Today our topic is for center of mass formula for continuous distribution of mass of the body. This topic is conceptual but important for whole topic in physics. So we will learn from basic concept to depth .Once you will learn the basic concept, Physics will be easy for you in every topic and your concept will developed the way to think about problems. In previous post we have study about basic concept of center of mass, When body is discrete, Means two or more body masses are separate. You can refer postcenter of massBut in this topic we will study center of mass formula, When body mass is distributed continuously, Hence the previous post formula will not be applicable, So have to drive new center of mass formula, So lets start and enjoy learning center of mass formula.
As we have discussed here body mass will be continuous distribution over a shape and length, not in discrete manner. So we required here concept of Integration meaning.
So that you can well understand without any doubt.
Hence first see the Integration meaning, Then will continue for center of mass formula derivation.
calculus is the important branch of mathematics, greatest invention for mankind and invented by Issac Newton .when he was stuck at a point where normal mathematics was not supporting to carry out his experiment and derivation after rigorous hard work for two years, Newton finally invented Calculus and you know without knowledge of mathematics we can’t understand physics well . mathematics is tool of physics or backbone of physics so we need to understand calculus application here .calculus is amazing branch of mathematics simple and complex both, but we are studying here physics so we will not study here in depth as per our need we will see application.
Now come to the Integration meaning. You generally see two symbol is frequently use in Physics and mathematics one Delta(∆) and other is d. Now see the meaning of both.
∆→ Meaning change for large number, When written as ∆xmeans change in x, but its value will be large.
d→ meaning change for very very small, When written as dx means change in x, but its value is very very small.
Here this concept will be use to calculate center of mass formula. In this topic you will learn how to take very small value to calculate mass of uniform distributed body. Apart from here this concept will be use in rotation mechanics also and everywhere in Physics as well as in mathematics.
So this concept is very important,Hence lets start.
Center of mass formula continuous mass distribution
Here the center of mass formula derive Xcm = ∫xdm/∫dm is for continuous mass distribution.Hence if continuous mass distributed then you can not apply individual or discrete center of mass formula Xcm = (m1x1+m2x2+m3x3+………..)/(m1+m2+m3…….).
Similarly Ycm = ∫ydm/∫dm or for z axis Zcm = ∫zdm/∫dm .
Now one important point what is meaning of ∫dm, This is sum of all small masses and it will be equal to total mass. So we can write ∫dm = M
Xcm = ∫xdm/M.
Now problem here in this above equation ∫xdm/M is that there are two variable x and dm of different type one is length and other is small mass so it can’t not be integrated unless you not convert into same type.
Hence to do integration above equation we have to change one type of variable,So we will have to change dm value in the form of x to do this one new term is introduce see below.
Linear mass density
Linear mass density is important concept try to understand this concept will be use in physics many topic. It is denoted by (𝞴) its meaning is mass per unit length.
𝞴 = M/L (mass/length) now important concept 𝞴 is always constant for uniform mass distribution, uniform means equally mass distributed over the length.
Hence for small dm above shown its length will be dx so we can write 𝞴 = dm/dx or
dm = 𝞴dx now we have got small mass dm value dm = 𝞴dx this was the key point of concept. Now put value of dm in above equation we will get.
Xcm = ∫xdm/M = ∫x𝞴dx/M here 𝞴 is constant so it will come out from integration hence
Xcm = 𝞴∫xdx/M now limit will be from one end of rod 0 to other end of rod L
Xcm = 𝞴∫xdx/M = 𝞴/2M*⎡x² ⎦ now put value of 𝞴 = M/L and limit after simplification we will
Xcm = L/2 this is the value of center of mass formula for uniform mass distributed rod .
Now mass distribution of rod can be uniform or non uniform both if non uniform then mass density will be not constant, Means value of 𝞴 will be not constant.
Question for you suppose non uniform mass distribution of same rod and its mass density is given 𝞴 = 2x then find center of mass.Try yourself if you have got the above concept it will be solve easily.
Center of mass formula for semicircular wire
Since it is a wire hence one dimension, So its linear mass density 𝞴 = M/L = M/𝞹R because it is a semicircle.
Again 𝞴 = dm/dx or dm = 𝞴dx now problem is that here dx is not know, So for that just see the small triangle dm arc. We know for small value of d𝞱 we can write Rd𝞱 = dx small length of mass.
So our small mass dm = 𝞴Rd𝞱 now put dm value in Ycm above equation we will get.
Ycm = ∫ydm/∫dm , put y = Rsin𝞠 in this equation we will get finally
Ycm = ∫Rsin𝞠𝞴Rd𝞠/M = R²*M/𝞹R*M∫sin𝞠d𝞠 f after integration we will get finally
Center of mass formula for semicircular wire Ycm = 2R/𝞹.This is important formula try to remember this formula.
Now we will continue in next post.I hope you have enjoyed to learn Center of mass formula for uniform distributed mass. if you like comment and share thanks for reading and sharing keep reading more and more.