Rigid Body

As shown in Figure, circle A with radius $R_1$ rotates about its fixed center $O_1$ with constant angular velocity $\omega_1$. Circle B with radius $R_2$ rolls without slipping on the outside of circle A, with constant angular velocity $\omega_2$ about its own center $O_2$.

As shown in the figure, two coaxial thin-walled cylinders A and B have radii of $R$ and $2R$, respectively. A small cylinder with a radius of $R/2$ is placed between them. Cylinders A and B rotate uniformly with angular velocities $\omega_1$ and $\omega_2$ in opposite directions. There is no slipping at the contact points D (with A) and C (with B).

An equilateral triangle ABC with side length $l$ undergoes plane motion. At a certain instant, the velocity of point A is $\vec{v}_A = \omega_0 \vec{AC}$, and its acceleration is $\vec{a}_A = 2\omega_0^2 \vec{AC}$. The magnitude of the velocity of point B is $v_B = l\omega_0/2$, and the magnitude of its acceleration is $a_B = l\omega_0^2/2$. Here, $l$ and $\omega_0$ are constants.

In the mechanism shown, rods OA and AB have equal length, $\overline{OA} = \overline{AB} = a$. Slider B moves with a constant velocity $v$. At the instant shown, OA is horizontal and perpendicular to the vertical rod AB.

Two point masses, 1 and 2, with masses $m_1$ and $m_2$ ($m_1 > m_2$) respectively, are on a smooth horizontal table. They are connected by a light, inextensible string of length $L$. Initially, mass 1 is held fixed while mass 2 revolves around it in a circle. Then, mass 1 is released, and mass 2 follows the trajectory shown in the figure.

A thin massless rod AC moves in a vertical plane. Its end A is in contact with the inner wall of a hemispherical bowl of radius $R$. A point B on the rod rests on the rim of the bowl. The center of the hemisphere, $O_1$, lies on the plane of the rim. At the instant when the radius $O_1A$ makes an angle $\theta$ with the vertical, the speed of end A is half the speed of end C, i.e., $v_A = v_C/2$.

A planar mechanism is shown in the figure. The lengths of the rods are $\overline{OA} = \overline{BC} = \sqrt{3}r$ and $\overline{AB} = 2\overline{CD} = 2r$. Rod OA rotates about a fixed pivot O with angular velocity $\omega$ (clockwise), and rod CD rotates about a fixed pivot D with angular velocity $2\omega$ (clockwise). At the instant shown, rod OA is vertical and perpendicular to rod AB, which is horizontal. Rod CD is also horizontal. The angle between rod BC and the horizontal is $60^\circ$.

A light rigid rod of length $L$ has two small balls, A and B, fixed at its ends. Their masses are $m_A = M$ and $m_B = 2M$, respectively. The system is placed on a smooth horizontal plane. At a certain instant, the speeds of the two balls are $v_A = v$ and $v_B = 2v$.