Moment of Inertia via Integration Practice Problems

A uniform thin bar with a mass of m_b and length of l_b is positioned along the y-axis. The bar is capable of rotating around an axis that passes through a point a distance of b away from one of its extremities. Determine the moment of inertia of the bar relative to this axis.

A 10 kg rectangular wooden plank with dimensions 180 cm by 40 cm is used as a seesaw. What is the moment of inertia of the plank when it rotates about its centre of mass?

A 15.0 cm radius bicycle wheel has a mass of 1.5 kg. What is the moment of inertia of the wheel about an axis passing through a point on the rim of the wheel?

The linear density of a thin bar with length l _b and mass m _b varies with position in accordance with the function λ(y) = ay ³, where y is the distance in meters from one end of the bar and an is a constant. Determine the relationship between the constant a and the bar's mass and length.

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.

Consider a wooden beam used in construction, having a length 𝐿 and a uniform mass 𝑀. This beam is suspended horizontally, with an axis of rotation passing through its midpoint and perpendicular to its length. Derive the formula for the moment of inertia of the beam about this central axis.

Determine the moment of inertia of a slender rod of length L, which has a linear mass density that increases uniformly from λ at one end to 2λ at the other, about an axis perpendicular to the rod and passing through its center.