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The Centre of Gravity may be viewed as a force that pulls objects downward and more specifically towards the centre of the Earth. Things can occasionally fall over and topple due to gravity, especially if they are high up and unstable.
- Tightrope walkers are the best people to grasp gravity.
- They frequently wobble from side to side while tiptoeing across the high wire to amuse us, but they seldom ever fall.
- They are able to maintain their grip on the rope due to their innate understanding of the physics of forces.
- The Latin or Greek terms for "centrum" and "gravitation" are the source of the English "center" and "gravitation," respectively.
- The location where gravity operates on the body is known as the center of gravity (CG) of any thing.
- The geometric center of an item with homogeneous density is referred to as the centroid.
Key Terms: Centre of Gravity, concept of centre of gravity, location of centre of gravity, properties, calculation, application.
What Is the Center Of Gravity?
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The centre of gravity (C.G.) of a body is the location where the algebraic sum of the moments of weights of all the constituent particles is equal to zero.
- At this stage, it is possible to consider that the complete body's weight will act regardless of where the body is located.
- In simple words, the centre of gravity is an imaginary point at which the total weight of an object is said to be concentrated.
- It is used to determine how a moving body will behave when being affected by gravity.
Following are some real-life examples of the concept of centre of gravity:
- When a ball is balanced on the tip of a finger, the exact point where the ball makes contact with the finger is where the ball's center of gravity can be observed.
- The force that makes sure the Earth and the other planets orbit the sun with the proper alignment.
- The force that prevents your drink from floating to the top of your glass but instead making it settle at the bottom.
- The force that causes a car to roll backwards even when the gas pedal is not engaged.
Read more: Gravity Formula
Formula for Centre Of Gravity
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Onе of thе most prevalent instances of a distributеd forcе is exemplified by an objеct's wеight. Whеn considеring a body that occupiеs a spеcific rеgion in spacе, thе weight of an infinitesimally small volumе еlеmеnt dV is represented as dW =ρ dV, where ρ denotes thе wеight dеnsity (wеight pеr unit volumе).
- Consеquеntly, thе cumulativе wеight W of thе еntirе body can bе thought of as thе summation of an infinitе array of parallеl forcеs dW.
- Thе centre of wеight, also known as thе cеntеr of gravity, for thе body corrеsponds to a point G.
- This point is whеrе thе cоllеctivе weight W is effectively concentrated to act.
- To dеtеrminе thе coordinatеs of this point G, onе can utilizе thе rеsultant momеnt gеnеratеd by thе distributed weight.
- This momеnt is equal to thе momеnt of W about thе coordinatе axеs.
- This approach allows for thе calculation of thе precise location of thе cеntеr of wеight or cеntеr of gravity for thе givеn body.
Centre of Gravity For Symmetrical Bodies
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The following methods can be used to identify the center of gravity for basic shaped objects:
- The center of gravity is the location at which an item balances when it is supported by a string or an edge.
- The following is another easy physical method used to locate the center of gravity:
- The center of gravity of the plate may be found by suspending it from point A with a cord and then securing the cable at point C.
Centre of Gravity for Symmetrical Bodies
When the plate is hung from A, the line AD is vertical, and when hanging from C, the line CE is vertical. The plate's center of gravity is located where AD and CE meet. The center of gravity of any hung item is located just underneath that point.
Centre of Gravity For Asymmetrical Bodies
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When the mass is not distributed evenly, we apply calculus to identify the center of gravity. We use the symbol S dw to represent the integration of a continuous function with regard to weight. The center of gravity is then established as follows:
cg * W = SxdW
Here, x denotes the separation from a reference line, dw is a weight increment, and W denotes the overall weight of the item.
W = mg
However, we are aware that an object's mass is equal to
m = ρV
By combining the two equations, we arrive at the solution.
W = gρV
Solving
dW = gρdV
dw = g * ρ (x, y, z) * dxdydz
The center of gravity functional form of the mass distribution,
cg * W = g * SSSx * ρ (x, y, z)dxdydz
A triple integral over dx, dy, and dz is denoted in the equation by the symbol SSS.
