A wheel of radius 30 cm is rotating at a rate of 2.0 revolutions every 0.080 s. (a) Through what angle, in radians, does the wheel rotate in 1.0 s? (b) What is the linear speed of a point on the wheel's rim? (c) What is the wheel's frequency of rotation?

A centrifuge has a rotational inertia of

$$6.5 \times 10 ^ { - 3 } \mathrm { kg } \cdot \mathrm { m } ^ { 2 }$$

. How much energy must be supplied to bring it from rest to 420 rad/s (4000 rev/min)?

A uniform sphere with mass $28.0 \mathrm{~kg}$ and radius $0.380 \mathrm{~m}$ is rotating at constant angular velocity about a stationary axis that lies along a diameter of the sphere. If the kinetic energy of the sphere is $176 \mathrm{~J}$, what is the tangential velocity of a point on the rim of the sphere?

A $60.0-\mathrm{kg}$ woman stands at the rim of a horizontal turntable having a moment of inertia of $500 \mathrm{~kg} \cdot \mathrm{m}^2$ and a radius of $2.00 \mathrm{~m}$. The turntable is initially at rest and is free to rotate about a frictionless, vertical axle through its center. The woman then starts walking around the rim clockwise (as viewed from above the system) at a constant speed of $1.50 \mathrm{~m} / \mathrm{s}$ relative to the Earth. (b) How much work does the woman do to set herself and the turntable into motion?

The angular position of a point on the rim of a rotating wheel is given by $\theta = 4.0t - 3.0t^2 + t^3$, where $\theta$ is in radians and t is in seconds. What are the angular velocities at $t = 4.0 \text{~s}$.

A flywheel with a radius of 0.300 m starts from rest and accelerates with a constant angular acceleration of $0.600 \mathrm { rad } / \mathrm { s } ^ { 2 }$. Compute the magnitude of the tangential acceleration, the radial acceleration, and the resultant acceleration of a point on its rim after it has turned through $60.0 ^ { \circ }$.

The angular position of a point on the rim of a rotating wheel is given by $\theta = 4.0t - 3.0t^2 + t^3$, where $\theta$ is in radians and t is in seconds. What is the instantaneous angular acceleration at the beginning of this time interval?

The angular position of a point on the rim of a rotating wheel is given by $\theta = 4.0t - 3.0t^2 + t^3$, where $\theta$ is in radians and t is in seconds. What are the angular velocities at $t = 2.0 \text{~s}$.

Linear vs. Angular Speed A wheel of radius 30 centimeters revolves at a rate of 10 revolutions per second. A dot is painted at a point $P$ on the rim of the wheel (see figure).

Use a graphing utility to graph the function in part (a).

You are analyzing the motion of a large flywheel that has radius 0.800 m. In one test run, the wheel starts from rest and turns with constant angular acceleration. An accelerometer on the rim of the flywheel measures the magnitude of the resultant acceleration a of a point on the rim of the flywheel as a function of the angle $\theta-\theta_{0}$ through which the wheel has turned. You collect these results:

$$\begin{matrix} \mathrm{\theta-\theta_{0}(\mathrm{rad})} & \text{0.50} & \text{1.00} & \text{1.50} & \text{2.00} & \text{2.50} & \text{3.00} & \text{3.50} & \text{4.00}\\\mathrm{a\left(\mathrm{m} / \mathrm{s}^{2}\right)} & \text{0.678} & \text{1.07} & \text{1.52} & \text{1.98} & \text{2.45} & \text{2.92} & \text{3.39} & \text{3.87}\\\end{matrix}$$

Construct a graph of $a^{2}\left(\operatorname{in} m^{2} / s^{4}\right)$ versus $\left(\theta-\theta_{0}\right)^{2}$ in $\left(\mathrm{rad}^{2}\right)$. When the flywheel has turned through an angle of $90.0^{\circ}$, what is the angle between the linear velocity of a point on its rim and the resultant acceleration of that point?

The New York Wheel is the world's largest Ferris wheel. It's 183 meters in diameter and rotates once every 37.3 minutes. Find the magnitudes of

(a) the average velocity and

(b) the average acceleration at the wheel's rim, over a 5.00-min interval.

(c) Compare your answer to

(b) with the wheel's instantaneous accelerations.

An electric turntable $0.750$ m in diameter is rotating about a fixed axis with an initial angular velocity of $0.250$ rev$/$s and a constant angular acceleration of $0.900$ rev$/$s$^2$.
($c$) What is the tangential speed of a point on the rim of the turn- table at $t=0.200$ s?

A merry-go-round has a radius of 2.0 m and a moment of inertia 300 kg·m². A boy of mass 50 kg runs tangent to the rim at a speed of 4.0 m/s and jumps on. If the merry-go-round is initially at rest, what is the angular velocity after the boy jumps on?

A wheel has an angular acceleration of α = (0.5 θ) rad/s^2, where θ is in radians. Determine the magnitude of the velocity and acceleration of a point P located on its rim after the wheel has rotated 2 revolutions. The wheel has a radius of 0.2 m and starts at ω0 = 2 rad/s.