These notes for CBSE Class 11 Physics, Chapter 7, cover the System of Particles and Rotational Motion. Key concepts include the Centre of Mass, defined as the point where the entire mass of a system can be considered concentrated. The notes explain its position for systems of particles and two-particle systems, noting it's independent of the body's state for rigid bodies and may not lie within the object. Rotational motion is defined, along with rigid bodies. Moment of Inertia (I) is introduced as the rotational equivalent of mass, dependent on the axis and mass distribution. The Radius of Gyration (K) is also explained. The Parallel Axes Theorem and Perpendicular Axes Theorem are detailed for calculating moment of inertia. Finally, the chapter introduces Torque, Angular Momentum, and the principle of Conservation of Angular Momentum, crucial for understanding rotational dynamics and aiding in exam revision.
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Centre of mass of a system is the point that behaves as whole mass of the system is
concentrated at it and all external forces are acting on it.
For rigid bodies, centre of mass is independent of the state of the body i.e., whether it is in rest
or in accelerated motion centre of mass will rermain same.
If a system consists of n particles of masses m1, m2
, m3 ,… mn having position vectors rl, r2,
r3,… rn. then position vector of centre of mass of
Choosing O as origin of the coordinate axis.
(ii) Position of centre of mass from m2 = (m1d) / m1 + m2
iii) If position vectors of particles of masses m1 and m2 are r1 and r2respectively, then
(iv) If in a two particle system, particles of masses m1 and m2 moving with velocities v1 and
v2respectively, then velocity the centre of mass
(v) If accelerations of the particles are a1, and a1respectively, then acceleration of the centre of
mass
(vi) Centre of mass of an isolated system has a constant velocity.
(vii) It means isolated system will remain at rest if it is initially rest or will move with a same velocity if it is in motion initially.
(viii) The position of centre of mass depends upon the shape, size and distribution of the mass of the body.
(ix) The centre of mass of an object need not to lie with in the object.
(x) In symmetrical bodies having homogeneous distribution mass the centre of mass coincides with the geometrical centre the body.
(xi) The position of centre of mass of an object changes translatory motion but remains unchanged in rotatory motion,
A rigid body performs a pure translational motion, if each particle the body undergoes the same displacement in the same direction in given interval of time.
A rigid body performs a pure rotational motion, if each particle of the body moves in a circle, and the centre of all the circles lie on a straight line called the axes of rotation.
If the relative distance between the particles of a system do not changes on applying force, then it called a rigtd body. General motion of a rigid body consists of both the translational motion and the rotational motion.
The inertia of rotational motion is called moment of inertia. It is denoted by L. Moment of inertia is the property of an object by virtue of which it opposes any change in its state of rotation about an axis.
The moment of inertia of a body about a given axis is equal to the sum of the products of the masses of its constituent particles and the square of their respective distances from the axis of rotation.
Its unit is kg.m2 and its dimensional formula is [ML2m].
The moment of inertia of a body depends upon
• position of the axis of rotation
• orientation of the axis of rotation
• shape and size of the body
• distribution of mass of the body about the axis of rotation.
The physical significance of the moment of inertia is same in rotational motion as the mass in linear motion.
The root mean square distance of its constituent particles from the axis of rotation is called the radius of gyration of a body.
It is denoted by K.
Radius of gyration
The product of the mass of the body (M) and square of its radius gyration (K) gives the same moment of inertia of the body about rotational axis.
Therefore, moment of inertia I = MK2 ? K = ?1/M
The moment of inertia of any object about any arbitrary axes is equal to the sum of moment of
inertia about a parallel axis passing through the centre of mass and the product of mass of the
body and the square of the perpendicular distance between
the two axes.
Mathematically I = ICM + Mr2
where I is the moment of inertia about the arbitrary axis, IcM is moment of inertia about the
parallel axis through the centre of mass, M is the total mass of the object and r is the
perpendicular distance between the axis.
The moment of inertia of any two dimensional body about an axis perpendicular to its plane
(Iz) is equal to the sum of moments of inertia of the body about two mutually perpendicular
axes lying in its own plane and intersecting
each other at a point, where the perpendicular axis passes through it.
Mathematically Iz = Ix + Iy
where Ix and Iy are the moments of inertia of plane lamina about perpendicular axes X and Y respectively which lie in the plane lamina an intersect each other.
Theorem of parallel axes is applicable for any type of rigid body whether it is a two
dimensional or three dimensional, while the theorem of perpendicular is applicable for
laminar type or two I dimensional bodies only.
(i) ? = ?0 + ?t
(ii) ? = ?0t + 1/2 ?t2
(iii) ?2 = ?0
2 + 2??
where ? is displacement in rotational motion, ?0 is initial velocity, omega; is final velocity and a is acceleration.
Torque or moment of a force about the axis of rotation
? = r x F = rF sin? n It is a vector quantity.
If the nature of the force is to rotate the object clockwise, then torque is called negative and if rotate the object anticlockwise, then it is called positive.
Its SI unit is ‘newton-metre’ and its dimension is [ML2T-2].
In rotational motion, torque, ? = I?
where a is angular acceleration and 1is moment of inertia.
The moment of linear momentum is called angular momentum.
It is denoted by L.
Angular momentum, L = I ? = mvr
In vector form, L = I ? = r x mv
Its unit is ‘joule-second’ and its dimensional formula is [ML2T-1].
Torque, ? = dL/dt
If the external torque acting on a system is zero, then its angular momentum remains conserved.
If ?ext 0, then L = I(?) = constant ? I1?1== I2?2
Total effect of a torque applied on a rotating body in a given time is called angular impulse. Angular impulse is equal to total change in angular momentum of the system in given time.
Total effect of a torque applied on a rotating body in a given time is called angular impulse.
Angular impulse is equal to total change in angular momentum of the system in given time.
The Centre of Mass of a system is a point where the entire mass of the system can be considered to be concentrated, and all external forces act on it.
Moment of Inertia (I) is the inertia of rotational motion, opposing any change in the state of rotation about an axis. It depends on mass distribution and the axis of rotation.
The Radius of Gyration (K) is the root mean square distance of the constituent particles of a body from the axis of rotation.
The Parallel Axes Theorem states that the moment of inertia about any axis is equal to the moment of inertia about a parallel axis through the centre of mass plus the product of the mass and the square of the distance between the axes (I = ICM + Mr^2).
The Perpendicular Axes Theorem applies to 2D bodies, stating that the moment of inertia about an axis perpendicular to the plane (Iz) is the sum of the moments of inertia about two perpendicular axes in the plane (Iz = Ix + Iy).
Torque (?) is the rotational equivalent of force, causing or tending to cause rotation. It is calculated as the cross product of the position vector and the force vector (? = r x F).
Angular momentum of a system is conserved if the net external torque acting on the system is zero.
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