A 50.0-kg child stands at the rim of a merry-go-round of radius 2.00 m, rotating with an angular speed of 3.00 rad/s. What is the minimum force between her feet and the floor of the carousel that is required to keep her in the circular path?

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 average angular acceleration for the time interval that begins at $t = 2.0 \text{~s}$ and ends at $t = 4.0 \text{~s}$?

At t = 3.00 s a point on the rim of a 0.200-m-radius wheel has a tangential speed of 50.0 m/s as the wheel slows down with a tangential acceleration of constant magnitude $10.0 \mathrm { m } / \mathrm { s } ^ { 2 }$. (a) Calculate the wheel’s constant angular acceleration. (b) Calculate the angular velocities at t = 3.00 s and t = 0. (c) Through what angle did the wheel turn between t = 0 and t = 3.00 s? (d) At what time will the radial acceleration equal g?

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?

A dart is thrown horizontally with an initial speed of 10 m/s toward point P, the bull's-eye on a dart board. It hits at point Q on the rim, vertically below $P, 0.19$ s later. What is the distance $P Q$ ?

The water side of the wall of a $100$-$\mathrm{m}$-long dam is a quarter circle with a radius of $10 \mathrm{~m}$. Find the hydrostatic force on the dam and its line of action when the dam is filled to the rim.

Circle the antecedent in each sentence, and underline the pronoun in parentheses that agrees with it.

Example: Where did your $\boxed{\text{{brothers}}}$ buy (his, $\underline{{their}}$) new skis?

The basketball hoop outside has icicles hanging from (its, their) rim.

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).

A point on the rim of a $0.75$-m-diameter grinding wheel changes speed uniformly from $12$ m/s to $25$ m/s in $6.2$ s. What is the angular acceleration of the grinding wheel during this interval?

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 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.

Suppose a person with a mass of $58 \mathrm{~kg}$ jumps on a trampoline and compresses the fabric a distance of $40 \mathrm{~cm}$. How high will the person then bounce above the rim of the trampoline if the spring constant of the trampoline is $1250 \mathrm{~N} / \mathrm{m}$ ?

Monica stands at the edge of a circular platform that is slowly rotating on a frictionless axle. She then walks toward the opposite edge, passing through the platform's center. Describe the motion of the platform as Monica makes her trip.