Rigid Body Dynamics

A light, inextensible rope passes over a fixed, disk-shaped pulley of mass $m$ and radius $R$. The ends of the rope are attached to objects of mass $m_1$ and $m_2$, with $m_1 > m_2$. The rope does not slip on the pulley, and friction at the pulley's axle is negligible.

As shown in Figure, a flywheel can rotate freely about a horizontal axis through its center O. A force of $F = 100$ N is applied at point P. The distance from the center is $OP = r = 50$ cm. The position vector $\vec{r} = \vec{OP}$ makes a 30° angle with the horizontal. The force $\vec{F}$, which is in the plane of the flywheel, makes a 45° angle with the horizontal.

As shown in Figure, a uniform disk with radius $R=15$ cm and mass $M=3.0$ kg is mounted on a fixed horizontal axle through its center. A light string is wrapped around its circumference with no slipping. A constant downward force $F=5.0$ N is applied to the string.

As shown in Figure, the disk from the previous problem ($M=3.0$ kg, $R=0.15$ m) has a string wrapped around it. An object with weight $W=5.0$ N is hung from the string. The system is released from rest.

A pair of forces that are equal in magnitude and opposite in direction, but whose lines of action do not coincide, is called a force couple. The perpendicular distance between their lines of action, $l$, is called the couple arm.

A child of mass M is standing on the edge of a stationary, freely rotating disk. The disk has radius R and moment of inertia I. The child throws a stone of mass m horizontally and tangentially to the edge of the disk. The velocity of the stone relative to the ground is v.

When a bicycle moves forward on horizontal ground with a speed $v$, what are the instantaneous speeds of the wheel's axle, the highest point on the wheel, and the lowest point on the wheel? Assume there is no slipping between the wheel and the ground.

A flywheel, which can be treated as a solid cylinder, has a diameter of 0.30 m and a mass of 5.0 kg. A rope is wound around its edge. A constant force pulls one end of the rope, causing it to accelerate uniformly from rest. After 0.50 s, its rotational speed reaches 10 r/s.

Two objects of mass $m_1$ and $m_2$ are connected by a string over a pulley as shown. Block $m_1$ hangs vertically, while block $m_2$ is on a horizontal surface with a coefficient of kinetic friction $\mu$. The pulley has a moment of inertia $I$ and a radius $r$. The string does not slip on the pulley.