Intro to Moment of Inertia Practice Problems

A non-uniform thin spoke has a total length L. Its mass per unit length depends on the position x measured from the left end and is described by the function dm/dx = Ax where A has units kg/m². i) Determine its moment of inertia about a perpendicular axis through its left end (x = 0) using I = ∫r²dm. ii) Does its moment of inertia differ from the moment of inertia of a uniform spoke?

A child's entertainment equipment has a 100.0 cm long, thin and uniform rod of mass 0.500 kg. Determine the moment of inertia of the rod pivoted about its center such that it spins perpendicular to its length. 

You are given a thin spoke of mass 3.0 kg that is 100.0 cm long to be used in an experiment. Determine its moment of inertia if it rotates about a perpendicular axis through i) one of its ends ii) its center iii) a parallel axis through the spoke. 

A uniform square slab of side a and mass m_s placed in the horizontal plane revolves around a vertical axis passing through its center. Write the expression of the moment of inertia.

We can use any arbitrary shape with uniform composition and radius = length = height = width to easily prove that when an object is x times large than the model, its mass and volume are x³ times greater than those of the model. How many times greater is the object's moment of inertia relative to the model (exclude point masses and parallel axes outside an object)?

Four spheres of masses of 300 g each are attached to each other in the shape of a square as depicted in the figure below. Consider an axis passing through the centre of the square. Determine the moment of inertia of the square about this axis.

Consider three masses of 150 g each are joined to each other with massless rods forming a circular ring as portrayed in the figure below. Determine the circular ring's kinetic energy if its rotation speed is 6.0 rev/s about the central axis.

A rectangle houses four masses, as shown below. The masses are linked using rods of negligible mass. Calculate the system's moment of inertia about an axis (diagonal) through masses 1 and 3.

A triangle houses three spheres (point masses), as shown below. The spheres are linked using bars of negligible mass. Determine the moment of inertia for the system of masses about an axis perpendicular to the page and through mass 2.

A bar with a moment of inertia of 3.5 kg•m² is spinning on its axis at an initial angular velocity of 1.3 rad/s. The image below shows the torque applied to the bar. Find the angular velocity of the bar at t = 2.5 s.

A 75.0 kg skater spins on the ice with arms outstretched. The skater's body is considered a 50.0 kg cylinder, with a radius of 0.300 m and a height of 1.60 m. Each arm is modeled as a 3.00 kg thin rod, 50.0 cm long, attached to the cylinder. Calculate what the moment of inertia of the skater will be.

What is the moment of inertia of a vinyl record with a diameter of 15 cm and a mass of 30 g when it rotates around an axis perpendicular to its centre?