Slide 1 : CONCEPT OF CENTRE OF MASS.
TORQUE.
ANGULAR MOMENTUM.
EQUATIONS OF ROTATIONAL MOTION.
MOMENT OF INERTIA.
THEOREMS OF PERPENDICULAR &PARALLEL AXIS
The Concept of a ‘System’,Internal Forces and External Forces : The Concept of a ‘System’,Internal Forces and External Forces A collection of any number of particles interacting with one another are said to form a system.
All the forces exerted by various particles of the system on one another are called internal forces.
Concept of Centre of Mass : Concept of Centre of Mass We may define centre of mass of a body or a system of bodies as a point at which the entire mass of the body/system of bodies, is supposed to be concentrated.
Centre of Mass of a Two Particle System : Centre of Mass of a Two Particle System Consider a system of two particles of mass m1 and m2 located at A and B respectively, where
Let C be the position of centre of mass of the system of two particles. It would lie on the line joining A and B. Let be the position vector of centre of mass.
Centre of Mass of a Two Particle System : Centre of Mass of a Two Particle System To evaluate suppose
are velocities of particles m1 and m2
Respectively at any instant t. Then
Let = external force applied on particle of mass m1
= external force applied on particle of mass m2
Centre of Mass of a Two Particle System : Centre of Mass of a Two Particle System = internal force on m1 due to m2
= internal force on m2 due to m1
Linear momentum of particle m1 is
According to Newton’s second law, the rate of change of linear momentum of this particle is equal to total force acting on this particle,
Centre of Mass of a Two Particle System : Centre of Mass of a Two Particle System which is
Centre of Mass of a Two Particle System : Centre of Mass of a Two Particle System Similarly, the equation of motion of second particle may be written as
Adding (3) and (4), we get
Centre of Mass of a Two Particle System : Centre of Mass of a Two Particle System Where = total external force on the system of two particles.
Using (1),we get
Centre of Mass of a Two Particle System : Centre of Mass of a Two Particle System Multiplying numerator and denominator of left side by (m1 + m2 ),we get
Centre of Mass of a Two Particle System : Centre of Mass of a Two Particle System From (7), we get
Hence position vector of centre of mass of a two particle system is such that the product of total mass of the system and position vector of centre of mass is equal to sum of the products of masses of the two particles and their respective position vectors.
CENTRE OF MASS : CENTRE OF MASS Discussion
Motion of total mass of the system is described under the effect of external forces only.
The internal forces between the particles forming the system cancel out in pairs.
Motion of the system can be studied either by applying Newton’s second law of motion to individual particles of the system or by applying the second law of motion to total mass located at the centre of mass.
CENTRE OF MASS : CENTRE OF MASS Discussion
If the centre of mass of two particles of the system were at the origin Hence centre of mass of a system of two particles lies always on the straight line joining these particles.
Centre of mass of two particle system lies closer to the heavier particle.
CENTRE OF MASS : CENTRE OF MASS Discussion
When the two particles are of equal masses i.e. m1=m2=m say, then
i.e. position vector of centre of mass of two particles of equal masses is the average of the position vectors of the two particles.
Centre of Mass of a System of N Particles : Centre of Mass of a System of N Particles Let O be the origin of a rectangular co-ordinate system XYZ.
Let a system consist of n particles of masses m1,m2,m3,….. mn whose position in the co-ordinate system are given respectively by the position vectors
Let be the position vector of centre of mass of this system of n particles.
Centre of Mass of a System of N Particles : Centre of Mass of a System of N Particles
CENTRE OF MASSIMPORTANT POINTS. : CENTRE OF MASSIMPORTANT POINTS. 1 The position of the centre of mass of a system is independent of the choice of co-ordinate system.
CENTRE OF MASSIMPORTANT POINTS. : CENTRE OF MASSIMPORTANT POINTS. 2. The position of the centre of mass depends on the shape and size of the body and the distribution of its mass. Hence it may lie within or outside the material of the body. e.g, centre of mass of a uniform circular ring lies at the centre of the ring, where there is no mass.
CENTRE OF MASSIMPORTANT POINTS. : CENTRE OF MASSIMPORTANT POINTS. 3. In symmetrical bodies with uniform distribution of mass, the centre of mass coincides with the geometrical centre or centre of symmetry of the body.
CENTRE OF MASSIMPORTANT POINTS : CENTRE OF MASSIMPORTANT POINTS 4. The centre of mass changes its position only under the translatory motion. There is no effect of rotatory motion on centre of mass of a body.
CENTRE OF MASSIMPORTANT POINTS. : CENTRE OF MASSIMPORTANT POINTS. 5. If we know the centre of mass of different parts of the system and their masses, we can get the combined centre of mass by treating various parts as point objects whose masses are concentrated at their respective centers of masses.
Co-ordinates of the Centre of Mass : Co-ordinates of the Centre of Mass Each position vector (of ith particle) can be expressed in terms of its components
in the form:
If x,y,z are the co-ordinates of the centre of mass of the system, we may write
Co-ordinates of the Centre of Mass : Co-ordinates of the Centre of Mass .
Examples of the motion of centre of Mass : Examples of the motion of centre of Mass The system, as a whole, may be changing shapes or orientations due to internal forces, but it will have no effect on the trajectory of centre of mass of isolated system.
Examples of the motion of centre of Mass : Examples of the motion of centre of Mass 1. When a radioactive nucleus initially at rest decays, the fragments fly off in different directions obeying the principles of momentum and energy conservation. As the decay occurs spontaneously, no external forces are involved.
