Learn center of mass formula in physics, practice problems to finding an object's center of mass, by balancing it and find the position of center of mass of the particles.
Note: This assumption, of all mass concentrated at center of mass is not applicable for all cases.
Position of center of mass of \(n\) particles of masses \(m_1, m_2 ,m_3.......m_n\) and position vectors \(\vec r_1 , \vec r_2 ,.........\vec r_n \) is given by,
\(\vec r _{cm} = \dfrac{m_1 \,\vec r _1 + m_2 \,\vec r_2 + m_3 \,\vec r_3 +....m_n\,\vec r_n}{m_1+m_2 +m_3+....m_n}\)
\(x_{cm} = \dfrac{m_1 \,x_1+m_2 \,x_2+m_3 \,x_3+....m_n \,x_n}{m_1 + m_2 + m_3 +....m_n}\)
\(y_{cm} = \dfrac{m_1 \,y_1+m_2 \,y_2+m_3 \,y_3+....m_n \,y_n}{m_1 + m_2 + m_3 +....m_n}\)
\(z_{cm} = \dfrac{m_1 \,z_1+m_2 \,z_2+m_3 \,z_3+....m_n \,z_n}{m_1 + m_2 + m_3 +....m_n}\)
Stepwise algorithm for calculation of center of mass:
Step 1- Choose the origin to be the point with respect to which the position of center of mass is to be calculated.
Step 2- Find the co-ordinates of the particles.
Step 3- Use the formula for \(x_{cm},\,y_{cm} \,\text{and} \,\,z_{cm}\) to find the co-ordinates of center of mass.
Note: Center of mass is the mass weighted center of the object. It is nearer to the heavier mass.
A \((7.2,0)\)
B \((9.2,0)\)
C \((8.4,2)\)
D \((5,6)\)
Position of center of mass of \(n\) particles of masses \(m_1, m_2 ,m_3.......m_n\) and position vectors \(\vec r_1 , \vec r_2 ,.........\vec r_n \) is given by,
\(\vec r _{cm} = \dfrac{m_1 \,\vec r _1 + m_2 \,\vec r_2 + m_3 \,\vec r_3 +.........m_n\,\vec r_n}{m_1+m_2 +m_3+.............m_n}\)
\(x_{cm} = \dfrac{m_1 \,x_1+m_2 \,x_2+m_3 \,x_3+............m_n \,x_n}{m_1 + m_2 + m_3 +..........m_n}\)
\(y_{cm} = \dfrac{m_1 \,y_1+m_2 \,y_2+m_3 \,y_3+............m_n \,y_n}{m_1 + m_2 + m_3 +..........m_n}\)
\(z_{cm} = \dfrac{m_1 \,z_1+m_2 \,z_2+m_3 \,z_3+............m_n \,z_n}{m_1 + m_2 + m_3 +..........m_n}\)
Stepwise algorithm for calculation of center of mass:
Step 1- Choose the origin to be the point with respect to which the position of center of mass is to be calculated.
Step 2- Find the co- ordinates of the particles.
Step 3- Use the formula to find the co- ordinates of center of mass.
Note: Center of mass is the mass weighted center of the object. It is nearer to the heavier mass.
A \(\left(\dfrac{4}{3}\,m,\dfrac{1}{\sqrt3}\,m\right)\)
B \(\left(\dfrac{1}{2}\,m,\dfrac{1}{\sqrt6}\,m\right)\)
C \(\left(\dfrac{2}{1}\,m,\dfrac{1}{3}\,m\right)\)
D \(\left(\dfrac{5}{1}\,m,\dfrac{6}{\sqrt3}\,m\right)\)
Discrete Particles
Position of center of mass of \(n\) particles of masses \(m_1, m_2 ,m_3.......m_n\) and position vectors \(\vec r_1 , \vec r_2 ,.........\vec r_n \) is given by
\(\vec r _{cm} = \dfrac{m_1 \,\vec r _1 + m_2 \,\vec r_2 + m_3 \,\vec r_3 +.........m_n\,\vec r_n}{m_1+m_2 +m_3+.............m_n}\)
\(x_{cm} = \dfrac{m_1 \,x_1+m_2 \,x_2+m_3 \,x_3+............m_n \,x_n}{m_1 + m_2 + m_3 +..........m_n}\)
\(y_{cm} = \dfrac{m_1 \,y_1+m_2 \,y_2+m_3 \,y_3+............m_n \,y_n}{m_1 + m_2 + m_3 +..........m_n}\)
\(z_{cm} = \dfrac{m_1 \,z_1+m_2 \,z_2+m_3 \,z_3+............m_n \,z_n}{m_1 + m_2 + m_3 +..........m_n}\)
Step 1
Choose the origin to be the point with respect to which the position of center of mass is to be calculated.
