Rigid Body

The angular velocity of a flywheel uniformly decreases from 900 rev/min to 800 rev/min in 5 seconds.

As shown in the figure, a large horizontal disk of radius R rotates about a fixed vertical axis O with a constant angular velocity ω. A second vertical axis O₁ is fixed on the disk at a distance $|OO_1| = R/2$. A rigid rod O₁P of length $l = R/2$ is pivoted at O₁ and rotates with a constant angular velocity ω' = ω relative to the disk. The position of the rod relative to the disk is given by the angle φ between the line segment OO₁ and the rod O₁P. At time t=0, φ=0, meaning O, O₁, and P are collinear.

A light rod AD rests on a semi-cylinder of radius $R$ and a horizontal ground surface, as shown in Figure. End A is on the ground, and the rod is tangent to the cylinder at point B. The angle between the cylinder radius OB and the vertical is $\theta$.

As shown in Figure, rod OA of length R rotates in a vertical plane about a horizontal axis through point O. Its endpoint A is attached to a light, inextensible string that passes over fixed pulleys B and C to a block M. Point B is directly above O at a distance H. At a certain instant, the angular velocity of the rod is $\omega$ and the angle between the string segment BA and the vertical line OB is $\alpha$.

A thin rod AB of length $l$ has its ends A and B constrained to move on the x and y axes, respectively. Point P is on the rod at a distance $\alpha l$ from end B, where $0 < \alpha < 1$.

A pushrod AB slides in a vertical guide K, driven by a cam M rotating about axis O with constant angular velocity $\omega$. At the instant shown, the contact point A is at a distance $r$ from O (OA=r). The angle between the normal $n$ to the cam surface at A and the line OA is $\alpha$.

As shown in the figure, disk B with radius $R_2$ rolls without slipping inside a stationary circular disk A with radius $R_1$. The disk A rotates around the center $O_1$ with a constant angular velocity $\omega_1$. Disk B rotates about its own center $O_2$ with angular velocity $\omega_2$.

As shown in Figure, a moving gear with radius $r$ is driven by a crank arm $OO_1$ to roll along a fixed gear with radius $R$. The crank arm rotates about the axis $O$ with a constant angular velocity $\omega_0$.

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).