Find the moment of inertia of a uniform solid cone about its central axis, which is the axis of symmetry perpendicular to the base. The cone has a base radius $R_{b}$, height $h$, and mass $m$ . Assume the cone is made of aluminum.

Hint: Consider the cone as composed of a stack of infinitesimally thin disks, each having a different radius that decreases linearly from the base to the tip as you move along the height.

You oversee projects for a business that produces goods. A thin, uniform rod measuring 60.0 cm in length and weighing 0.400 kg is one of the machine parts being assembled on the assembly line. One of your engineers has suggested that you bend the rod at its center into a V-shape with a 30.0^o angle at its vertex in order to lessen the moment of inertia. What would this bent rod's moment of inertia be at its vertex, which is an axis perpendicular to the plane of the V?

A bicycle wheel is made with a rim and 6 thin rods. The wheel is placed along a horizontal plane and is mounted on a fixed frictionless vertical axle passing through the wheel's hub. The rim's diameter is 58 cm, and its mass is 1.250 kg. The rim is connected to the hub by the rods, as shown in the figure. The mass of a rod is 0.100 kg, and its length is 29 cm. Calculate the moment of inertia about the vertical axle.

Four-point masses are arranged at the vertices of a rectangle. The masses are connected by rigid massless rods, as shown in the figure. Given that, m = 11 kg and m' = 2.3 kg determine the moment of inertia of the mass distribution about the y-axis that bisects the rectangle into two symmetrical pieces.

A solid cylinder of mass 2 kg and radius 0.5 m is rotating about its central axis at 3.0 rev/s. Assuming the cylinder to be a uniform object, determine its rotational kinetic energy.

A rotating turbine has a diameter of 2.0 m and a mass of 300 kg. If the turbine is spinning at a maximum angular velocity of 1800 rpm, determine the energy stored in the turbine. Assume the turbine is disc-shaped.

Calculate the kinetic energy of a spherical asteroid due to its daily rotation about its axis in a geostationary orbit around Earth. Assume the asteroid has a mass of 1.2$\times$10¹³ kg and is located at an altitude of 3.6$\times$10⁴ km above the Earth's surface. The radius of the asteroid = 2.4$\times$10³ m.

How much torque is supplied by an electric motor in a grinder that outputs 1.2 kW when operating at a rotational speed of 2000 rpm?

A 15 kg flywheel has the shape of a solid disk with a radius of 8.00 cm. Determine the amount of steady torque required to accelerate the flywheel from rest to the engine rotational speed of 3600 rev/min within 3.2 s. 

A rigid, uniform bar is attached by its lowest end to a frictionless horizontal hinge. The bar is initially perpendicular to the horizontal plane. The bar of length 1.0 m and mass 0.250 kg is released from rest and rotates in a vertical plane. Calculate the speed of the highest end of the bar when it reaches the horizontal plane.

A soccer ball is kicked along a 23° inclined plane with a speed of v = 5.0 m/s at the foot of the incline. Determine the time it takes for the soccer ball (a hollow sphere) to climb the incline and return to its original position.