6.8.2 - Centre of gravity
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Definition and Importance of Centre of Gravity
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Today, we'll learn about the centre of gravity or CG. Can anyone tell me what they think the centre of gravity means?
Is it like the balance point of an object?
Exactly! The CG is the point where the total weight of the body acts. When an object is balanced at this point, it will remain stable.
So, if you hold an object at its CG, it won’t tip over?
Correct! If the CG is above the base of support, the object is in equilibrium. Remember, we can locate the CG even in irregular shapes.
Can you show us how to find the CG practically?
Sure! Let’s do a simple experiment with a piece of cardboard to locate its CG by balancing it on a pencil.
To recap: The CG is crucial in mechanics because it helps us understand stability and balance.
Methods to Locate the Centre of Gravity
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Now, let’s delve into how we can locate the centre of gravity. One common method is the balance method. Can anyone explain how we could do this?
We could use a pencil or a finger to balance the object!
Exactly! When the object balances horizontally, the point of support is where the CG is located.
What about irregular shapes? Can we still apply this method?
Great question! Yes, we can repeat the balancing from different points on the object. The intersection of the vertical lines from different balancing points gives us the CG.
What if we have a shape that is very complex?
In such cases, we have specific formulas and methods, but the basic principle remains the same. The idea is to find where all the weight acts together as one.
Summarizing, locating the CG helps us understand a lot about how objects behave under different forces.
Application and Significance of Centre of Gravity
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Let’s now discuss why the centre of gravity is significant in our daily lives and engineering practices.
Does CG affect how stable a car is?
Absolutely! Cars with a lower CG are less likely to roll over during sharp turns. Understanding CG helps in design and safety.
What about sports? Do athletes consider CG?
Yes! Gymnasts and divers use their CG to achieve balance and control during flips and rotations.
Can CG change if the shape of the object changes?
Yes, changes in mass distribution will change the CG! That’s vital to think about in dynamic situations.
In summary, CG is crucial for stability in design, performance, and everyday tasks.
Introduction & Overview
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Quick Overview
Standard
This section explains how the centre of gravity represents the average location of weight distribution in an object, detailing methods for locating it experimentally. The significance of the centre of gravity is also discussed in terms of mechanical and rotational equilibrium.
Detailed
Centre of Gravity
The centre of gravity (CG) of an object is defined as the point at which the object can be balanced. This section elaborates on the principles surrounding the CG in the context of mechanical equilibrium and rotational dynamics. By understanding the position of CG, one can predict how an irregular object will behave when subjected to gravitational forces.
- **Definition of Centre of Gravity: ** The CG is the point where the resultant torque due to gravity acting on the body is zero. For uniform objects, the CG typically coincides with the geometric centre.
- Balancing Experiment: The CG can be experimentally located using methods like balancing an object on a point, thereby finding where it can rest without tipping over. An example provided illustrates balancing a cardboard cutout on the tip of a pencil.
- Torque Consideration: Each particle of the object contributes to its weight, and the CG is the point where their collective moments (torques due to gravitational forces) balance out.
- Importance in Equilibrium: The CG plays a crucial role in determining the stability and equilibrium conditions of objects. The CG is vital in assessing whether an object will tip over or remain stable under applied forces.
By understanding CG, one can make predictions about the behaviour of various objects in real-life scenarios, including engineering applications, sports, and architecture.
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Balancing a Cardboard
Chapter 1 of 3
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Chapter Content
Many of you may have the experience of balancing your notebook on the tip of a finger. Figure 6.24 illustrates a similar experiment that you can easily perform. Take an irregular-shaped cardboard having mass M and a narrow tipped object like a pencil. You can locate by trial and error a point G on the cardboard where it can be balanced on the tip of the pencil. (The cardboard remains horizontal in this position.) This point of balance is the centre of gravity (CG) of the cardboard.
Detailed Explanation
This chunk explains how to find the center of gravity (CG) by balancing an irregular object. When you balance the cardboard on a pencil, you are finding the point G where the weight of the cardboard acts straight down. At this point, the gravitational pull is evenly distributed, resulting in stability. This means that the forces acting on the cardboard, like its weight, and the force from the pencil together create no torque, keeping the cardboard level.
Examples & Analogies
Think of a seesaw at a playground. If you sit in the exact middle (CG), the seesaw stays balanced. If you sit off-center, one side will go down while the other side rises. The center of gravity ensures stability in various objects, and is similar to how balance works on a seesaw.
Total Gravitational Torque
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Chapter Content
If ri is the position vector of the ith particle of an extended body with respect to its CG, then the torque about the CG, due to the force of gravity on the particle is τττττi = ri × mi g. The total gravitational torque about the CG is zero, i.e., τ τ τ τgii im = = × =∑ ∑ r g 0 (6.33). We may therefore, define the CG of a body as that point where the total gravitational torque on the body is zero.
Detailed Explanation
Here, we learn that the center of gravity (CG) can be mathematically defined through torque. When all the particle weights (m_i g) create torques (τ_i), for the CG, the total effect of these torques must balance out to zero. This means that if you were to sum all the torques created by the weights at their respective distances (r_i) from the CG, it would cancel out, indicating stability and balance.
Examples & Analogies
Imagine trying to balance a long stick on your finger. If the CG of the stick is directly over your finger (like the point where all weights balance out), it stands up straight. If it's not, it will tip over. The concept of balancing torques relates directly to how we find this CG in real-life situations.
Center of Gravity vs. Center of Mass
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Chapter Content
We notice that in Eq. (6.33), g is the same for all particles, and hence it comes out of the summation. This gives, since g is non-zero, ∑mi ir = 0. Remember that the position vectors (ri) are taken with respect to the CG. Now, in accordance with the reasoning given below Eq. (6.4a) in Sec. 6.2, if the sum is zero, the origin must be the centre of mass of the body. Thus, the centre of gravity of the body coincides with the centre of mass in uniform gravity or gravity-free space.
Detailed Explanation
In this segment, we explore the relationship between the center of gravity and the center of mass. Since the gravitational field (g) does not change across small objects, the CG aligns with the center of mass (CM). We summarize that when an object's mass is uniformly distributed (like a solid ball), its CG is also the point where mass is evenly distributed about it.
Examples & Analogies
Picture a perfectly balanced seesaw where the mass on both sides is equal. The point in the middle where it balances is both the center of gravity and the center of mass, demonstrating how evenly distributed mass translates to balance.
Key Concepts
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Centre of Gravity (CG): The point where the weight of a body is balanced.
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Balance Stability: An object remains stable when CG is within its base of support.
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Torque: The rotational effect of a force applied at a distance from an axis.
Examples & Applications
Balancing a cardboard cutout on the tip of a pencil to find its CG.
Analysis of a car's stability and performance based on CG location.
Memory Aids
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Rhymes
Balance in the center, the weight we find, measure it carefully, to stability bind!
Stories
Imagine a circus performer balancing on a tightrope; they must find their centre of gravity to avoid falling.
Memory Tools
CG - Careful Geometry: Remember to think of geometric shapes for uniform objects.
Acronyms
CG
'Central Gravitational point' where balance is key.
Flash Cards
Glossary
The point where the total weight of a body is assumed to act.
A measure of the force causing an object to rotate.
The point at which an object can be balanced without tipping over.
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