1!Phys 2101 Gabriela González 2!Our definition for torque produced by a given force F about a point P as τ = F r sinθ# allows us to add up torques only when the forces are all in the same plane. We now define torque as a vector, so we can add up the torques produced by arbitrary forces about a point P: In this way, the torque is also a vector, and has magnitude and direction. €  τ = r × F2!3!The angular momentum of a particle with respect to a point is defined as : The angular momentum of a system of particles is the sum of the angular momenta of each particle. €   = r × p 4!For a single particle, F=ma can also be written as… Torque and angular momentum about any point (but the same point!) €  F net=d p dt €  τ = r × F = r ×d p dt=ddt r × p ( )−d r dt× p  τ =d  dt3!5!No forces, no linear momentum: conservation of linear momentum. No torques, no angular momentum: conservation of angular momentum. total linear momentum total angular momentum €  F net=d P dt €  τ net=d L dt6! €  F net=d P dt=d M v com( )dt= Md v comdt= M a com € d L dt= τ net= I α = Id ω dt=d I ω ( )dt €  P = M v com L = I ω4!7! €  L = I ω 8!A 4.0 kg particle moves in an xy plane. At the instant when the particle's position and velocity are r= (2.0 i+ 4.0 j ) m and v= -4.0 j m/s, the force on the particle is F = -3.0 i N. At this instant, determine: (a) the particle's angular momentum about the origin, (b) the particle's angular momentum about the point x = 0, y = 4.0 m, (c) the torque acting on the particle about the origin, and (d) the torque acting on the particle about the point x = 0, y = 4.0 m. r v F x y5!9!A 30 kg child stands on the edge of a stationary merry-go-round of mass 100 kg and radius 2.0 m. The rotational inertia of the merry-go-round about its axis of rotation is 150 kg·m2. The child catches a ball of mass 1.0 kg thrown by a friend. Just before the ball is caught, it has a horizontal velocity of 12 m/s that makes an angle of 37° with a line tangent to the outer edge of the merry-go-round, as shown in the overhead view of the figure. What is the angular speed of the merry-go-round just after the ball is caught? 10!A 1.0 g bullet is fired into a 0.50 kg block that is mounted on the end of a 0.60 m nonuniform rod of mass 0.50 kg. The block–rod–bullet system then rotates about a fixed axis at point A. The rotational inertia of the rod alone about A is 0.060 kg·m2. Assume the block is small enough to treat as a particle on the end of the rod. (a) What is the rotational inertia of the block–rod–bullet system about point A? (b) If the angular speed of the system about A just after the bullet's impact is 4.5 rad/s, what is the speed of the bullet just before the