Centre of Gravity of Different Objects
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The centre of gravity of different objects are mentioned below:
| The shape of the body | Centre of Gravity |
|---|---|
| A ring | The hollow centre of the ring |
| A circular disc | The disk’s centre |
| A cylinder | The middle point of the axis |
| A cube | The point where the diagonals intersect |
| A triangular plate | The point where the medians intersect |
| A square or rectangular plate | The point where the diagonals intersect |
| Cylinder | The middle point of the axis |
| Cone or pyramid | The point that is at 1/4th the length of the apex from the centre of the base. |
Properties of Centre of Gravity
The properties are listed below:
- The centre of mass and gravity are the same only in a homogeneous gravitational field, where g (gravitational force) is constant.
- But g changes depending on the item. In these circumstances, the centre of gravity will shift toward an area where the item is subject to a greater gravitational field than the uniform field.
- The geometric centre of the majority of uniform objects serves as their centre of gravity.
- The centre of gravity in some things can also be situated outside of the geometric centre.
- The centre of gravity is frequently situated outside of the physical body in hollow or asymmetrical shaped things. Rings and chairs are two examples.
Read more: Difference between Centre of Gravity & Centroid
Difference Between Centre Of Gravity And Centre Of Mass
The Center of Gravity and the Center of Mass are not the same, unlike common belief, and this is incorrect. There are various differences between the two concepts, and they are as follows:
| Parameter | Centre of Mass | Centre of Gravity |
|---|---|---|
| Definition | The point at which the mass of a system is centred | The point at which the force of gravity appears to be acting |
| Calculation | Consider only the mass of the system | Consider both the mass and the distribution of gravitational forces on the system |
| Depend On | Only the mass distribution of the system | Both the mass distribution and the local gravitational field |
| Location | May or may not coincide with the geometric centre | Always coincides with the geometric centre |
| Example | A uniform rectangular plank has its centre of mass at its geometric centre | A car on a hill has its centre of gravity shifted towards the downhill side |
| Application | Used in mechanics and dynamics | Used in the analysis of gravitational fields |
Also Read:
| Related Articles | ||
|---|---|---|
| Gravitation | Gravitational Potential energy | Weightlessness |
Things to Remember
- The place where all of the body's mass is concentrated is known as the center of gravity or center of mass.
- The formula W=S x dw may be used to determine the center of gravity.
- The centroid is the location at which an object's density is evenly distributed across its whole body at its geometric center.
- The centroid may be determined either by taking the mean of the median or by utilizing the plumb line approach.
- These two phrases have been connected to the ideas of an object's weight, density, and balance.
- The centre of mass and gravity are the same only in a homogeneous gravitational field, where g (gravitational force) is constant.
Previous Year Questions
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Sample Questions
Ques. How does the balance change when the center of gravity changes? (3 marks)
Ans. The object's stability is influenced by its center of gravity.
- An item is more stable the lower its center of gravity is.
- The likelihood of an object collapsing when pushed increases with its height.
- Due to their low center of gravity, racing vehicles can make sharp turns without tipping over.
Ques.. The coordinates of vertices of a triangle ABC are A(1, 2), B(4, 6), and C(-3, -2). Three particles of masses 1 kg, 2 kg, and m kg are placed at the vertices of the triangle. If the coordinates of the centre of gravity are (3/5, 2). Find the mass m. (5 marks)
Ans. Let the coordinates of centre of gravity be (x, y), then for three particle system
x = (m1x1 + m2x2 + m3x3) / (m1 + m2 + m3)and
y = (m1y1 + m2y2 + m3y3) / (m1 + m2 + m3)
Given
x = 3/5 and y = 2
x1 = 1, x2 = 4, and x3 = – 3
y1 = 2, y2 = 6, and y3 = – 2
m1 = 1 kg, m2 = 2 kg, and m3 = m
On substituting the values in x coordinate, we get
x = (m1x1 + m2x2 + m3x3) / (m1 + m2 + m3)
⇒ 3/5 = (1 x 1 + 2 x 4 + 3 x -3) / (1 + 2 + m)
⇒ 3/5 = (9 - 3m) / (3 + m)
⇒ 9 + 3m = 45 - 15m
⇒ m = 2 kg
Ques. Two point masses, m1 = 5 Kg and m2 = 2Kg, are located at x = 2 m and x = 6 m respectively. Find the Centre of gravity. (3 marks)
Ans. Formula for the Centre of gravity is equal to centre of mass when gravity is constant
X cm = m1x1 + m2x2+…../M
m1 = 5Kg, m2 = 2Kg and x = 2 m and x = 6 m.
