14. Torque & Rotational Dynamics

A hollow cylinder (hoop) is rolling on a horizontal surface at speed v = 3.0 m/s when it reaches an 18° incline. How far up the incline will it go?

A cord is wrapped around the rim of a solid uniform wheel 0.250 m in radius and of mass 9.20 kg. A steady horizontal pull of 40.0 N to the right is exerted on the cord, pulling it off tangentially from the wheel. The wheel is mounted on frictionless bearings on a horizontal axle through its center. Find the magnitude and direction of the force that the axle exerts on the wheel.

To get a flat, uniform cylindrical satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 10–61. Suppose that the satellite has a mass of 3600 kg and a radius of 4.0 m, and that the rockets each add a mass of 250 kg. What is the steady force required of each rocket if the satellite is to reach 28 rpm in 5.0 min, starting from rest?

A particle of mass m uniformly accelerates as it moves counterclockwise along the circumference of a circle of radius R: $\vec{r}=\hat{i}R\cos \theta +\hat{j}R\sin \theta$ with $\theta ={\omega }_{0}t+\frac{1}{2}\alpha t^2$, where the constants ω₀ and α are the initial angular velocity and angular acceleration, respectively. Determine the object’s tangential acceleration ${\overrightarrow{a}}_{\tan }$ and determine the torque acting on the object using $\vec{\tau }=I\vec{\alpha }$.

If you increase the length of the lever arm ($r$) while applying force ($F$) to turn a wrench, how does the amount of force required to produce the same torque ($\tau$) change? (Recall: $\tau =rF$.)

A pulley of radius $R$ and moment of inertia $I$ has two ropes wrapped around it on opposite sides. The tensions in the ropes are $12 N$ and $10 N$. What is the angular acceleration $\alpha$ of the pulley?

Which equation correctly expresses the relationship between $\tau$ (torque) and $\alpha$ (angular acceleration) for a rigid body rotating about a fixed axis?

Which of the following is an effective way to increase the amount of $\tau$ (torque) produced by a wind turbine?

Suppose that piano has a long, thin bar ran through it (totally random), shown below as the vertical red line, so that it is free to rotate about a vertical axis through the bar. You push the piano with a horizontal 100 N (blue arrow), causing it to spin about its vertical axis with 0.3 rad/s² . Your force acts at a distance of 1.1 m from the bar, and is perpendicular to a line connecting it to the bar (green dotted line). What is the piano's moment of inertia about its vertical axis?

A solid uniform disk of mass 21.0 kg and radius 85.0 cm is at rest flat on a frictionless surface. Figure 10–76 shows a view from above. A string is wrapped around the rim of the disk and a constant force of 35.0 N is applied to the string. The string does not slip on the rim. When the cm has moved a distance of 5.2 m, determine how fast the disk is spinning (in radians per second.

A 2.00-kg textbook rests on a frictionless, horizontal surface. A cord attached to the book passes over a pulley whose diameter is 0.150 m, to a hanging book with mass 3.00 kg. The system is released from rest, and the books are observed to move 1.20 m in 0.800 s. What is the tension in each part of the cord?

A 1.6-kg grindstone in the shape of a uniform cylinder of radius 0.20 m acquires a rotational rate of 22 rev/s from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.

A large spool of rope rolls on the ground with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance ℓ, holding onto it, Fig. 10–70. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

<IMAGE>

Assume that a 1.00-kg ball is thrown solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 10–57. The ball is accelerated uniformly from rest to 8.5 m/s in 0.38 s, at which point it is released. Calculate the force required of the triceps muscle. Assume that the forearm has a mass of 3.7 kg and rotates like a uniform rod about an axis at its end.