Mechanics problem - 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.

 

[LowlyPion]

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.

 

[thomate1]

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.