Mod 10 - Module 10 ROTATIONAL KINEMATICS Reference Book...

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Module 10: ROTATIONAL KINEMATICS Reference Book: Properties of Matter – Dr. Tofazzal Hossain MOMENT OF INERTIA: According to Newton’s first law of motion, “a body must continue in its state of rest or of uniform motion along a straight line, unless acted upon by an external force.” This inertness or inability of a body to change by itself its position of rest or of uniform motion is called inertia. Exactly in the same manner, in case of rotational motion, also we find that, a body free to rotate about an axis opposes any change desired to be produced in its state of rest or of rotation, showing that it possesses inertia for this type of motion. It is the rotational inertia of the body, which is called moment of inertia . In case of linear motion, the inertia of a body depends on wholly on its mass. In case of rotational motion, the inertia depends not only on its mass of the body but also on the effective distance of its particles from the axis of rotation. So, two bodies of the same mass may possess different moments of inertia. A rigid body can be considered as a system of particles in which the relative positions of the particles do not change. The moment of inertia of a single particle I can be expressed as where m = the mass of the particle, and r = the shortest distance from the axis of rotation to the particle. KINETIC ENERGY OF ROTATION AND MOMENT OF INERTIA: The kinetic energy of system of particles of masses m 1 , m 2 …….m n is defined as K.E = 2 1 - - - - - - - - - - 2 1 2 1 2 1 2 2 3 3 2 2 2 2 1 1 n n v m v m v m v m + + + + = 2 1 2 1 mv n i = Substituting v = ϖ r [where ϖ is the angular speed of the rotating body], we have K.E = ( 29 2 1 2 2 1 2 1 2 1 ϖ ϖ = = = n i i i i i n i r m r m 1
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in which ϖ is the same for all particles. The quantity in the parentheses tells us how the mass of the rotating body is distributed about its axis of rotation. This quantity is known as moment of inertia. We may now write, 2 1 i n i i r m I = = dm r I = 2 Thus, K. E = 2 2 1 ϖ I TORQUE: The ability of a force F to rotate a body depends not only on the magnitude of its tangential component t F but also on just how far from the axis of rotation at O it is applied. To include both these factors, we define a quantity called torque τ as ] sin F ][ r [ φ = τ , ……………….. [1] Two equivalent ways of computing the torque are ] 3 ...[ , ......... F r ] F ][ sin r [ ] 2 ..[ , ......... rF ] sin F ][ r [ t = φ = τ = φ = τ where r is the perpendicular distance between the axis of rotation at O and an extended line running through the vector F . This extended line is called the line of action of F and r is called the moment arm of F . TABLE 1: ANALOGS IN ROTATIONAL AND LINEAR MOTION LINEAR MOTION ROTATION ABOUT A FIXED AXIS Displacement X Angular displacement θ velocity dt dx v = Angular velocity dt d θ = ϖ Acceleration dt dv a = Angular acceleration dt d ϖ = α Mass [translational inertia] M Rotational inertia I Force Ma F = Torque α = τ I Work = Fdx W Work θ τ = d W Kinetic energy 2 Mv 2 1 Kinetic energy 2 I 2 1 ϖ Power Fv P =
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