Chapter 7- System Of Particles And Rotational Motion | class 11th | revision notes physics

Systems of Particles and Rotational Motion Class 11 Notes Physics Chapter 7

We know that force, energy and power are associated with rotational motion. These and other aspects of rotational motion are covered in this chapter. We shall see that all important aspects of rotational motion either have already been defined for linear motion.

Centre of Mass

Centre of mass is a very special point. The concept of centre of mass of a system enables us, in describing the overall motion of the system by replacing the system by an equivalent single point, where the entire mass of the body or system is supposed to be concentrated.

Now, suppose we have a system of n particles of masses m₁, m₂, m₃ …m_n respectively along a straight line at distances x₁, x₂, x₃ …x_n from the origin respectively. Then the centre of mass of the system is given by

X_cm=m1x1+m2x2+…mnxnm1+m2+…mnm1x1+m2x2+…mnxnm1+m2+…mn

X_cm=m1x1+m2x2+…mnxnMm1x1+m2x2+…mnxnM

where M is the total mass of the system.

Centre of Gravity

The centre of gravity is that point of the body, where the whole weight of the body is supposed to be concentrated. We may define the centre of gravity of a body as that point where the total gravitational torque acting on the body is zero.

Now, suppose we have a system of n particles of masses w₁, w₂, w₃ …w_n respectively along a straight line at distances x₁, x₂, x₃ …x_n from the origin respectively. Then the centre of gravity of the system is given by

Xcd=w1x1+w2x2+…wnxnw1+w2+…wnXcd=w1x1+w2x2+…wnxnw1+w2+…wn

Motion of centre of mass

We can write equation of Centre of Mass for a system of particles as follow.

MR_cm=m1r1+m2r2+m3r3+…+mnrnm1r1+m2r2+m3r3+…+mnrn

Differentiating the two sides of the equation with respect to time, we get

MdRdt=m1dr1dt+m2dr2dt+…+mndrndtMdRdt=m1dr1dt+m2dr2dt+…+mndrndt

The rate of change of position is velocity. So we can replace dR/dt with v_cm where v_cm is the velocity of the centre of mass.

Mv_cm=m1v1+m2v2+…+mnvnm1v1+m2v2+…+mnvn

Mdvcmdt=m1dv1dt+m2dv2dt+…+mndvndtMdvcmdt=m1dv1dt+m2dv2dt+…+mndvndt

Change in velocity is acceleration, so we get

Ma_cm=m1a1+m2a2+…+mnanm1a1+m2a2+…+mnan

where a₁, a₂, …..a_n are the acceleration of first, second, and ….n^th particle respectively and a_cm is the acceleration of the centre of mass of the system of particles.

Linear Momentum of a System of Particles

The linear momentum of a particle is defined as

P=mv

and according to Newton’s second law,

F=dPdtF=dPdt

i.e. the rate of change of linear momentum of a particle is equal to the net force acting on the object. Using equation of motion of center of mass we can write

Mvcm=n∑i=1miviMvcm=∑i=1nmivi

n∑i=1Pi=n∑i=1mivi∑i=1nPi=∑i=1nmivi

L.H.S. is the summation of linear momentum of n particles of the system, which is equal to product of the total mass of the system and velocity of the center of mass of the system.

P=MvcmP=Mvcm

Differentiating the above equation w.r.t. time, we get

dPdt=Mdvcmdt=macm=FextdPdt=Mdvcmdt=macm=Fext

Now, if the net external force on the system is zero, the linear momentum of the system, is conserved and the centre of mass will move with constant velocity.

Rigid Body

A rigid body is a collection of large number of particles, moving in a constrained manner. The constraint is that separation between any two particles of the system does not change.

Motion of Rigid Body

(i). Translation

If any line drawn on the rigid body remains parallel to itself throughout the motion, then the body is said to be in pure translation. In pure translation, all the particles of the body have equal velocity and acceleration at all instants and they cover equal distance and displacement in equal time.

(ii). Rotation

If any line drawn on the rigid body does not remain parallel to itself throughout its motion, then the body is said to be rotating. For example, the ceiling fan, bicycle wheel or a football rolling on ground.

(iii). Axis of Rotation

An imaginary line drawn perpendicular to the plane of motion of different points of the body and passing through the stationary point is called the axis of rotation.

