The angular speed of a circular hoop is constant and is given by w. The hoop has a mass M and radius a and is rolling without slipping along the horizontal axis in the xy plane. Calculate the total angular momentum of the hoop about the origin when the hoop's center is at a distance d= 3a from the origin. (а) Мао? (b) ZMao (c) 2Ma@ (d) 3Mao
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- A disk of radius R and thickness t has a mass density that increases from the center outward, given by p = po (r/R), where r is the distance from the axis of the disk. What is the moment of inertia about the disk axis in terms of M and R? O MR 2MR O sMP 3MR 5 O 2MR 3 5MR 3This problem illustrates the two contributions to the kinetic energy of an extended object: rotational kinetic energy and translational kinetic energy. You are to find the total kinetic energy Ktotal total of a dumbbell of mass m when it is rotating with angular speed ω and its center of mass is moving translationally with speed v. (Figure 1)Denote the dumbbell's moment of inertia about its center of mass by Icm. Note that if you approximate the spheres as point masses of mass m/2 each located a distance r from the center and ignore the moment of inertia of the connecting rod, then the moment of inertia of the dumbbell is given by Icm=mr2, but this fact will not be necessary for this problem. Find the total kinetic energy Ktot of the dumbbell. Express your answer in terms of m, v, Icm, and ω.A uniform solid ball rolls without slipping down a plane which is inclined at 31° to the horizontal. If the ball has a radius r=0.4m, a mass m=0.1 kg and starts from rest, find: a) the speed of the ball after it travels 2m down the incline. b) at this point, what is the angular momentum of the ball? c) If the coefficient of friction between the ball and the plane is 0.25, what is the maximum angle of inclination that allows the ball to roll without slipping?
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