Center of Gravity Depends on Shape:
Is a unique location in space that is the average position of the system’s mass. This place is called the center of mass,After developing kinematics of rotational motion, the rotational motion dynamics were developed. In this the parallel concepts are applied to linear dynamics which contains some unique concepts and equations. The main important concept in rotational dynamics is a torque. It also involves the study of work, power, and kinetic energy like in linear motion, moment of inertia, center of mass and equations for a description of the combination of linear and angular motion.watch Center of Gravity Depends on Shape to learn more.
Center of Gravity:
The terms center of mass or gravity are used in a constant field of gravity which shows a specific point in the system. It is used to describe a response of the system towards the external force and torque. If we see in on dimension plane then it is similar to seesaw at a pivot point. It contains the total mass of the body at one point.watch Center of Gravity Depends on Shape to learn more.
This clock uses the principles of center of mass to keep balance on a finger.
Center of Mass Definition:
A point where whole of the body’s mass can be assumed to be located or concentrated is called center of mass.
- The point can be real or imaginary, for example, in the case of a hollow or empty box the mass is physically not located at the center of mass point. This mass is supposed to be located at the center of mass in order to simplify calculations.
- The motion of the center of mass characterizes the motion of the entire object. The center of mass may or may not be the same to the geometric center if a rigid body is considered. It is considered as a reference point for many other calculations of mechanics.
The center of mass of a rigid body is a point whose position is fixed with respect to the body as a whole. The point may or may not lie in the body. The position of the center of mass of a rigid body depends on:
- Shape of the body
- Distribution of mass in the body
Center of Mass Equation:
Consider two particles A and B of masses m1 and m2, respectively. Take the line joining A and B as the X-axis. Let the coordinates of the particles at time ‘t’ be x1 and x2. Suppose no external force acts on the system. The particles A and B, however, exert forces on each other and these particles accelerate along the line joining them. Suppose the particles are initially at rest and the force between them is attractive.
The center of mass at time t is situated at X = m1x1+m2x2m1+m2
As time passes, x1 and x2 change and hence, X changes and the center of mass moves along the X-axis. Velocity of the center of mass at time t is,
Vcm→ = dxdt = m1v1→+m2v2→m1+m2
The acceleration of the center of mass is
acm→ = dVcm→dt = m1a1→+m2a2→m1+m2………(1)
Suppose the magnitude of the forces between the particles is F. As the only force acting on A towards B, is F, its acceleration is
a1 =F⃗ m1
The force on B is (-F) and hence,
a2 = −F⃗ m2
Substituting this in equation (1)
acm→ = m1(F⃗ m1)+m2(−F⃗ m2)m1+m2 = 0
This means, the velocity of the center of mass does not change with time. But as we assumed initially, the particles are at rest. Thus,v1→ = v2→ = 0 then Vcm has to be zero. Hence, the center of mass remains fixed and does not change with time.
Thus, if no external force acts on a two-particle system and its center of mass is at rest, initially it remains fixed even when the particles individually move and accelerate.
If the external forces do not add up to zero, the center of mass is accelerated and is given by,
If we have a single particle of mass m on which a force Fext→ acts, its acceleration would be the same as Fext→m. Thus, the motion of the center of mass of a system is identical to the motion of a single particle of mass equal to the mass of given system, acted upon by the same external forces that act on the system.
The formula for center of mass in three dimension
R represents the center of mass
‘M’ the mass of the body, m (i) = masses, r(i) = positions
This can be customized for x and y coordinates also. Then the equations would be,
Xcm = ∑m(i)xiM
Ycm = ∑m(i)yiM
Subject : Physics
Lesson : Gravitation
Topic:Center of Gravity Depends on Shape
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