a) By modelling the ballerina with her arms and legs tucked in as a uniform eylinder of mass 60kg and radius 13em, find her angular momentum after she has tucked her arms in. The moment of inertia of a uniform eylinder of mass M and radius R is given by I = ¿MR b) Hence, estimate her moment of inertia with respect to her axis of rotation when her arms and legs were outstretched.
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- A yo-yo is made of two uniform disks, each of mass M and radius R, which are glued to a smaller central axle of negligible mass and radius 0.5R (see figure). A string is wrapped tightly around the axle. The yo-yo is then released from rest and allowed to drop downwards, as the string unwinds without slipping from the central axle. a) Find the moment of inertia, I, of the yo-yo with respect to an axis through the common centers of the disks, in terms of the mass and radius. b) Calculate the magnitude of the linear velocity V, in meters per second, of the yo-yo after it has fallen a distance 0.46 m.Four point masses, m¡ = 2 kg, m2 = 3 kg, m3 = 4 kg and m4 5 kg are on the x axis as shown in the figure below. mi m2 m3 1.0 2.0 3.0 4.0 x (m) Find the moment of inertia of the above system about an axis of rotation that is perpendicular to the page and passes through x = 4 m. Find the moment of inertia of the above system about an axis of rotation that is perpendicular to the page and passes through x = 3 m.A yo-yo is made of two uniform disks, each of mass M and radius R, which are glued to a smaller central axle of negligible mass and radius 0.5R (see figure). A string is wrapped tightly around the axle. The yo-yo is then released from rest and allowed to drop downwards, as the string unwinds without slipping from the central axle. Part (a) Find the moment of inertia, I, of the yo-yo with respect to an axis through the common centers of the disks, in terms of the mass and radius. Part (b) Find the linear speed V of the yo-yo, after it has descended a distance H. Part (c) Calculate the magnitude of the linear velocity V, in meters per second, of the yo-yo after it has fallen a distance 0.39 m
- Modern wind turbines generate electricity from wind power. The large, massive blades have a large moment of inertia and carry a great amount of angular momentum when rotating. A wind turbine has a total of 3 blades. Each blade has a mass of m = 5500 kg distributed uniformly along its length and extends a distance r = 44 m from the center of rotation. The turbine rotates with a frequency of f = 12 rpm. a)Calculate the total moment of inertia of the wind turbine about its axis, in units of kilogram meters squared. b)Enter an expression for the angular momentum of the wind turbine, in terms of the defined quantities. c)Calculate the angular momentum of the wind turbine, in units of kilogram meters squared per second.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?An oversized yo-yo is made from two identical solid disks each of mass M = 2.10 kg and radius R = 10.0 cm. The two disks are joined by a solid cylinder of radius r = 4.00 cm and mass m = 1.00 kg as in the figure below. Take the center of the cylinder as the axis of the system. R. M (a) What is the moment of inertia of the system? Give a symbolic answer. (Use any variable or symbol stated above as necessary.) moment of inertia %3D
- two uniform solid spheres,A and B have the same mass.Each spbere,A and B has their own axis of rotation,the radius of sohere B is twice of sphere A.Which one of the following is true? (give I = 2/5mr^2) a)the moment of inertianof A IS 1/2 B b)the moment of inertianof A is 5/2 of B c) the moment of inertia of A is 1/4 B d)the two spheres have equal moments of inertiaof Sides a and b has a mass M. Four point-like balls, each of rnass m = each corner of the plate as indicated in the figure. What is the moment of inertia of this object if the axis of M are glued to rotation is through the end of one sidt, like a door, as indicated in the figure by the blue fine? (A) Isoor=M (a² + b²) (B) Isoor= M(a² + b) (C) Idoor M(a² + b*) (D) Isoor = }M(a²+8) (F) Isoor = Ma² %3D (G) Isoor Ma? %3D (H) Idor Ma² A rectangular plate with four umall point-like balls glued to each corner. The blue line represents the axis of rotationA space craft may be modeled as a uniform disk. Suppose the disk shaped craft has a mass of 2500 kg and a radius of 5.67 ngedalla par pplica meters. (a) What is the moment of inertia of the spacecraft? (b) Two rocket engines on opposite sides of the craft each apply an identical tangential force to impart a uniform angular acceleration in the counterclockwise sense. Suppose the craft acceleration from rest to an angular velocity of 1.00 revolutions per second in the counter clockwise sense over a period of 30.0 seconds. What is this final angular velocity in radians per second? (c) What is the angular acceleration of the craft over the period of uniform angular ac- celeration? (d) What net torque is needed to achieve the angular acceleration in part (c)? (e) What force is applied by each rocket engine during the period of uniform angular accel- eration?
- A uniform solid sphere of mass 12.0 kg and radius 7.0 cm rotates at 300 revolutions per minute (rpm) on an axis passing through its center. Calculate: a) Its moment of inertia b) Its rotational kinetic energy c) The angular momentum (L), that is, the magnitude of the product Iw, in the appropriate IS units d) Based on the general definition of the moment of inertia (I = Mr²), determine the radius of gyration of the sphere around the axis that passes through its center.Twin skaters approach one another as shown in the figure below and lock hands. (a) Calculate their final angular velocity, given each had an initial speed of 1.60 m/s relative to the ice. Each has a mass of 70.0 kg, and their centers of mass are 0.850 m from their locked hands. You may approximate their moments of inertia to be that of point masses at this radius. rad/s(b) Compare the initial and final kinetic energy. Ki Kf =Calculate the moment of inertia of a skater given the following information. (a) The 92.0-kg skater is approximated as a cylinder that has a 0.120-m radius. kg · m2(b) The skater with arms extended is approximately a cylinder that is 86.0 kg, has a 0.120-m radius, and has two 0.850-m-long arms which are 3.00 kg each and extend straight out from the cylinder like rods rotated about their ends.