Two boxes of mass $10\ kg$ and $18\ kg$ are on either side of the pivot, as shown below.

At what distance $d$ should the $18\ kg$ mass be kept so that the two sides are balanced about the pivot?

A ladder of mass $5\ kg$ is resting on a frictionless wall. However, the friction between the foot of the ladder and the floor is non-zero.

What is the normal force acting on the ladder due to the wall? (Hint: Consider the foot of the ladder as a pivot.)

A cylindrical object is fixed about a line passing through its axis. It is set into motion by applying a torque. The angular velocity of the cylinder increases to $0.8\ rad/s$ in $2\ s$. If the moment of inertia of the cylinder is equal to $4\ kgm^2$, what is the average torque acting on the cylinder?

A massless spring joining two identical blocks is rotating, as shown below.

If Max increases the angular velocity of the system, which of the below statements is TRUE?

A disk with a moment of inertia $\frac{MR^2}{2}$ is rotating with an angular velocity of $ω_o$. It slows down due to frictional forces between the disk and the axle. If the disk rotates $N$ times before stopping completely, what is the magnitude of the frictional torque acting on the disk?

Questions 6 and 7 are based on the below information.

A thin stick of length $30\ cm$ is made to rotate either about point $X$ or $Y$. The forces acting on the stick are as follows.

What is the net torque acting on the stick about an axis passing through $X$?

A thin stick of length $30\ cm$ is made to rotate either about point $X$ or $Y$. The forces acting on the stick are as follows.

How much is the net torque acting on the stick for an axis passing through $Y$?

A block of mass $3m_o$ is tied with a pulley of mass $m_o$ and radius $r_o$, as shown below.

The system is released from rest. What will be the angular acceleration of the pulley when the block starts moving? (Assume the pulley is in the form of a disk with moment of inertia $\frac{1}{2} m_o r_o^2$.)

Three objects of masses $2 \ kg, 3 \ kg,$ and $1.5 \ kg$ are in a 2D coordinate system having coordinates $(1,1),(0,-2),$ and $(3,4)$. What is the moment of inertia of the system consisting of these three particles for an axis passing through the origin?

(Assume the coordinates are in meters)

A solid cylinder having moment of inertia about an axis passing through its center is $I_cm = \dfrac{1}{2} \ MR^2$. A torque $τ=10 \ Nm$ is applied to the cylinder, and its angular acceleration $(α_1)$ is noted. The cylinder is then constrained to move about an axis passing through its circumference. A torque $ τ’=3τ $ is applied to the cylinder, and subsequent angular acceleration $(α_2)$ is noted. What is the ratio $α_1:α_2$?

The change in the angular velocity of a rotating system is shown in the below graph.

The system starts at rest and rotates for $t = 50 \ s$ with uniform acceleration. It then slows down with twice the acceleration. What is the angular displacement of the system from start to stop?

A thin solid disk (in x-y plane) is rotating with a speed of $ω=3 \ rad/s$ (clockwise as seen in a cartesian coordinate system). It is moving about an axis passing through its center of mass. The radius of the disk is $3 \ cm$. A particle of mass m is at a distance of $2 \ cm$ from the center of the disk. The particle starts moving from point $(2,0)$. What is the $x –$ component of linear velocity of the particle after $t = \dfrac{1}{3} \ s$?