Chapter 10 Dynamics of Rotational Motion PDF

Chapter 10 Dynamics of Rotational Motion Lecture 2 Sec. 10.5-10.6 PowerPoint® Lectures for General Physics I and Engineering I – Physics...

Chapter 10 Dynamics of Rotational Motion Lecture 2 Sec. 10.5-10.6 PowerPoint® Lectures for General Physics I and Engineering I – Physics Department, UAEU Customized by: © Physics Department, UAEU, 2014. Learning Goals Define the angular momentum of a particle or a rigid body. Learn how the angular momentum changes with time. Use the angular momentum conservation law to describe rotation. Customized by: © Physics Department, UAEU, 2014. 10.5 Angular Momentum A particle with mass 𝑚, moving with velocity 𝑣 at a position vector 𝑟 relative to the origin, has an angular momentum: 𝐿 =𝑟×𝑝 Its magnitude is: 𝐿 = 𝑚𝑣𝑟 𝑠𝑖𝑛∅ The right-hand rule shows that its direction is along the + z axis (perpendicular to the plane formed by 𝑟 and 𝑣 ). 𝑚2 SI units: 𝑘𝑔. 𝑠 Customized by: © Physics Department, UAEU, 2014.  Newton's 2nd law analogy The time rate of change of angular momentum is the total torque on it. 𝑑𝐿 𝑑𝑟 𝑑𝑣 = × 𝑚𝑣 + 𝑟 × 𝑚 𝑑𝑡 𝑑𝑡 𝑑𝑡 𝑑𝐿 = 0 + 𝑟 × 𝑚𝑎 𝑑𝑡 𝑑𝐿 =𝑟×𝐹 𝑑𝑡 𝑑𝐿 =𝜏 Analog to Newton’s second law 𝑑𝑡 Customized by: © Physics Department, UAEU, 2014.  Angular Momentum of a Rigid Body The angular momentum of 𝑖th particle of rigid body is: 𝐿𝑖 = 𝑚𝑖 𝑟𝑖 𝜔 𝑟𝑖 = 𝑚𝑖 𝑟𝑖 2 𝜔 𝐿= 𝐿𝑖 = 𝑚𝑖 𝑟𝑖 2 𝜔 The angular momentum of a rotating body is: 𝐿 = 𝐼𝜔 Customized by: © Physics Department, UAEU, 2014. Direction of angular momentum Customized by: © Physics Department, UAEU, 2014. Example 10.9 A turbine fan in a jet engine has a moment of inertia of 2.5 𝑘𝑔. 𝑚2 about its axis of rotation. As the turbine starts up, its angular velocity is given by 𝜔𝑧 = 40𝑡 2. a) Find the fan’s angular momentum as a function of time, and find its value at t = 3.0 s. b) Find the net torque on the fan as a function of time, and find its value at t = 3.0 s. Customized by: © Physics Department, UAEU, 2014. 10.6 Conservation of Angular Momentum When the net external torque acting on a system is zero, the total angular momentum of the system is constant (conserved). 𝑑𝐿 =0 𝑑𝑡 𝐿 is constant 𝐿1 = 𝐿2 𝐼1 𝜔1𝑧 = 𝐼2 𝜔2𝑧 Zero net external torque Customized by: © Physics Department, UAEU, 2014. Example 10.10 A physics professor stands at the center of a frictionless turntable with arms outstretched and a 5.0-kg dumbbell in each hand. He is rotating about the vertical axis, making one revolution in 2.0 𝑠. Find his angular velocity if he pulls the dumbbells into his stomach. His moment of inertia (without dumbbells) is 3.0 kg. 𝑚2 with arms outstretched and 2.2 kg. 𝑚2 with his hands at his stomach. The dumbbells are 1.0 m from the axis initially and 0.20 m at the end. Customized by: © Physics Department, UAEU, 2014. Example 10.12 A door 1.00 m wide, of mass 15 kg, can rotate freely about a vertical axis through its hinges. A bullet with a mass of 10 g and a speed of 400 m/s strikes the center of the door, in a direction perpendicular to the plane of the door, and embeds 1 itself there. 𝐼𝑑𝑜𝑜𝑟 = 𝑀𝑑2 3 a) Find the door’s angular speed. b) Is the kinetic energy conserved. Customized by: © Physics Department, UAEU, 2014.  Class Activities Customized by: © Physics Department, UAEU, 2014. Q1 A spinning figure skater pulls his arms in as he rotates on the ice. As he pulls his arms in, what happens to his angular momentum L and kinetic energy K? a) L and K both increase. b) L stays the same; K increases. c) L increases; K stays the same. d) L and K both stay the same. Customized by: © Physics Department, UAEU, 2014. Q2 A ball is attached to one end of a string, and you hold the other end and whirl the ball in a circle around your hand. If the ball moves at constant speed, a) Is its linear momentum 𝑝 constant? b) Is its angular momentum 𝐿 constant? Customized by: © Physics Department, UAEU, 2014. Q3 The tree bodies in the figure below have the same mass and radius. They roles down to the bottom. Which body reaches the bottom first? a) The sphere. b) The solid cylinder. c) The hollow cylinder. d) All reach the bottom at the same time. Customized by: © Physics Department, UAEU, 2014.  Summary The angular momentum a particle is: 𝐿 =𝑟×𝑝 The net external torque on a system is: 𝑑𝐿 𝜏= 𝑑𝑡 If the net external torque is zero the angular momentum is conserved, 𝐼1 𝜔1 = 𝐼2 𝜔2 Customized by: © Physics Department, UAEU, 2014.