explain the centre of mass for n- numbers for particles
Answers
Answer:
.The centre of mass (CoM) is the point relative to the system of particles in an object. This is that point of the system of particles that embarks the average position of the system in relation to the mass of the object. At the centre of mass, the weighted mass gives a sum equal to zero.
Answer:
Centre of mass of a body or system of a particle is defined as, a point at which whole of the mass of the body or all the masses of a system of particle appeared to be concentrated.
When we are studying the dynamics of the motion of the system of a particle as a whole, then we need not bother about the dynamics of individual particles of the system. But only focus on the dynamic of a unique point corresponding to that system.
We can extend the formula to a system of particles.The equation can be applied individually to each axis,
X_{com}X
com
= \frac{∑_{i=0}^n~ m_i x_i }{M}
M
∑
i=0
n
m
i
x
i
Y_{com}Y
com
= \frac{∑_{i=0}^n~ m_i y_i}{M}
M
∑
i=0
n
m
i
y
i
Z_{com}Z
com
= \frac{∑_{i=0}^n~m_i z_i }{M}
M
∑
i=0
n
m
i
z
i
The above formula can be used if we have point objects. But we have to take a different approach if we have to find the center of mass of an extended object like a rod. We have to consider a differential mass and its position and then integrate it over the entire length.
X_{com}X
com
= \frac{∫~x ~dm}{M}
M
∫ x dm
Y_{com}Y
com
= \frac{∫~y ~dm}{M}
M
∫ y dm
Z_{com}Z
com
= \frac{∫~z ~dm}{M}
M
∫ z dm
Suppose we have a rod as shown in the figure and we have to find its center of mass.
Centre of mass of rod
Let the total mass of the rod be MM and the density is uniform. Also we assume that the breadth of the rod is negligible i.e. the center of mass lies on the x-axis. We consider a small dx at a distance from the origin. Therefore,
dmdm = \frac{M}{l}~ dx
l
M
dx
Using the equation for finding center of mass,
X_{com}X
com
= \frac{∫~\frac{M}{l}~ dx ~.x}{M}
M
∫
l
M
dx .x
X_{com}X
com
= \frac{∫~ dx ~.x}{l}
l
∫ dx .x
Integrating it from 00 to ll we get,
X_{com}X
com
= \frac{l}{2}
2
l
Using the above method we can find the center of mass for any geometrical shape. You can try out for a semi circular ring or a triangle. So if a force is applied to that extended object it can be assumed to act through the center of mass and hence it can be converted to a point mass.