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Physics: Rotational Motion

Topics are arranged as per SYLLABI published by Maharashtra State Board of Secondary and Higher Secondary Education.

In rotation of a rigid body about a fixed axis, every particle of the body moves in a circle, which lies in a plane perpendicular to the axis and has its centre on the axis.
The following table summarizes the rotational motion analogous of the basic concepts for linear or translatory motion.

Linear Motion Rotational Motion
Distance/displacement (s)t Angle or angular displacement Rotational Motion
Linear velocity Rotational Motion Angular velocity Rotational Motion
Linear acceleration Rotational Motion Angular acceleration Rotational Motion
Mass (m) Moment on inertia (I)
Linear momentum, p = mv Angular momentum, L = Iω
Force, F = ma Torque, Rotational Motion
Also, Force Rotational Motion Also, torque, Rotational Motion
Translational KE, Rotational Motion Rotational KE, Rotational Motion
Work done, W = Fs Work done, Rotational Motion
Power, P = Fv Power, Rotational Motion
Linear momentum of a system is conserved when no external force acts on the system. Angular momentum of a system is conserved when no external torque acts on the system.
Equation of translator motion Equation of rotational motion

It is a quantity expressed by, a body’s tendency to resist angular acceleration, which is the sum of the products of the mass of each particle in the body with the square of its distance from the axis of rotation.

  • It is analogues of mass in linear motion.
  • Independent of the magnitude of the angular velocity.
  • It is a characteristic of the rigid body and the axis about which it rotates.
The moment of inertia of a body about a given axis of rotation is defined as the sum of the products of the masses of the particles of the body and the squares of their respective distances from the axis of rotation.
Let’s assume body is made up of n discrete particles of masses m1, m2,m3 .... mn situated at respective distance of r1, r2, r3 ...... rn from the axis of rotation, the moment of inertial of the body is given as : Rotational Motion
For a rigid body having a continuous and uniform distribution of mass, the moment of inertia is : Rotational Motion
Its dimensions are Rotational Motion. SI unit is Kg - m2

Moment Of Inertia Of Some Object:

Body Axis of Rotation Moment of Inertia (I)
Uniform circular ring of mass M and radius R About an axis passing through centre and perpendicular to its plane. MR2
About an diameter 0.5 MR2
About an tangent in its own plane 1.5 MR2
About a tangent perpendicular to its plane 2MR2
Uniform Circular Disc of mass M and radius R. About an axis passing through centre and perp. To its plane. 0.5MR2
About an diameter 0.25 MR2
About an tangent in its own plane 1.25 MR2
About a tangent perpendicular to its plane 1.5MR2
Solid sphere of radius R and mass M. About an diameter. 0.4MR2
About a tangential axis 1.4 MR2
Spherical shell of radius R and mass M About an diameter. 0.67MR2
About a tangential axis 1.67 MR2
Long thin rod of Length L About an axis through C.G. and perpendicular to rod. Rotational Motion
About an axis through one end and perpendicular to rod. Rotational Motion

Consider a rigid object rotating about a fixed axis at a certain angular velocity. Since every particle in the object is moving, every particle has kinetic energy. To find the total kinetic energy related to the rotation of the body, the sum of the kinetic energy of every particle due to the rotational motion is taken.
Let us assume there are n particles, K.E. is represented as Rotational Motion

Let us consider a body like a sphere, disc or a wheel rolling on a plane surface without slipping. A rolling body undergoes translational motion about the centre of mass and rotational motion about an axis passing through the centre of mass.
Total K.E. of a rolling body = Transitional K.E. + Rotational K.E.
Rotational Motion Where m = mass of the body
R = Radius of the body
v = Translational speed of the body
I = M.I. of the body about an axis of rotation passing through the centre of mass.
K = Radius of gyration of the body

Radius of gyration of a given body about a given axis of rotation is the normal distance of a point from the axis, where if whole mass of the body is placed then its moment of inertia will be exactly same as it has with its actual distribution of mass. Rotational Motion or Rotational Motion
S.I. Unit of radius of gyration is metre.
The value of radius of gyration shall depend upon shape and size of the body, position and configuration of the axis of rotation and also on distribution of mass of the body w.r.t. the axis of rotation.

By comparing linear motion and rotational motion formulae we find that role played by moment of inertia in rotational motion is same as the role played by mass in translational motion.
In translational motion, mass of a body represents its inertia, ie reluctance to undergo a change in its state of rest or of uniform translational motion. Obviously, moment of inertial also represents inertia of a body in rotational motion (i.e. its reluctance to undergo a change in its state of rotation).Moment of inertia is also called rotational inertia.

  • The turning effect of a force applied at a point on a rigid body about the axis of rotation.
  • Torque (Rotational Motion) is a pseudo vector directed along the axis of rotation. Its direction is given by the right hand fist rule as in the case with angular momentum. Unit of torque is N-m.
  • It’s dimensional formula is Rotational Motion
  • If a body executes rotator motion, it is not necessary that a torque acts on it.
  • Torque is necessary for producing angular acceleration only.

