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Center of Mass : Introduction 0/4
Momentum Conservation & Collision 0/4
Center of mass of continuous objects
To calculate Center of mass of continuous objects, first we calculate Center of mass of a very small object and then integrate it to find out Center of mass of complete object.
dm is a mass of elementary object. x is the axis coordinate of Center of mass of elementary object.
Elementary object may be any particle, ring, Rod, disk.
Center of mass of A Rod
let’s take elementary object as particle in this case. We will take a particle of mass dm and integrate it over the length of the Rod to find out Center of mass of rod. We assume that mass of the Rod is evenly distributed over the length.
We can assume that Centre of mass of particle dm lies at X.
Center of mass of semi-circular Ring
Let’s take a semi-circular ring, which lies in XY plane. We assume that mass of ring is evenly distributed over its area.
Let’s apply integration on this dm mass in y direction. (Centre of mass in X and Z direction are zero due to symmetry.)
Center of mass of semi-circular disc
In this case our elementary object will be a ring of mass DM and width dr. we know that Centre of mass of the ring of radius R lies at .
As mass of half disc is M,
And centre of mass of ring,
Center of mass of hollow hemisphere
In this case are elementary object is a ring of radius and width .
Center of mass of a solid hemisphere
To calculate Center of mass of a solid hemisphere we will take our element object as a disc.
In this case, centre of mass of disc = y
Center of mass of hollow cone
Let’s take an elemental object as a ring of mass dm and width dl.
Let’s assume, mass density per area =
Elementary object = Ring.
Center of mass will lie at 2H/3 from top or H/3 from bottom.
Center of mass of a solid cone
Elementary object = Disc
Center of mass lies at 3H/4 from top or H/4 from bottom.
Find out the distance of centre of mass of the given system from the centre.
In this case we have a half ring and a half disc. Let’s consider half ring and half disc as a point mass or a particle which is situated at their respective Center of mass.
Now we can find out Centre of mass of the system,
Find out distance of centre of mass from centre. Assume that mass density per unit area is uniform.
We can consider this system as below for calculating center of mass,
Q6. Center of mass of non-uniform rod.
Find out the distance of centre of mass of the rod from point o.