suppose its C.G. is at , and the perpendicular distance between and the vertical line through is .

The moment of the weight about is

If , the weight produces a turning effect, causing the lamina to rotate. The lamina continues rotating until it reaches equilibrium. At equilibrium,

Since , this requires

Thus, the line of action of passes through . Because weight acts vertically downward, must lie vertically below . Repeating the suspension from another point produces another vertical line, and the intersection of the lines locates .

Apparatus: Irregular cardboard lamina, retort stand, pin, plumb line, pencil and ruler.

Method:

1. Make three small holes at well-separated points near the boundary of the lamina.
2. Suspend the lamina freely from the first hole.
3. Hang a plumb line from the same point and allow it to become stationary.
4. Draw the vertical line indicated by the plumb line.
5. Repeat the procedure from the second and third holes.
6. The three lines should meet at, or very close to, one point.
7. Mark their common intersection as the C.G.

Using three suspension points improves reliability because any small triangle formed by the lines reveals experimental error.

Symmetrical lamina:

• Its C.G. can often be predicted from geometry.
• The C.G. lies on every axis of symmetry.
• For a uniform lamina with two or more axes of symmetry, their intersection gives the C.G.
• Suspension can be used mainly to verify the geometrical result.

Unsymmetrical lamina:

• Its C.G. usually cannot be located by simple symmetry.
• Two or preferably three suspension lines are required.
• The intersection of these vertical lines gives the C.G.
• Greater care is needed because no geometrical centre is generally available for direct verification.

A single suspension establishes only that the C.G. lies somewhere on one vertical line below the suspension point. It does not identify one exact point.

When the lamina is suspended from a second point:

• A second vertical line containing the C.G. is obtained.
• The C.G. must lie on both lines.
• Therefore, their intersection identifies its position.

A third suspension line is often drawn as a check. Ideally, all three lines pass through the same point.

Suspension pin:

• Supports the lamina while allowing it to rotate freely.
• Provides the point about which moments act.
• At equilibrium, the C.G. lies vertically below this point.

Plumb line:

• Establishes the true vertical direction using the weight of its bob.
• Shows the line passing downward through the suspension point.
• Enables this vertical line to be transferred onto the lamina.

Together, the pin and plumb line identify a line on which the C.G. must lie.

Important precautions include:

• Allow free rotation: The lamina must not touch the stand, table or wall.
• Wait for rest: Mark the line only after oscillations have stopped.
• Use small holes: Large holes make the suspension point uncertain.
• Reduce friction: The lamina should hang freely from the pin.
• Use separated points: Suspension holes should be well apart to produce clearly intersecting lines.
• Avoid parallax: View the string directly from the front while marking.
• Use a thin string and sharp pencil: This reduces line-thickness error.
• Keep the lamina rigid: Bending changes its mass distribution and can shift its C.G.

For a uniform rectangle, the C.G. lies at the intersection of its diagonals. Measured from one corner, its coordinates are

and

Thus, the predicted C.G. is from either vertical side and from either horizontal side.

To verify it:

• Suspend the lamina from a hole near one corner and draw the plumb-line direction.
• Repeat from another corner.
• Mark the intersection of the suspension lines.
• Compare this point with the intersection of the diagonals.

The points should coincide within experimental uncertainty.

For a uniform lamina, the mass on one side of an axis of symmetry is balanced by an equal mass distribution on the other side. Therefore, the C.G. lies on the axis.

• Circular lamina: Every diameter is an axis of symmetry, so the C.G. is at the centre of the circle.
• Square lamina: The diagonals and the lines joining opposite side midpoints are symmetry axes. Their intersection is the C.G.
• Rectangular lamina: The two lines joining opposite side midpoints are symmetry axes, and the diagonals also intersect at the same central point.

The suspension method can experimentally confirm these geometrical locations.

Yes. The C.G. need not lie within the material of an object; it represents the effective point of action of the total weight.

For example, the C.G. of a uniform circular ring lies at its geometric centre, where no material is present.

Experimental demonstration:

• Suspend the ring from one point and extend the plumb-line direction across its open region.
• Suspend it from a second point and draw another vertical line.
• The extended lines intersect at the centre of the ring.

Thus, the C.G. lies in empty space inside the ring.

Suspension method:

• The lamina is hung successively from different points.
• Vertical lines are marked using a plumb line.
• Their intersection gives the C.G.
• It is particularly suitable for irregular laminas.

Balancing method:

• The lamina is balanced on a narrow edge or pointed support.
• When balanced on a point, the vertical line through the C.G. passes through the support.
• It provides a quick verification but can be sensitive to friction and the width of the support.

The suspension method usually gives a clearer graphical location, while balancing provides a useful independent check.

Geometrical symmetry alone does not guarantee that the C.G. is at the geometric centre. The mass distribution must also be symmetrical.

If one region is denser or carries an attached mass:

• That region contributes more weight.
• The C.G. shifts toward the heavier region.
• The geometric centre may no longer be the balance point.

For discrete parts, the C.G. coordinate can be expressed as

with a similar expression for . Therefore, experimental suspension is valuable when uniformity is uncertain.

Possible causes include:

• The lamina or plumb line was still oscillating when a line was marked.
• The suspension holes were too large.
• Friction prevented the lamina from hanging freely.
• Thick pencil lines or a thick string reduced precision.
• Parallax occurred while transferring the plumb-line position.
• The lamina touched the support or became bent.

The C.G. may be estimated at the centre of the small error triangle. The experiment should then be repeated with smaller holes, a fine string, a sharp pencil and sufficient settling time. A smaller triangle indicates better precision.

Theoretical position:

1. Draw both diagonals of the square.
2. Mark their intersection as the theoretical C.G.
3. If the side length is , the point is at coordinates from any chosen corner.

Experimental position:

1. Make small holes near three different corners.
2. Suspend the square from the first hole and mark the plumb-line direction.
3. Repeat from the other two holes.
4. Mark the common intersection as the experimental C.G.

Comparison:

Measure the separation between the two positions. Ideally, . A small nonzero value may result from line thickness, non-uniform cardboard, parallax or inaccurate suspension.

Removing material destroys the original uniform symmetry and reduces the mass on the cut side. Consequently, the C.G. shifts away from the removed portion and toward the side containing more remaining mass.

The new C.G. can be found by:

• Making two or three holes near different points on the remaining lamina.
• Suspending it from each hole in turn.
• Drawing the corresponding plumb-line directions.
• Marking their intersection as the new C.G.

The original geometric centre should not be assumed to remain the C.G. after the material is removed.

Aim: To locate the C.G. of symmetrical and unsymmetrical laminas experimentally.

Procedure:

1. Select a uniform symmetrical lamina and an irregular lamina.
2. Make at least three small, well-separated holes in each.
3. Suspend each lamina from one hole at a time.
4. Hang a plumb line from the same point and mark its vertical direction after equilibrium is reached.
5. Identify the common intersection of the lines.
6. For the symmetrical lamina, compare this point with the intersection of its symmetry axes.

Observations:

• In the symmetrical lamina, the experimental C.G. is close to the geometrical centre.
• In the unsymmetrical lamina, the C.G. is determined by the suspension-line intersection and need not match the visual centre.

Conclusion: A freely suspended body's C.G. lies vertically below its suspension point. Symmetry predicts the C.G. only when the mass distribution is also uniform and symmetrical.