Step 2
Find the co- ordinates of the particles.
Step 3
Use the formula to find the co- ordinates of center of mass.
Note: Center of mass is the mass weighted center of the object. It is nearer to the heavier mass.
A \(\left(\dfrac{44}{9},4,\dfrac{52}{9}\right)\)
B \(\left(\dfrac{63}{9},4,\dfrac{52}{12}\right)\)
C \(\left(\dfrac{57}{9},4,\dfrac{47}{9}\right)\)
D \(\left(\dfrac{6}{2},4,\dfrac{4}{8}\right)\)
The same is applicable if we allow a body to be balanced on a hinge. In this case, center of mass is directly below the hinge point.
Steps for finding center of mass:
Step 1: Hinge or pivot the body at any point A and let it balance itself.
Step 2: Draw a vertical line passing through the hinge/pivot. The center of mass will lie on this line AB.
Step 3: Hinge or pivot the body at another point C.
Step 4: Draw a vertical line passing through C. Center of mass will also lie on this line CD.
Step 5: Intersection of AB and CD will be the position of center of mass .
The body is same in step (1) & (2) and (3) (4) & (5).
1. Square Sheet:
Center of mass is at the geometrical center.
2. Disk, Ring, Hollow sphere and Solid sphere:
Center of mass is at the geometrical center.
3. Rectangular Sheet:
Center of mass is at the geometrical center.
A E
B A
C Between E and O
D O
For calculation of center of mass of a combination of bodies, we can consider different bodies to be point masses, situated at their respective center of masses.
A \(\left(\dfrac{M_2 (R_1 + R_2)}{M_1 + M_2} , 0\right)\)
B \(\left(\dfrac{M_1 (R_1 + R_2)}{M_1 + M_2} , 0\right)\)
C \(\left(\dfrac{M_2 (M_1 + M_2)}{R_1 + R_2} , 0\right)\)
D \((0,0)\)
Mass = Volume × Density
For a rod, \(\text{Volume} = A × \ell\)
\(\therefore M = A × \ell × \rho\)
Keeping density and area same,
\(M \propto \ell\)
\(\therefore \) If we cut a rod into two parts keeping the length in the ratio \(1: n\), the masses will also be divided in the same ratio.
\(\dfrac{M_1}{M_2} = \dfrac{\ell_1 }{\ell_2} = \dfrac{1}{n}\)
For a disk, \(\text{Volume} = \pi \,R^2 \,t\)
\(\therefore M = \pi \,R^2 \,t \, \rho\)
Keeping density and thickness same,
\( M \propto R^2\)
If we have two disks of same density and thickness of radii in ratio \(1:n \,,\) then the masses of the disks will be in the ratio \(1:n^2\).
\(\dfrac{M_1}{M_2 } = \dfrac{(R_1)^2}{(R_2)^2} = \left(\dfrac{1}{n}\right)^2\)
Volume of solid sphere \( = \dfrac{4}{3} \pi \,R^3\)
\(\therefore M = \dfrac{4}{3} \pi \,R^3 × \rho\)
Keeping density same,
\(M\propto R^3\)
If we have two spheres of same density of radii in the ratio 1: n, then the masses of the disks will be in the ratio \(1: n^3\).
\(\dfrac{M_1}{M_2} = \left(\dfrac{R_1}{R_2 }\right)^3 = \left(\dfrac{1}{n}\right)^3\)
A \(\dfrac{M}{4}\)
B \(2M\)
C \(\dfrac{M}{2}\)
D \(4M\)
A \(\left(\dfrac{-R}{3},0\right)\)
B \(\left(\dfrac{-R}{2},0\right)\)
C \(\left(0,\dfrac{-R}{3}\right)\)
D \(\left(\dfrac{-R}{3}, \dfrac{-R}{3}\right)\)