Rotational Kinetic Energy An object rotating about some axis with an angular speed, , has rotational kinetic energy even though it may not have any translational kinetic energy Each particle has a kinetic energy of K i = 1/2 m i v i 2 Since the tangential velocity depends on the distance, r, from the axis of rotation, we can substitute v i =  i r

2 Fig 10.6

3 Rotational Kinetic Energy, cont The total rotational kinetic energy of the rigid object is the sum of the energies of all its particles Where I is called the moment of inertia

4 Rotational Kinetic Energy, final There is an analogy between the kinetic energies associated with linear motion (K = 1/2 mv 2 ) and the kinetic energy associated with rotational motion (K R = 1/2 I  2 ) Rotational kinetic energy is not a new type of energy, the form is different because it is applied to a rotating object The units of rotational kinetic energy are Joules (J)

5 Moment of Inertia The definition of moment of inertia is The dimensions of moment of inertia are ML 2 and its SI units are kg. m 2 We can calculate the moment of inertia of an object more easily by assuming it is divided into many small volume elements, each of mass  m i

6 Moment of Inertia, cont We can rewrite the expression for I in terms of  m With the small volume segment assumption, If  is constant, the integral can be evaluated with known geometry, otherwise its variation with position must be known

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9 Moment of Inertia of a Uniform Solid Cylinder Divide the cylinder into concentric shells with radius r, thickness dr and length L Then for I Fig 10.8

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Torque The moment arm, d, is the perpendicular distance from the axis of rotation to a line drawn along the direction of the force d = r sin 

Definition of Torque Torque, , is the tendency of a force to rotate an object about some axis Torque is a vector  = r F sin  = F d F is the force  is the angle between the force and the horizontal (the line from the axis to the position of the force) d is the moment arm (or lever arm)

26 Torque, cont. The horizontal component of the force (F cos  ) has no tendency to produce a rotation Torque will have direction If the turning tendency of the force is counterclockwise, the torque will be positive If the turning tendency is clockwise, the torque will be negative

27 Torque Unit The SI unit of torque is N. m Although torque is a force multiplied by a distance, it is very different from work and energy The units for torque are reported in N. m and not changed to Joules

28 Torque as a Vector Product Torque is the vector product or cross product of two other vectors

29 Vector Product, General Given any two vectors, and The vector product is defined as a third vector, whose magnitude is The direction of C is given by the right-hand rule Fig 10.13

30 Properties of Vector Product The vector product is not commutative If is parallel (  = 0 o or 180 o ) to then This means that If is perpendicular to then

31 Vector Products of Unit Vectors The signs are interchangeable For example,

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34 Net Torque on an object The force F 1 will tend to cause a counterclockwise rotation about O The force F 2 will tend to cause a clockwise rotation about O  net      F 1 d 1 – F 2 d 2

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Force vs. Torque Forces can cause a change in linear motion, which is described by Newton’s Second Law F = Ma. Torque can cause a change in rotational motion, which is described by the equation 

39 The Rigid Object In Equilibrium The net external force must be equal zero The net external torque about any axis must be equal zero

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42 Fig 10.16(b) & (c)

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Rotational motion of a rigid object under a net torque The magnitude of the torque produced by a force around the center of the circle is  = F t r = (ma t ) r The tangential acceleration is related to the angular acceleration  =  (ma t ) r =  (mr  ) r =  (mr 2 )  Since mr 2 is the moment of inertia of the particle,  = I  The torque is directly proportional to the angular acceleration and the constant of proportionality is the moment of inertia

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54 Fig 10.18(a) & (b)

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60 Work in Rotational Motion Find the work done by a force on the object as it rotates through an infinitesimal distance ds = r d  The radial component of the force does no work because it is perpendicular to the displacement Fig 10.19

61 Work in Rotational Motion, cont Work is also related to rotational kinetic energy: This is the same mathematical form as the work-kinetic energy theorem for translation If an object is both rotating and translating, W =  K +  K R

62 Power in Rotational Motion The rate at which work is being done in a time interval dt is the power This is analogous to P = Fv in a linear system

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