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Angular Kinetics II KIN 335 Spring 2005 What you should know • Definition of Moment of Inertia • Moment of inertia and movement performance. • Definition of Angular Momentum • Angular impulse-momentum relationship • Newton’s law of angular motions. • Comparison between linear and angular kinetics Moment of inertia • Inertia – The property of an object that resists ______ in motion – Linear inertia Î quantified as an object’s _______ – More massive object Î Difficult to speed up, slow down, or change the direction – Angular inertia Î quantified as an object’s ________________ – More angular inertia Î Difficult to speed up or slow down the rotation or change the axis of rotation of an object Moment of inertia • Moment of inertia – The quantity that describes angular inertia Ia = _________ ¾ Ia = moment of inertia about axis a ¾ mi = mass of particle i ¾ ri = radius (distance) from particle i to axis of rotation a – Unit : kg·m2 Ia = __________ ¾ mi = mass of particle i ¾ ka = radius of gyration about axis a ¾ radius of gyration : a length measurement that represents how far from the axis of rotation all of the object’s mass must be concentrated to create the same resistance to change in angular motion as the object had in its original shape. Characteristics of Moment of inertia • It’s function of _____ and __________________ • More sensitive to the radius. Why? • Objects may have many different moments of inertia because an object may rotate about many different axes of rotation. • A human’s moment of inertia about any axis is ________ Îmore than one value depending on the body postures. • Figure skater, divers, gymnasts, skaters, long jumper, ….. Corking a wooden bat? • The illegal practice of cutting out a cylinder of wood from the barrel of the bat and replacing it with lighter material. The effect is to decrease the bat's mass and moment of inertia about an axis through its handle - bat can be accelerated more effectively. • tuck vs. layout position of a diver or gymnast – Athlete rotates about an axis through his/her center of gravity. Tuck concentrates body mass closer to axis; Layout body mass concentrated away from axis; changes in moment of inertia ultimately affect body's rate of rotation (see conservation of angular momentum discussions) • changes in position of a runner's leg during swing phase – Hip represents an important axis of rotation for the entire leg as it is swung forward. Runner naturally tucks leg (flexes knee) early in swing the effect is to concentrate the mass of the leg more closely to the hip axis, thereby reducing the moment of inertia of the leg about an axis through the hip. Advantage - less resistance to angular acceleration of the leg during swing. Angular Momentum • Momentum – Quantity of motion ¾ “…Chicago Bulls Gaining Momentum…” – Linear motion Î L= ____ (unit : kg·m/s) – Angular motion Î H= I·ω (unit : _______) ¾ Quantity of __________________ ¾ Function of moment of inertia and angular velocity ¾ Vector quantity • Angular momentum of human body – The sum of the angular momentum of all the body segments gives an approximation of the angular momentum of the entire body. Newton’s 1st law • For linear motion – Every body continues in its state of rest, or of uniform motion in a straight line, unless it is compelled to change that state by force impressed upon it. Î Law of inertia • For angular motion – The angular momentum of an object remains ________ unless a net external torque is exerted on it. (ΣT)⋅(∆t) =∆(I⋅ω)Î 0 =∆(I⋅ω) Î _______________ – Controlling _________________ Î speed up or slow down angular velocity. – Conservation of angular momentum – figure skater, diver, gymnasts,…. • Backward one and a half diving • Layout – Tuck – Layout Newton’s 2nd law • For linear motion – If a net external force is exerted on an object, the object will accelerate in the direction of the net external force, and its acceleration will be proportional to the net external force and inversely proportional to its mass. Σ F = m⋅a • For angular motion – If a net external torque is exerted on an object, the object will accelerate angularly in the direction of the net external torque, and its angular acceleration will be directly proportional to the net external torque and inversely proportional to its moment of inertia. _________________ Impulse-Momentum Relationship • Rewriting of Newton’s 2nd Law ΣF = m⋅a Î ΣF = m⋅∆v/∆t Î(ΣF)⋅(∆t) = m⋅∆v – Impulse = (ΣF)⋅(∆t) – Change of momentum = m⋅∆v • Angular motion – Angular Impulse = ____________ – Change of angular momentum = ________ = I⋅ ∆ω (ΣT)⋅(∆t) =∆(I⋅ω)=I⋅∆ω ÎΣT =I⋅(∆ω/∆t)Î ΣT = I⋅α • Force = the rate of change of _______ momentum • Torque = the rate of change of _______ momentum Newton’s 3rd law • For linear motion – For every force exerted by one body on another, the other body exerts an equal force back on the first body but in the opposite direction – Key note : the forces are ______ but _______ direction • For angular motion – For every torque exerted by one body on another, the other body exerts an equal torque back on the first body but in the opposite direction – Key note : the torques acting on the two objects have ____________________ but opposite direction Newton’s 3rd law • Internal forces • Internal muscle torques • Tightrope walker Linear vs. Angular Kinetics Liner Angualr Quantity Symbol SI Unit Quantity Inertia (mass) m Kg Moment of Inertia Force F N Torque T = F ·r Linear Momentum L = m·v Angular Momentum H = I·ω Impulse (ΣF)(∆t) N·s Angular Impulse Symbol SI Unit I= Nm·s Q1. At the instant of takeoff, a 60-kg diver’s angular momentum about his transverse axis is 20 kg·m2/s. His radius of gyration about the transverse axis is 1.0 m at this instant. During the dive, the diver tucks and reduces his radius of gyration about the transverse axis is to 0.5 m. a. At takeoff, what is the diver’s angular velocity about the transverse axis? b. After the diver tucks, what is his angular velocity about the transverse axis? Q. What average amount of force must be applied by the elbow flexors inserting at an average perpendicular distance of 1.5 cm from the axis of rotation at the elbow over a period of 0.3 seconds to stop the motion of the 3.5 kg arm swing with an angular velocity of 5 rad/s when k = 20 cm ? k = radius of gyration.