Work done by a force F, acting on a particle of a body rotating about a fixed axis: the particle describes a circular path with centre C on the axis : arc Rotational Motion gives the displacement of the particle.
Let’s assume mass m is moving with constant velocity along a line parallel to the x-axis, away form the origin. It’s angular momentum wrt origin remains constant. L =mvy

Both theorems are related to moment of inertia.

Parallel Axis Theorems Perpendicular Axis Theorem
Moment of inertia of a body about a given axis I is equal to the sum of moment of inertia of the body about a parallel axis passing through its centre of mass ICM and the product of mass of body (M) and square of normal distance d between two axes. The sum of moment of inertia of a plane laminar body about two mutually perpendicular axes lying in its plane is equal to the moment of inertia about an axis passing through the point of intersection of these two axes and perpendicular to the plane of laminar body.
If Ix and Iy be moment of inertia of the body about two perpendicular axes in its own plane and Iz be the moment of inertia about an axis passing through point O and perpendicular to the plane of lamina, then Iz = Ix + Iy
Applicable for any type of rigid body whether it is a two-dimensional or three-dimensional. Applicable for laminar type / planer or two-dimensional bodies only.
The point of intersection of the three axes (x,y and z) may be any point on the plane of body (it may even lie outside the body). This point may or may not be the center of mass of the body.

Let us assume a small object of uniform density rolls up a curved surface with an initial velocity v. It reaches up to a maximum height of Rotational Motion with respect to the initial position.
Now we need to find, which is this object?
In this scenario, Loss of Kinetic Energy = Gain in Potential Energy
Rotational Motion ; put given value of h in this equation.
Rotational Motion
Rotational Motion
This represents moment of inertia for Disc (with axis of rotation passing through C.G and perpendicular to its plane).

Body Axis Figure I
Thin circular ring, radius R Perpendicular to plane at centre MR2
Thin circular ring, radius R Diameter Rotational Motion
Thin rod, length L Perpendicular to rod, at mid point Rotational Motion
Circular disc, radius R Perpendicular to disc at centre Rotational Motion
Circular disc, radius R Diameter Rotational Motion
Hollow cylinder, radius R Axis of cylinder MR2
Solid cylinder, radius R Axis of cylinder Rotational Motion
Solid sphere, radius R Diameter Rotational Motion

The moment of linear momentum of a given body about an axis of rotation is called as its angular momentum. If p be the linear momentum of a particle and r is its position vector from the point of rotation then Rotational Motion.
When a body is projected at an angle with the horizontal in the uniform gravitational field of the earth, the angular momentum of the body about the point of projection increases, as it proceeds along the path.
Conservation of angular momentum and its applications: According to the law of conservation of angular momentum, if no external unbalanced torque is acting on system , then total vector sum of angular momentum of different particles of the system remains constant. Rotational Motion
If no external torque acts on a body, the angular momentum of the body remains constant.

  • Every point on a rotating body has an angular velocity and a linear velocity.
  • Objects farther from the axis of rotation will move faster.
  • The rotational inertia of an object depends not only on its mass distribution but also the location of the axis of rotation..
  • The moment of inertial of a cube about it’s body diagonal is minimum.
  • During pure rotational motion, the axis of rotation passes through centre of mass.
  • Axis of rotation is always perpendicular to plane of rotation.
  • If radius of gyration of the body is equal to its radius, then the body is a ring.
  • Theorem of perpendicular axis is applicable only to thin lamina like sheet, disc, ring, etc. While parallel axes theorem is applicable to all type of bodies.
  • Total K.E. is maximum for ring and minimum for solid sphere for same mass and moving with same speed.
  • If metallic disc is melted and is moulded in the form of a sphere, then M.I. will decrease.
  • Angular momentum is an axial vector, as it points along the axis of rotation.
  • If there had been a single propeller in a helicopter, then due to non-conservation of angular momentum, the helicopter would turn in the reverse direction.
  • When a person standing on a rotating platform raises his arms, M.I. of the system increases. Hence ω decreases.
  • Angular Impulse, is the product of torque and time for which the torque acts.
  • Axis of rotation of the rolling body is parallel to the plane on which it rolls.
  • Acceleration of a rolling body does not depend on its mass.
  • Theorem of perpendicular axis is applicable only to thin lamina like sheet, disc, ring etc. whose thickness is very small as compared to other dimensions (length, breadth or radius).
  • Graph between M.I. (I) and Radius of Gyration (K) is parabola [I=MK2]
  • Graph between Rotational K.E. (ER) and angular velocity (ω) is a parabola Rotational Motion.
  • Graph between Angular momentum (L) and angular velocity is a straight line. L = Iω
  • Graph between Angular momentum (L) and K.E. of rotation (E) will be parabola.