A Classical Approach of Newtonian Mechanics

8 ROTATIONAL MOTION 8.5 Centre of mass X Now that we have defined the vector product of two vectors, let us find a use for this ...

8 ROTATIONAL MOTION 8.5 Centre of mass M  v P  r^ O   Figure 71: Rigid rotation. If the object under consideration is con ...

8 ROTATIONAL MOTION 8.5 Centre of mass where V is the volume of the object. According to Eq. (8.20), the centre of mass of a bod ...

8 ROTATIONAL MOTION 8.5 Centre of mass a ̧ ̧ ̧ dx dy dz ± ± ̧ ̧ ̧ z dz ̧ dz ̧ dy ̧ dy ̧ ̧ ̧ z dz y z geometric centre a x a h zc ...

8 ROTATIONAL MOTION 8.6 Moment of inertia (^) ̧ (^0). (8.24) 2 i 0 i 1 h 0 a (^2) (1 − z/h) (^2) dz Performing the y-integrals, ...

8 ROTATIONAL MOTION 8.6 Moment of inertia X i X × × × ̧ ̧ ̧ Here, the quantity I is termed the moment of inertia of the object, ...

8 ROTATIONAL MOTION 8.6 Moment of inertia ̧ ̧ = M b 2 ring Figure 75: The moment of inertia of a ring about a perpendicular symm ...

8 ROTATIONAL MOTION 8.6 Moment of inertia x Figure 76: The perpendicular axis theorem. ̧ ̧ x^2 dx dy (^) (^) (8.38) Iy = M (^) ̧ ...

8 ROTATIONAL MOTION 8.6 Moment of inertia ̧ ̧ ̧ ̧ ̧ ̧ y ring x z Figure 77: The moment of inertia of a ring about a coplanar sym ...

8 ROTATIONAL MOTION 8.6 Moment of inertia ̧ ̧ ̧+d M. The above equation can be expanded to give IJ = M [(x^2 + y^2 ) − 2 d x + d ...

8 ROTATIONAL MOTION 8.6 Moment of inertia q ̧ 2π σ dσ ̧b 2π σ dσ h i 0 ring = M b (^2) ring axis M b I = 2 M b 2 original axis n ...

8 ROTATIONAL MOTION 8.7 Torque The moment of inertia of a solid cylinder of mass M and radius b about the cylindrical axis is I ...

8 ROTATIONAL MOTION 8.7 Torque Figure 79: A rotating bicycle wheel. the tangential motion of the wheel, yields M ̇v = f sin θ, ( ...

8 ROTATIONAL MOTION 8.7 Torque O Figure 80: Definition of the length of the level arm, l. velocity, then torque must be analogou ...

8 ROTATIONAL MOTION 8.7 Torque Since force is a vector quantity, it stands to reason that torque must also be a vector quantity. ...

8 ROTATIONAL MOTION 8.7 Torque f torque  P r O pivot Figure 81: Torque about a fixed point. between the directions of r and f. ...

8 ROTATIONAL MOTION 8.8 Power and work 8.8 Power and work Consider a mass m attached to the end of a light rod of length l whose ...

8 ROTATIONAL MOTION 8.8 Power and work In other words, the rate at which a torque performs work on the object upon which it acts ...

8 ROTATIONAL MOTION 8.9 Translational motion versus rotational motion ̧ ̧ (^) τ ≡ r × f = I α Work W = f·dr Work W = τ·dφ Power ...

8 ROTATIONAL MOTION 8.10 The physics of baseball pivot J’ J’ bat b centre of mass h l v ball J J  Figure 82: A schematic baseb ...