A man stands on a merry-go-round that is rotating at 2.5 rad/s. If the coefficient of static friction between the man's shoes and the merry-go-round is

$$μ_S = 0.5,$$

how far from the axis of rotation can he stand without sliding?

Using the parallel axis theorem, what is the moment of inertia of the rod of mass $m$ about the axis shown below?

A wheel has a constant angular acceleration of 5.0 rad/s². Starting from rest, it turns through 300 rad. (a) What is its final angular velocity? (b) How much time elapses while it turns through the 300 radians?

A neutron star of mass

$$2 × 10^{30} kg$$

and radius 10 km rotates with a period of 0.02 seconds. What is its rotational kinetic energy?

The angular velocity of a flywheel with radius 1.0 m varies according to ω(t) = 2.0t. Plot

$$a_c(t)$$

and

$$a_t(t)$$

from t = 0 to 3.0 s for r = 1.0 m. Analyze these results to explain when

$$a_c >> a_t$$

and when

$$a_c << a_t$$

for a point on the flywheel at a radius of 1.0 m.

On takeoff, the propellers on a UAV (unmanned aerial vehicle) increase their angular velocity from rest at a rate of ω = (25.0t)rad/s for 3.0 s. (a) What is the instantaneous angular velocity of the propellers at t = 2.0 s? (b) What is the angular acceleration?

A vertical wheel with a diameter of 50 cm starts from rest and rotates with a constant angular acceleration of 5.0 rad/s² around a fixed axis through its center counterclockwise. (a) Where is the point that is initially at the bottom of the wheel at t = 10 s? (b) What is the point's linear acceleration at this instant?

The angular velocity of a rotating rigid body increases from 500 to 1500 rev/min in 120 s. (a) What is the angular acceleration of the body? (b) Through what angle does it turn in this 120 s?

At its peak, a tornado is 60.0 m in diameter and carries 500 km/h winds. What is its angular velocity in revolutions per second?

Calculate the rotational kinetic energy of a 12-kg motorcycle wheel if its angular velocity is 120 rad/s and its inner radius is 0.280 m and outer radius 0.330 m.

A uniform rod of mass $1.0 \mathrm{~kg}$ and length $2.0 \mathrm{~m}$ is free to rotate about one end (see the following figure). If the rod is released from rest at an angle of $60^{\circ}$ with respect to the horizontal, what is the speed of the tip of the rod as it passes the horizontal position?

A rotating wheel has a constant angular acceleration. It has an angular velocity of 5.0 rad/s at time t=0 s, and 3.0 s later has an angular velocity of 9.0 rad/s. What is the angular displacement of the wheel during the 3.0-s interval? (a) 15 rad (b) 21 rad (c) 27 rad (d) There is not enough information given to determine the angular displacement.

(a) Calculate the rotational kinetic energy of Earth on its axis. (b) What is the rotational kinetic energy of Earth in its orbit around the Sun?

Evaluate the integrals using appropriate substitutions.

$$\int \frac { d x } { 1 + 16 x ^ { 2 } }$$