4) If a wire in the shape of the helix: x = 2 sin t ,y = 2cos t,z= 3t; 0 < t < 2n, and the density is a constant k. Find the moments of inertia about the x-, y- and z-axes.
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- W; = 0 %3D Wf initial final A flywheel consists of a thin uniform disk of mass m and radius R, free to rotate about a frictionless central axle. A small weight, also of mass m, is attached to the wheel's rim. The is wheel is held at rest (left), the small weight level with the axle, then released. What is the angular velocity of the wheel when the weight reaches its lowest point? 4g V 3R 8g V 5R 3g V 2R V28R 2gRA 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.A 366-N force is applied at A as shown. y = 370 mm. Determine (b) the smallest force applied at B that creates the same moment about D. Write numerical value and 2 decimal places. + for counterclockwise -100 mm- -200 mm y mm 25° A P D 125 mm C B
- Find the moment of inertia for a solid cube is bounded by planes x = F1, z = 71 , y = 3 and y = 5 about x – axisA solid cylinder (disk) and a hollow cylinder are rolling with the same center- of-mass velocity v = 2 m/s on a level surface towards an incline. Both R cylinders have the same radius R and mass M. The moments of inertia are: 1 Solid cylinder 1, MR² Hollow cylinder I, = MR² (a) What is true about the kinetic energy when the cylinders are rolling on level ground? (i) they have the same translational kinetic energy (ii) they have the same rotational kinetic energy (iii) they have the same total kinetic energy [1] only (i) [2] only (ii) [3] (i) and (ii) [4] (i) and (iii) [5] (i), (ii), and (iii) (b) Which conservation law will allow you to calculate the final height? Conservation of: [1] momentum [5] moment of inertia [2] mechanical energy [3] kinetic energy [4] angular momentum (c) Calculate the maximum height the solid cylinder will reach before it rolls down again; please use g = 10 m/s². (d) Calculate the maximum height the hollow cylinder will reach before it rolls down again.