APT ACADEMIC SOLUTIONS : APT ACADEMIC SOLUTIONS
Slide 2 : www.aptacads.com
Session : www.aptacads.com Session System of Particles
Session Objectives : www.aptacads.com Session Objectives Centre of Mass: An intuitive introduction?
Centre of Mass as an imaginary/mathematical point?
Centre of Mass of discrete system.
4. Centre of Mass of continuous system.
5. Motion of Centre of Mass.
Center of Mass – an Intuitive definition : www.aptacads.com So far we have considered the motion of POINT PARTICLES FINITE OBJECTS
can move as a whole
(translational motion)
and also
rotate about the “Centre of Mass” The “Centre of Mass” is that point where if we apply a force, the object will not rotate. What happens depends on where we apply the force Center of Mass – an Intuitive definition
Systems of Particles : www.aptacads.com Systems of Particles Centre of Mass: -
A special point that moves as if
(a) the total mass were concentrated there,
Systems of Particles : www.aptacads.com Systems of Particles Made of lots of little particles, the atoms with various forces among them forces
A Mathematical detour : www.aptacads.com A Mathematical detour Force on ith particle Force on the entire body Can we achieve a similar form for our system ?
A Mathematical detour : www.aptacads.com Claim Total force on all particles is same as the external force. Why?? Newton’s Third Law rescues us from the menacing sum! A Mathematical detour cancel out when summed over;
Slide 10 : www.aptacads.com For our purpose, mass remains constant. If we define a vector, This looks similar to a single particle equation!Doesn’t it??
Slide 11 : www.aptacads.com Acceleration of center of mass Imaginary/mathematical point
associated with the body. F Force acting on a point mass M whose position coordinate is given by This point is a kind of average ‘r’ in which different ri’s have weights proportional to their masses. Center of mass
Discrete Systems and their Centre of Mass : www.aptacads.com Discrete Systems and their Centre of Mass which in component version may be expressed as:
Discrete Systems and their Centre of Mass : www.aptacads.com Discrete Systems and their Centre of Mass
Slide 14 : www.aptacads.com How do we find the
Centre of Mass?
Centre of Mass (1D) : www.aptacads.com Centre of Mass (1D) M = m1 + m2 M xcm = m1 x1 + m2 x2 In general moment of M = moment of individual masses
Centre of Mass (2D) : www.aptacads.com Centre of Mass (2D) For a collection of masses in 2D rcm = ixcm +jycm So in a solid body we can find the CM by finding xcm and ycm
Slide 17 : www.aptacads.com Xcm = 16/15 = 1.07 m ycm = 20/15 = 1.33 m
Illustrative Example : www.aptacads.com Where is the center of mass of the arrangement of particles shown below? Illustrative Example
Solution : www.aptacads.com Solution
Illustrative Example : www.aptacads.com Illustrative Example Three particles of masses 1 kg, 2 kg and 3kg are placed at the three corners of a right angled triangle of sides 9cm, 12cm and 15cm as shown in the figure. Locate the position of center of mass of the system.
Solution : www.aptacads.com Solution Take the axes as shown in the figure. The coordinates of three particles are as follows:
Mass x-coordinate y-coordinate
1 kg 0 0
2 kg 12 0
3 kg 0 9 Hence, the coordinates of center of mass of the system are Thus, the center of mass is 4 cm to the right and 4.5cm above the 1 kg particle.
Centre of mass: Continuous Object : www.aptacads.com Centre of mass: Continuous Object
Continuous Systems and their Centre of Mass : www.aptacads.com Continuous Systems and their Centre of Mass
Illustrative Example : www.aptacads.com Illustrative Example A rod of length L and mass M is lying as shown in the figure. Find the position of centre of mass with the axis as shown in the figure.
