Circular Motion

A uniform rubber band of mass $m$, natural radius $r_1$, and stiffness coefficient $k$ is placed horizontally on a vertical cylinder of radius $r_2$ ($r_2 > r_1$). The mass distribution of the band remains uniform. The coefficient of static friction between the band and the cylinder is $\mu$. The cylinder rotates about its vertical axis with angular velocity $\omega$.

An inclined plane with side ratios 3:4:5 is fixed on a horizontal turntable. A wooden block is placed on the inclined plane and remains stationary at a distance $r = 40$ cm from the center of the turntable. The coefficient of static friction between the block and the plane is $\mu_s = 1/4$.

Two small balls are undergoing uniform circular motion.

A curved section of railway track has a radius of curvature $r$. The distance between the two rails is $L$, as shown in the figure.

A particle undergoes uniform circular motion. At time $t_1$, its velocity is $\vec{v}_1 = v_x \hat{i} + v_y \hat{j}$. At a later time $t_2$, its velocity is $\vec{v}_2 = -\vec{v}_1$. The time interval is $\Delta t = t_2 - t_1$.

An object is in uniform circular motion (UCM).

As shown in the figure, one end of a thin rope is fixed at point A. A weight B is attached to the rope at a distance $a$ from A, making the length of segment AB constant, $l_{AB} = a$. The other end of the rope passes over a fixed pulley at point C. Points A and C lie on the same horizontal line. The free end of the rope is pulled with a constant speed $v$. At the instant shown, the rope segments AB and BC make angles $\alpha$ and $\beta$ with the horizontal, respectively.

A particle undergoes uniform circular motion. At a given instant, its position vector is $\vec{r}$, its velocity is $\vec{v}$, and its acceleration is $\vec{a}$.

A small block of mass $m$ is carefully placed on the inner surface of a hollow, thin-walled cylinder of mass $M$ and radius $R$. Initially, the cylinder is at rest on a horizontal surface. The block is also at rest, at a vertical height $R$ from the horizontal surface, level with the cylinder's axis. There is no friction between the block and the inner wall of the cylinder. The cylinder rolls without slipping on the horizontal surface. The acceleration due to gravity is $g$.

A particle moves at a constant speed along a helix of constant pitch on the surface of a cylinder with radius $R$. The radius of curvature of this helix is $\rho$, and the period of motion of the particle's projection onto a plane perpendicular to the cylinder's axis is $T$.