(iv). Angle of Rotation (θ)

When the object rotates, its configuration changes, the angle by which any line drawn on the object rotates during the change in configuration is the angle of rotation.

While the body rotates, every point of the body moves in a circle, whose centre lies on axis of rotation, and every point experiences the same angular displacement during a particular time interval.

(v). Angular Velocity (ω)

The rate of rotation is measured by angular velocity. The angular velocity is defined as

ω=dθdtω=dθdt

The unit of the angular velocity is rad/s.

(v). Angular Acceleration (α)

The angular acceleration is defined as

α=dωdtα=dωdt

The unit of the angular acceleration is rad/s².

Dot Product and Cross Product

(i). The Dot Product of Two Vectors

The scalar product or dot product of any two vectors →AA→ and →BB→, denoted as →AA→ . →BB→ (read as →AA→ dot →BB→) is defined as

→AA→ . →BB→ = AB cosθ

Where A & B are magnitudes of vectors →AA→ and →BB→ respectively and θ is the smaller angle between them. Dot product is called scalar product as A, B and cosθ are scalars. Both vectors have a direction but their scalar product does not have a direction.

Properties

• Dot product is commutativeA . B = B . A
• Dot product is distributiveA . (B + C) = A . B + A . C
• Dot product of a vector with itself gives square of its magnitude A . A = AA cosθ = A
• A . (λB) = λ(A . B)where λ is a real number
• ˆii^ . ˆjj^ = ˆjj^ . ˆkk^ = ˆkk^ . ˆii^ = 0
• ˆii^ . ˆii^ = ˆjj^ . ˆjj^ = ˆkk^ . ˆkk^ = 1

(ii). The Cross Product of Two Vectors

The vector product or cross product of any two vectors →AA→ and →BB→, denoted as →AA→ x →BB→ (read as →AA→ cross →BB→) is defined as

→AA→ x →BB→ = AB sinθ

Where A & B are magnitudes of vectors →AA→ and →BB→ respectively and θ is the smaller angle between them. Cross product is called vector product as A, B and sinθ are scalars. Both vectors have a direction and their vector product has a same direction.

Properties

• The vector product is do not have Commutative Property.A×B = – (B×A)
• The following property holds true in case of vector multiplication(kA)×B= k(A×B) =A×(kB)
• If the given vectors are collinear thenA×B= 0
• Following the above property, We can say that the vector multiplication of a vector with itself would beA×A= |A||A|sin0 ˆnn^ = 0
• Also in terms of unit vector notationˆi׈i=ˆj׈j=ˆk׈k=0i^×i^=j^×j^=k^×k^=0
• From the above discussion it also follows thatˆi׈j=ˆk=−ˆj׈ii^×j^=k^=−j^×i^ˆj׈k=ˆi=−ˆk׈jj^×k^=i^=−k^×j^ˆk׈i=ˆj=−ˆi׈kk^×i^=j^=−i^×k^

Relation between Angular Acceleration and Linear Acceleration

Angular Acceleration is given by

α=dωdtα=dωdt …..(1)

Linear Acceleration is given by

a=dvdta=dvdt …..(2)

From eq (1) and (2),

a=d(r×ω)dta=d(r×ω)dt

a=r×dωdta=r×dωdt

a=r×αa=r×α

Relation between Angular Velocity and Linear Velocity

Let, any rigid body is rotating about any rotational axis with angular velocity (ω). If distance of a particle at a perpendicular be r from the fixed axis and linear velocity be v of any particle of thye rigid system then relation between them is given by

→v=→r×→ωv→=r→×ω→

Torque

The tendency of a force to rotate the body to which it is applied is called torque. The torque, specified with regard to the axis of rotation, is equal to the magnitude of the component of the force vector lying in the plane perpendicular to the axis, multiplied by the shortest distance between the axis and the direction of the force component.