Solution : www.aptacads.com Solution Consider an infinitesimally small element of length dx at a distance x from origin. Then, and yCM = 0
zCM = 0
Continuous Systems and their Centre of Mass : www.aptacads.com Continuous Systems and their Centre of Mass (1) The center of mass, of a symmetrical body of uniform density, lies at the geometrical centre. For instance, the center of mass of a uniform sphere is at its center, the center of a uniform cylinder is at the midpoint of its axis. (2) If the object can be divided into several parts, treat each of the part as a particle located at its own center of mass. (3)Choose the appropriate axis. If your system is a body with a line of symmetry, the center of mass will lie on this line. Choose one of axis along this line. Tip: To locate the center of mass of a system, make use of the following strategies :
Illustrative Problem : www.aptacads.com Illustrative Problem Find the location of center of mass of a uniform semicircular plate of radius R (see fig.).
Solution : www.aptacads.com Solution Choose X-Y axis (with the origin at O) By symmetry , center of mass must lie on Y-axis. Thus, Consider a thin strip of length 2x and, thickness dy as shown. Then, mass dm of the strip
Illustrative Example : www.aptacads.com Illustrative Example Two identical uniform rods of length are joined to form a L shaped frame as shown in the figure. Locate the center of mass of the frame.
Solution : www.aptacads.com Solution Let mass of each rod be m. Choose co-ordinate axis as shown.
The individual cm of each rod respective geometrical centers.
Replace rod OB with a point mass m at its geometrical center P.
Replace rod AO with a point mass m at its geometrical center P'.
The system is now reduced to a two point mass system.
Example : www.aptacads.com Example The fig. below shows a circular plate of radius R from which a disc of radius R/2 has been removed. Find the position of center of mass of this object.
Solution : www.aptacads.com Solution Choose X-Y axes as shown . ycm = 0 Symmetry
Solution : www.aptacads.com Solution Note: Centre of mass of a body may lie outside the body. (Why?)
Illustrative Question : www.aptacads.com Illustrative Question
Solution : www.aptacads.com Solution xCM = a/2
Hence, the centre of mass lies at the point of contact.
Motion of CM/System of particles : www.aptacads.com Motion of CM/System of particles System of n particles position vector of the centre of mass, velocity of this point. acceleration of this point.
Motion of CM/System of particles : www.aptacads.com Motion of CM/System of particles Momentum of center of mass
Motion of CM/System of particles : www.aptacads.com Motion of CM/System of particles Similarly ? For a system of particles, the dynamics of the Centre of Mass obeys Newton 2nd Law.
Motion of CM/System of particles : www.aptacads.com Motion of CM/System of particles Note: If external forces acting on a system add up to zero, then using simple kinematics we can make the following observations:
(1) The velocity of center of mass will remain constant.
(2) If the center of mass of an isolated system is at rest initially, it will continue to be at rest even if the particles of the system move.
Illustrative Example : www.aptacads.com Illustrative Example Two spherical balls having mass m1 = 2kg and m2 = 4kg are placed on a line joining their centres at a distance 2 m apart. They start moving under the influence of mutual gravitational force at time t = 0.
(a) Calculate the acceleration of their centre of mass.
(b) Calculate the initial position of their centre of mass.
(c) Calculate the final position of their centre of mass.
Solution : www.aptacads.com Solution
Solution : www.aptacads.com Solution YCM = 0
ZCM = 0
Illustrative Problem : www.aptacads.com Illustrative Problem Two balls having mass m1 = 2 kg and m2 = 4 kg are placed 2 m apart on the line joining their centres. At time t = 0 both balls are given a velocity of 2 m/s each such that they move towards each other. Assuming gravitational effect and friction to be negligible, find the
(a) velocity of their centre of mass
(b) initial position of their centre of mass
(c) position of their centre of mass at the instant the collide
Solution : www.aptacads.com Solution (c) Time taken to collide: \ Distance traveled by their cm in this time: sCM = vCM·t0
Illustrative Problem : www.aptacads.com Illustrative Problem A man of mass m1 is standing on a platform of mass m2 kept on a smooth surface. If the man moves a distance d with respect to the platform, find the displacement of the plat form with respect to ground.
Solution : www.aptacads.com Solution Let the displacement of the platform be S towards left (w.r.t ground frame) =>external horizontal force=0
& xcm doesn’t change