Problems on Centre of Mass

Many problems can be solved if the mass of the object is believed to be situated in a single location. When the proper position is determined, equations of forces and motion work the same way as when mass is spread out. The centre of mass is the term for this unique site. Its location is determined by an object or a group of objects whose centre of mass must be computed. It is usually the centroid for uniform shapes.

We do not need to think about the dynamics of individual atoms of the system when examining the dynamics of the motion of the system as a whole. Rather, we must concentrate on the dynamics of a single point that corresponds to that system. The motion of this one-of-a-kind point is similar to that of a single particle whose mass is equivalent to the total of all the system’s distinct particles. The result of all forces exerted by surrounding bodies on all particles of the system is also exerted directly on that particle. The centre of mass of the system of particles is this point.

Differences between Centre of Mass and Centre of Gravity

To explain a body’s translational motion, the centre of mass is where the body’s entire mass can be assumed to be concentrated. On the other hand, the centre of gravity is the place at which the sum of the gravitational forces acting on all of the body’s particles acts.

These two points are in the same location for many things. However, only when the gravitational field is homogenous across an item are they the same. The centre of gravity, for example, coincides with the centre of mass in a homogeneous gravitational field such as that of the earth on a tiny body.

Formulas to Solve Problems on the Centre of Mass

1. Pointed objects:

• For x-axis-

Xcom =   (∑i=0 n mi xi)/M

• For y-axis-

ycom =   (∑i=0 n mi yi)/M

• For z-axis-

Zcom =  ( ∑i=0 n mi zi)/M

2. To identify the centre of mass of an extended item such as a pole, we use a different approach. Then we examine a differential mass and its position and integrate it across the full length.

• Xcom =  1/M (∫xdm)

• Ycom =  1/M (∫ydm)

• Zcom =  1/M (∫zdm)

Here,

Xcom, ycom and zcom = Centre of Mass along x, y and z-axis, respectively.

M                      = Total mass of the system

n                           = Number of objects

mi                           = Mass of the ith object

xi                             = Distance from the x-axis of the ith object.

Locating Centre of Mass

The centre of mass of a body is determined experimentally by using gravity forces on the body and assuming that the centre of mass and centre of gravity are equal in the parallel gravity field near Earth’s surface.

A body possessing a symmetry axis and stable density will have its centre of mass on this axis. As a result, the centre of mass of a circular cylinder with constant density is located on the cylinder’s axis.

The centre of mass, the place where gravity can be considered to act, is frequently referred to as the centre of gravity in the setting of a completely uniform gravitational field. The motion of the complete system, as opposed to the motion of its pieces, can be studied by determining the system’s centre of mass.

The position of the object’s centre of mass is fixed in the case of a rigid body. The position of the centre of mass is a point in space among them that does not necessarily coincide with the position of any particular mass in the event of a loose distribution of masses in free space, such as released from a gun.

Problems on Centre of Mass

Problem 1:

Two objects of masses 10 kg and 4 kg are con­nected by a spring of tiny mass and placed on a frictionless horizontal table. An impulse gives a velocity of 14 m/s to the heavier object in the direction of the lighter object. What is the velocity of the centre of mass?

Solution:

Right after collision

vc= m1v1 + m2v2/m1 + m2

vc= (10 x 14 + 4 x 0)/(10+4)

vc= 10 m/s

Problem 2:

Due to mutual gravitational attraction, two bodies with different masses of 2 kg and 4 kg are travelling at velocities of 2 m/s and 10 m/s towards one other. So, what is the velocity of the centre of mass?

Solution:

Let’s take the direction of 4kg Q + Ve

VCM = ((m1 x v1) + (m2 v2))/(m1 + m2)

           (2 x (-2) + 4 x 10)/(2+4) 

           36/6 = 6

The velocity of the centre of mass is 6m/s in the direction of the 4kg block.

Conclusion

The centre of mass is a position inside or outside of an object that acts as though the object’s entire mass is concentrated there. All the points in rigid body dynamics maintain their mutual distance and move as if one point is following the other. The centre of mass simplifies the problem and makes it easier to analyse rotational, linear, spinning, periodic, and most other motions. We employ two stages of simplification by using the centre of mass for many rigid body problems. First, we will look at each object’s centre of mass. The centre of mass for the entire system is then calculated.