→τ=→r×→Fτ→=r→×F→

Relation between Torque and Angular Velocity

→L=→r×→PL→=r→×P→

Differentiating w.r.t. t, we get

d→Ldt=ddt(→r×→P)dL→dt=ddt(r→×P→)

d→Ldt=→r×d→Pdt+d→rdt×→PdL→dt=r→×dP→dt+dr→dt×P→

d→Ldt=→r×→F+→v×m→vdL→dt=r→×F→+v→×mv→

d→Ldt=→r×→F+(→v×→v)dL→dt=r→×F→+(v→×v→)

d→Ldt=→r×→F+0dL→dt=r→×F→+0 … [∴→v×→v=0∴v→×v→=0]

d→Ldt=τdL→dt=τ

Moment of Inertia

When any object rotate about any axis then it has tendency to resist its motion, this tendency of resistance is called moment of inertia. It is denoted by I and it’s SI unit is kg/m².

Moment of Inertia of any object is defined as product of mass of that object and square of perpendicular distance of rotational axis.

I=MR2I=MR2

Theorems of Perpendicular and Parallel Axes

(i). Theorems of Perpendicular Axes

It states that the moment of inertia of a planar body (lamina) about an axis perpendicular to its plane is equal to the sum of its moments of inertia about two perpendicular axes concurrent with perpendicular axis and lying in the plane of the body.

Consider a lamina in x-y plane as shown in the figure and suppose that it consists of n particles of masses m₁, m₂, m₃ …m_n at perpendicular distances r₁, r₂, r₃ …r_n respectively from the axis OZ and suppose the corresponding perpendicular distances of these particles from the OY are x₁, x₂, x₃ …x_n and from the axis OX are y₁, y₂, y₃ …y_n respectively.

Let I_x, I_y, and I_z be the moment of inertia of the lamina about axes OX, OY and OZ respectively. Now,

Ix=n∑i=1miy2iIx=∑i=1nmiyi2 …..(1)

Similarly Iy=n∑i=1mix2iIy=∑i=1nmixi2 …..(2)

and Iz=n∑i=1mir2iIz=∑i=1nmiri2 …..(3)

Adding equations (i) and (ii), we get

Ix+Iy=n∑i=1mi(y2i+x2i)Ix+Iy=∑i=1nmi(yi2+xi2)

From the figure, we can see

r2i=x2i+y2iri2=xi2+yi2

Ix+Iy=n∑i=1mir2i=IzIx+Iy=∑i=1nmiri2=Iz

Iz=Ix+IyIz=Ix+Iy

(ii). Theorems of Parallel Axes

It states that the moment of inertia of a body about any axis is equal to the sum of the moment of inertia of the body about a parallel axis passing through its centre of mass and the product of its mass and the square of the distance between the two parallel axes.

Consider a rigid body, as shown in the figure and suppose we know the moment of inertia of the body about axis AB, and want to find the moment of inertia of the body about EF which is at a perpendicular distance d from AB.

Suppose the rigid body is made up of n particles of masses m₁, m₂, m₃ … m_n at perpendicular distances r₁,r₂, r₃, …r_n respectively from the axis AB passing through the centre of mass C of the body. If ri is the perpendicular distance of the particle from the axes AB ; then

I=n∑i=1mir2iI=∑i=1nmiri2

I=n∑i=1mi(R+a)2I=∑i=1nmi(R+a)2

I=n∑i=1mi(R2+a2+2Ra)I=∑i=1nmi(R2+a2+2Ra)

I=n∑i=1miR2+n∑i=1mia2+n∑i=1mi2RaI=∑i=1nmiR2+∑i=1nmia2+∑i=1nmi2Ra

I=n∑i=1miR2+n∑i=1mia2+0I=∑i=1nmiR2+∑i=1nmia2+0

I=Icm+n∑i=1miR2I=Icm+∑i=1nmiR2

Rolling motion

Rolling Motion of a body is a combination of both translational and rotational motion of a round shaped body placed on a surface. When a body is set in rolling motion, every particle of body has two velocities – one due to its rotational motion and the other due to its translational motion, and the resulting effect is the vector sum of both velocities at all particles.

Rolling Motion is classified in two categories – Pure Rolling and Rolling with Sliding. Pure rolling is a case when there is no relative motion at point of contact of rolling body and the surface; and body is considered to be rotating about this point of contact frame.

Rotational Kinetic Energy

If the mass of i^th particle is m_i and its speed is v_i, its kinetic energy is

Ki=12miv2iKi=12mivi2

Ki=12mir2iω2Ki=12miri2ω2

Ki=12(mir2i)ω2Ki=12(miri2)ω2

Ki=12Iω2Ki=12Iω2