M = m1 + m2
⇒ M = 5 + 2 = 7
xcm = m1x1 + m2x2+…../M
⇒ xcm = m1x1 + m2x2/M
⇒ xcm = (5)(2)+(2)(6)/7
⇒ xcm = 22/7
Ques. Three bodies of masses 1 kg, 2 kg, and 3 kg are lying in the x-y plane at (0, 0), (2, 2), and (4, 0) respectively. What are the coordinates of the centre of gravity? (5 marks)
Ans. Let the coordinates of centre of gravity be (x, y), then for three particle system
x = (m1x1 + m2x2 + m3x3) / (m1 + m2 + m3) and
y = (m1y1 + m2y2 + m3y3) / (m1 + m2 + m3)
Given
x1 = 0, x2 = 2, and x3 = 4
y1 = 0, y2 = 2, and y3 = 0
m1 = 1 kg, m2 = 2 kg, and m3 = 3 kg
On substituting the values in x coordinate, we get
x = (m1x1 + m2x2 + m3x3) / (m1 + m2 + m3)
⇒ x = (1 x 0 + 2 x 2 + 3 x 4) / (1 + 2 + 3)
⇒ x = 8/3
On substituting the values in y coordinate, we get
y = (m1y1 + m2y2 + m3y3) / (m1 + m2 + m3)
⇒ x = (1 x 0 + 2 x 2 + 3 x 0) / (1 + 2 + 3)
⇒ x = 2/3
Therefore, the coordinates of the centre of gravity are (8/3, 2/3).
Ques. Two bodies of masses 1 kg and 2 kg are lying in the x-y plane at (-1, 2) and (2, 4) respectively. What are the coordinates of the centre of gravity? (5 marks)
Ans. Let the coordinates of centre of gravity be (x, y), then for two particle system
x = (m1x1 + m2x2) / (m1 + m2) and
y = (m1y1 + m2y2) / (m1 + m2)
Given
x1 = -1 and y1 = 2
x2 = 2, and y2 = 4
m1 = 1 kg
m2 = 2 kg
On substituting the values in x coordinate, we get
x = (m1x1 + m2x2) / (m1 + m2)
⇒ x = (1 x -1 + 2 x 2) / (1 + 2)
⇒ x = 1
On substituting the values in the y coordinate, we get
y =(m1y1 + m2y2) / (m1 + m2)
⇒ y = (1 x 2 + 2 x 4) / (1 + 2)
⇒ y = 10/3
Therefore, the coordinates of centre of gravity are (1, 10/3)
Ques. Can thе Center of Gravity Changе? (1 mark)
Ans. Yеs, thе cеntеr of gravity can change if thе distribution of mass changеs. For еxamplе, if you add or remove weight from onе sidе of an objеct, its cеntеr of gravity will shift in that dirеction.
Ques. How is thе Cеntеr of Gravity Usеd in Vehicle Dеsign? (2 marks)
Ans. In vеhiclе dеsign, thе cеntеr of gravity is critical for stability and handling. A lower cеntеr of gravity improvеs stability and rеducеs thе risk of rollovеrs. Enginееrs adjust thе distribution of mass in vеhiclеs to optimizе thеir performance and safеty.
Ques. Can thе Cеntеr of Gravity Bе Outsidе an Objеct? (1 mark)
Ans. No, thе cеntеr of gravity is always within thе confinеs оf thе object itsеlf. It's a propеrty of thе objеct's mass distribution.
Ques. How Doеs thе Cеntеr of Gravity Rеlatе to thе Cеntеr of Mass? (2 marks)
Ans. The tеrms "cеntеr of gravity" and "cеntеr of mass" arе oftеn usеd interchangeably, especially in еvеryday languagе. While they're related concеpts, thе cеntеr of mass rеfеrs to thе point whеrе thе mass of an objеct is concentrated, whilе thе cеntеr of gravity is thе point whеrе thе gravitational forcе acts.
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