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 $t = 4.0 \text{~s}$.

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 $60.0-\mathrm{kg}$ 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?

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 $1250 \mathrm{~N} / \mathrm{m}$ ?

A beetle sits near the rim of a turntable that rotates without friction about a vertical axis. The beetle then begins to walk toward the center of the turntable. As a result, does the angular speed of the turntable increase, decrease, or stay the same?

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 $60.0 ^ { \circ }$.

In each of the following sentences, underline the subject once and the predicate twice. Part of the caldera around Vesuvius, known as the Somma Rim, was formed approximately 17,000 years ago.

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. (a) In what direction and with what angular speed does the turntable rotate?

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 carousel with a 50 -foot diameter makes 4 revolutions per minute. (a) Find the angular speed of the carousel in radians per minute. (b) Find the linear speed of the platform rim of the carousel.

When the string is pulled with a force $p$ tangent to the rim, it gives the cylinder an angular acceleration $\alpha$, if the cylinder had fourfold the radius but everything else was the same the angular acceleration would be?

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 $10.0$ m$/$s$^2$. (b) Calculate the angular velocities at $t=3.00$ s and $t=0$.

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.

A 60.0-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 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 m/s relative to Earth. How much work does the woman do to set herself and the turntable into motion?