Q2: Find the moments of inertia about the x-axis, y-axis, and z-axis (Ix, Iy, and I₂) for the region lying between the hemisphere z = √√4-x² - y² and the xy-plane. Given that the density p(x, y, z) = k.z where k is a constant. Draw the region.

A bicycle wheel of mass M (all at the edges) and radius r is fixed about an axis through its center and is initially at rest. You wrap the wheel with a cord and attach a hanging weight of mass m to the cord. You release the weight and allow it to fall a distance d. The weight falls smoothly, and the cord does not slip over the wheel as it unwinds. a. (Write an expression for the angular acceleration α of the wheel in terms of the tension T of the cord.b. Write an expression for the linear acceleration a of the weight in terms of the tension T of the cord.c. (Using parts (a) and (b) find the linear acceleration in terms of M, m, and g.d. Write an expression for the linear speed v of the weight after it falls a distance d.e.  For M = 7.5 kg, r = 33 cm, m = 250 g, and d = 1.5 m, what is the rotational kinetic energy of the wheel afterthe weight has fallen?

A ship's wheel can be approximated as an annular cylinder (different from a ring) of mass 6.7 kilograms, inner radius 25 centimeters, and outer radius 32 centimeters; and eight spokes of mass 0.50 kilograms and length 49 centimeters.   a. What is the moment of inertia? Include units in your answer. More information. b. Disagreeing over which way to go, the captain and the helmsman try to turn the wheel in opposite directions. The captain applies a force of 311 newtons at the inner radius, while the helmsman applies a force of 287 newtons at the outer radius. What is the magnitude of the angular acceleration of the wheel? Include units in your answer.

a.) A tension force, T, of magnitude 5.0 N is applied to the end of a bar at a distance r = 1.5 m from the axis of rotation O (represented by the black dot). The angle between the force and the distance vector, θ=25o. Calculate the magnitude and direction of the torque vector.   b.) In the previous problem, let’s make some changes.  The bar is now a different length and reoriented within 3-D space such that the r is a vector that has components (3,2,0) in the xyz coordinate frame, and F is a vector with components (4,5,0). What is the torque exerted on the rigid body about a pivot located at the origin? Express your answer in i, j, k, components.  Assume all distances are represented in meters and forces in newtons.