Important Centre of Mass Formulas for JEE Main and Advanced

Before we look at the centre of mass formulas lets us quickly look at what the concept means. Centre of mass of a body is defined as a point at which whole of the mass of the body appears to be concentrated. Centre of mass of few useful configurations are given below.

Centre of Mass Formulas

1. A system of two point masses m1r1 = m2r2. The centre of mass lies closer to the heavier mass.

System of two point masses

2. Triangle (at the centroid)

yc=h/3

Triangle (at the centroid)

3. A semi-circular disc

yc= 4r/3π

 A semi-circular disc

4. Semicircular ring

yc= 2r/π

Semicircular ring

5. Hemispherical shell

yc= r/2

Hemispherical shell

6. Solid Hemisphere

yc =3r/8

Solid Hemisphere

7. Cone.

The height of the centre of mass from the base is h/4 for the solid cone and h/3 for the hollow cone.

Motion of the Centre of Mass and Conservation of Momentum

1. Velocity of centre of mass

Vcm=m1v1+m2v2——mnvnM\vec{V}_{cm}=\frac{m_{1}\vec{v_{1}}+m_{2}\vec{v_{2}}——-m_{n}\vec{v_{n}}}{M} pcm=Mvcm\vec{p}_{cm}=M\vec{v_{cm}}

2. Acceleration of centre of mass

acm=m1a1+m2a2——mnanM\vec{a}_{cm}=\frac{m_{1}\vec{a_{1}}+m_{2}\vec{a_{2}}——-m_{n}\vec{a_{n}}}{M} acm=FextM\vec{a}_{cm}=\vec{F}_{ext}{M}

Impulse

J=Fdt=Δp\vec{J}=\int \vec{F}{dt}=\Delta\vec{p}

Collision

Collision

Momentum conservation m1v1+m2v2 =m1v’1 + m2v’2

Elastic Collision= ½ m1v12 + ½ m2v22 = ½ m1v’1 2+½m2v’22

Coefficient of restitution

e=(v1v2)v1v2e=\frac{-(v’_{1}-v’_{2})}{v_{1}-v_{2}}

e=0, completely elestic collision

e=1, completely inelastic collision

If v2=0 and m1≪ m2 then v’1 = -v1

If v2=0 and m1 ≫ m2 then v’2 = 2v1

Elastic Collision with m1=m2 then v’1 = v2 and v’2 = v1