# Mechanics problem - centre of mass

• thehammer
In summary, Three uniform rods, each of length L = 22cm, form an inverted U. The vertical rods each have a mass of 14g; the horizontal rod has a mass of 42g. Using the equation for the centre of mass, x(com) = L/2, we can find the x coordinate of the system's centre of mass to be 0.11m. To find the y coordinate, we can view the system from the side and imagine a see-saw with a weight of 42g on one end and a weight of 28g on the other end, with the balance point being the y coordinate of the centre of mass.
thehammer

## Homework Statement

Three uniform rods, each of length L = 22cm, form an inverted U. The vertical rods each have a mass of 14g; the horizontal rod has a mass of 42g. What are (a) the x coordinate and (b) the y coordinate of the system's centre of mass?

## Homework Equations

x(com) = 1/M $$\int$$ x dm
y(com) = 1/M $$\int$$ y dm

dm/M = dx/L
dm/M = dy/L

## The Attempt at a Solution

I used the result of the integration done to find that the centre of mass of a uniform rod in terms of its length: x(com) = L/2.

x(com) = 0.22 m / 2 = 0.11 m

However, I've no idea about what the y coordinate of the centre of mass would be. I am completely confused and I cannot find any examples of such a situation in my book. I have a mechanics examination tomorrow and need to clear up any misunderstood concepts so I would greatly appreciate any help.

Try viewing it from the side.

Pretend you have a 42g object sitting on a see-saw that weighs 28g and is 22 cm long, and they have asked you to find the balance point with no weight at the other end.

The y coordinate of the centre of mass can be found by taking into account the distribution of mass along the vertical and horizontal rods. Since the vertical rods have a mass of 14g each, their combined mass is 28g. This means that the horizontal rod, with a mass of 42g, must be split into two equal parts to maintain balance. This gives us a mass of 21g for each of the horizontal sections.

Using the formula for the centre of mass, we can find the y coordinate as:

y(com) = 1/M \int y dm

= (28g * 0.11m + 21g * 0.22m) / (28g + 21g)

= (3.08g*m + 4.62g*m) / 49g

= 7.7cm

Therefore, the centre of mass of the system is located at (0.11m, 7.7cm). It is important to note that the units for the y coordinate have been converted to centimeters to match the units of the given lengths. It is also important to double check the calculation to ensure that it is accurate.

I hope this explanation helps to clear up any confusion and good luck on your mechanics exam! Remember to always carefully consider the distribution of mass when calculating the centre of mass of a system.

## 1. What is the definition of centre of mass in mechanics?

The centre of mass is a point in an object or system that represents the average location of the mass of the object. It is the point where the total mass of the system can be concentrated to and still have the same external forces and torques acting on it.

## 2. Why is the centre of mass an important concept in mechanics?

The centre of mass is important because it allows us to simplify the analysis of a system by considering the motion of the entire system as a whole, rather than each individual part. It also helps us understand how external forces and torques affect the overall motion of an object or system.

## 3. How is the centre of mass calculated?

The centre of mass can be calculated by finding the weighted average position of all the individual mass elements in an object or system. This is typically done by using the formula:

xcm = (m1x1 + m2x2 + ... + mnxn) / (m1 + m2 + ... + mn)

Where xcm is the position of the centre of mass, mn is the mass of each individual element, and xn is the position of each individual element.

## 4. Can the centre of mass be located outside of an object?

Yes, the centre of mass can be located outside of an object. This is often the case for irregularly shaped objects or objects with non-uniform mass distributions. In these cases, the centre of mass may be located at a point where there is no actual mass present.

## 5. How does the centre of mass affect the stability of an object?

The centre of mass is directly related to an object's stability. If the centre of mass is located within the base of support, the object will be stable. However, if the centre of mass is located outside of the base of support, the object will be unstable and may topple over. This is why it is important to consider the position of the centre of mass in designing structures or objects to ensure stability.

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