Examples of the motion of centre of Mass : Examples of the motion of centre of Mass 2. We know that moon revolves around the earth in a circular orbit and the earth revolves around the sun in an elliptical orbit.
Examples of the motion of centre of Mass : Examples of the motion of centre of Mass The earth and moon exert gravitational forces of attraction on each other, which are internal forces only. Both the earth and moon are revolving about their common centre of mass, such that they are always on opposite sides of the common centre of mass.
Examples of the motion of centre of Mass : Examples of the motion of centre of Mass 3. When an object of finite size is thrown with some initial velocity at an angle with the horizontal, it follows a parabolic path. The centre of mass of such an object also follows the parabolic path, even if the object were to disintegrate in mid-air.
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Examples of the motion of centre of Mass : Examples of the motion of centre of Mass Example, when a fire cracker projected, explodes in mid air The centre of mass continues to follow parabolic path
Centre of Mass of Rigid Body : Centre of Mass of Rigid Body The centre of mass of a rigid body is defined as a point where the entire mass of the rigid body is supposed to be concentrated. The nature of motion of the rigid body shall remain unaffected, if all the forces acting on the body were applied directly on the centre of mass of the body.
Centre of Mass of Rigid Body : The centre of mass of a rigid body is at a fixed position with respect to the body as a whole. It may or may not lie within the body.
For rigid bodies of regular geometrical shapes and having uniform distribution of mass, the centre of mass is at their geometrical centre., Centre of Mass of Rigid Body
Centre of Mass of Rigid Body : Centre of Mass of Rigid Body
Centre of Mass of Rigid Body : Centre of Mass of Rigid Body
Equations of Rotational Motion : Equations of Rotational Motion .
Moment of a Force or Torque : Moment of a Force or Torque The moment of a force or the torque due to a force gives us the turning effect of the force about the fixed point/axis.
It is measured by the product of magnitude of force and perpendicular distance of the line of action of force from the axis of rotation.
ROTATIONAL M0TION : ROTATIONAL M0TION Moment of force or Torque
= force x perpendicular distance
where ? is smaller angle between
is unit vector along
Moment of a Force or Torque : Moment of a Force or Torque . The direction of is perpendicular to the plane containing and is determined by right handed screw rule.
The S.I. Unit of torque is N-m, which is equivalent to joule.
The dimensions of torque are [M1 L2 T-2].
Expression for Torque in Cartesian Co-ordinates : Expression for Torque in Cartesian Co-ordinates Consider a particle of mass m rotating in plane XY about the origin O. Let P be the position of the particle at any instant, where
Let the rotation occur under the action of a force applied at P, along
Expression for Torque : Expression for Torque In a small time dt, let the particle at P reach Q, where
In vector triangle OPQ,
Or
Small amount of work done in rotating the particle from P to Q is
Expression for Torque : Expression for Torque If Fx,Fy are rectangular components of force
and dx, dy are rectangular components of displacement then
Expression for Torque : Expression for Torque we get
Let the co-ordinates of the point P be (x,y).As is clear from
x = r cos ?
Expression for Torque : Expression for Torque y=r sin ?
Differentiating w.r.t ?, we get
Expression for Torque : Expression for Torque Again, differentiating w.r.t,?,we get
Expression for Torque : Expression for Torque
PHYSICAL MEANING OF TORQUE. : PHYSICAL MEANING OF TORQUE. We define torque as a quantity in rotational motion, which when multiplied by a small angular displacement gives us work done in rotational motion.
This quantity corresponds to force in linear motion, which when multiplied by a small linear displacement gives work done in linear motion . This is the physical meaning of torque.
Expression for Torque in Polar Co-ordinates. : Expression for Torque in Polar Co-ordinates. Suppose the line of action
of force makes an angle
? with X-axis.
? Fx=F cos ?
Fy = F sin ?
If x,y are the co-ordinates
of the point P, where
ROTATIONAL M0TION : ROTATIONAL M0TION Substituting these values , we get
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Expression for Torque in Polar Co-ordinates. : Expression for Torque in Polar Co-ordinates. Let be the angle
which the line of action
of makes with the
position vector
As is clear from Fig.
Putting in , we get
Expression for Torque : Expression for Torque Hence torque due to a force is the product of force and perpendicular distance of line of action of force from the axis of rotation.
MAXIMUM TORQUE : MAXIMUM TORQUE Torque due to a force is maximum, when r is maximum. For example, we can open or close a door easily by applying force near the edge of the door.
That is why a handle/knob is provided near the free edge of the plank of the door.
MAXIMUM TORQUE : MAXIMUM TORQUE Similarly ,to unscrew a nut fitted tightly to a bolt, we need a wrench with long arm.
MAXIMUM TORQUE : MAXIMUM TORQUE When the length of arm(r) is long, the force(F) required to produce a given turning effect(r x F) is smaller.
(ii) Torque will be maximum, when sin ?=max.=+1.Therefore, ?=900,i.e. when force is applied in a direction perpendicular to For example, it is easiest to open or close a door by applying force at the edge of the plank in a direction perpendicular to the plank of the door.
NEXT LECTURE : NEXT LECTURE TOPICS:
ANGULAR MOMENTUM
MOMENT OF INERTIA
THEOREMS OF PERPENDICULAR &PARALLEL AXIS
Slide 54 : For all your Physics Problems
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