Exam 10: Energy and Momentum in Rotating Systems

A 3.0-kg particle has a position vector given by $\overrightarrow{\mathbf{r}}=\left(2.0 t^{2} \hat{\mathbf{i}}+3.0 \hat{\mathbf{j}}\right)$ where $\overrightarrow{\mathbf{r}}$ is in metres and t is in seconds. What is the angular momentum of the particle, in kg.m²/s, about the origin at t = 2 s?

A force $\overrightarrow{\mathbf{F}}$ is applied to a cylindrical roll of paper of radius R and mass M by pulling on the paper as shown. The acceleration of the centre of mass of the roll of paper (when it rolls without slipping) is:

A uniform solid sphere rolls without slipping along a horizontal surface. What fraction of its total kinetic energy is in the form of rotational kinetic energy about the CM?

A 0.5 kg fish, hooked as shown below, starts to swim away at a speed of 3 m/s. The angular momentum of the fish relative to the hand holding the fishing rod is about:

A massless rope is wrapped around a uniform cylinder that has radius R and mass M, as shown in the figure. Initially, the unwrapped portion of the rope is vertical and the cylinder is horizontal. The linear acceleration of the cylinder is:

Stars originate as large bodies of slowly rotating gas. Because of gravity, these clumps of gas slowly decrease in size. The angular velocity of a star increases as it shrinks because of:

In the figure shown, a 1.6-kg weight swings in a vertical circle at the end of a string having negligible weight. The string is 2-m long. If the weight is released with zero initial velocity from a horizontal position, its angular momentum (in kg.m²/s) at the lowest point of its path relative to the centre of the circle is approximately:

Identical particles are placed at the 50-cm and 80-cm marks on a metre stick of negligible mass. This rigid body is then mounted so as to rotate freely about a pivot at the 0-cm mark on the metre stick. If this body is released from rest in a horizontal position, what is the angular speed of the metre stick as it swings through its lowest position?