Physics

How can Darcy explain why there is nonzero angular momentum before the skater starts to rotate around the pole?

"A particle moving in a straight line has angular momentum about any axis displaced from the path of the particle."

Darcy asks Wilhelmina why angular momentum is constant throughout the motion. Which of Wilhelmina's responses is correct?

"Because no net external torque acts on the skater, the total angular momentum must be conserved."

Darcy clicks "reset" and sets the initial speed to 0.50 m/s and the skater's arm extension to 60 cm. She clicks "start." Once the skater grabs hold of the pole, Darcy slowly decreases the arm extension. What does Darcy tell Wilhelmina she observed?

"As the skater pulls herself closer to the pole, her angular momentum remains constant, whereas both the linear and angular speeds increase."

Darcy asks how Wilhelmina can determine the skater's angular speed when she first grabs the pole, given vi and ri. What is Wilhelmina's response?

"Because the skater's angular momentum is conserved, and her linear speed and the distance r of her center of mass from the pole stay the same (because the skater keeps her arm extended to the same extent), I use v = ωr to get ω = vr, and plug in vi and ri."

Darcy asks Wilhelmina why angular speed increases as the skater pulls herself closer to the pole. Which of Wilhelmina's responses is correct?

"For an object with a fixed mass that is rotating about a fixed symmetry axis, angular momentum can also be written as Iω, where I is the moment of inertia of the skater. From conservation of angular momentum, Iiωi = Ifωf, and so when the skater pulls her arm in close to her body, her moment of inertia is reduced and her angular speed is increased."

How does Wilhelmina explain to Darcy the reason that the angular speed increases as the skater gets closer to the pole?

"From v = ωr, angular speed is ω = vr. Because the linear speed is constant and the distance between the skater and the pole decreases, ω increases."

With respect to rolling motion, neglecting any rolling friction, which of the following statements Chandra makes to Darcel is false?

"If the object moving down along an incline were to slip, the total mechanical energy of the object-Earth system would increase because of the nonconservative force of kinetic friction."

Wilhelmina sets the initial speed to 0.50 m/s and arm extension to 30 cm, then clicks "start." How does she explain to Darcy what she observes?

"The angular speed increases as the skater moves toward the pole and reaches its maximum value when she grabs the pole and begins moving in a circular path. The angular momentum is constant throughout the motion."

Darcel and Chandra decide to try a little experiment. They completely fill two identical cylindrical bottles with water. One is placed in a freezer until the water has turned to ice; then both bottles are placed on a short incline and allowed to roll to the bottom. Assume that the ice and bottle expand along the axis, so that the radii of both bottles are the same. Darcel asks Chandra to predict which will reach the bottom of the incline first, and to explain her reason. Which response is correct?

"The bottle with the liquid water will reach the bottom of the incline sooner. The ice-water bottle must spin only a portion of its mass to keep itself from sliding, but the bottle with liquid water must spin the entire mass of the bottle and water, so its effective moment of inertia is higher. Because they start at the same level, they both have the same potential energy; however, for the liquid, less of that available energy is transformed into translational kinetic energy." OR "The bottle with the liquid water will reach the bottom of the incline sooner. The ice-water bottle must spin all of its mass to keep itself from sliding, but the bottle with liquid water will only have to spin the bottle (and some of the water due to friction), so its effective moment of inertia is lower. Because they start at the same level, they both have the same potential energy; however, for the solid, less of that available energy is transformed into translational kinetic energy."

The speed of the object at the bottom of the incline is equal to the translational speed once the object reaches the bottom. Which of Chandra's statements is supported by her observations?

"The final speed depends on how the mass is distributed inside the object."

From their observations, Darcel and Chandra found that the speed of an object at the bottom of an incline depends on how the mass is distributed in the object. Now they consider the moments of inertia of the objects in the simulation, which are related to their mass distributions. Darcel asks Chandra what effect the moment of inertia has on the final speed. Which is the correct response?

"The larger the moment of inertia of the object, the slower it will be moving at the bottom of the incline."

Darcel clicks on the large solid sphere, then clicks "roll." He pays attention to the time the sphere takes to roll down the incline. Then he clicks on the small solid sphere and repeats. Which of Darcell's statements about his observations of the time it takes for the two objects to reach the bottom is correct?

"The time for the large sphere to reach the bottom of the incline is equal to the time for the small sphere."

Now Darcel compares the times it takes for the large solid sphere and the large hollow sphere to roll down the incline. What can he say about his observations of the time it take for the two objects to reach the bottom?

"The time for the solid sphere to reach the bottom of the incline is less than the time for the hollow sphere."

Now Chandra clicks on the large hollow sphere, and clicks "roll." She pays attention to the time the sphere takes to roll down the incline. Then she clicks on the small hollow sphere and repeats. Which statement about her observations of the times is correct?

"The time it takes for the large sphere to reach the bottom of the incline is equal to the time for the small sphere."

Darcel runs the simulation for the large version of each object. What can he say in general about the energies of the objects in the simulation as they roll down the incline? (Select all that apply.)

"The total energy of the system remains constant and is equal to the gravitational potential energy of the object at the top of the incline." "The total kinetic energy of a rolling object is the sum of the rotational kinetic energy about the center of mass and the translational kinetic energy of the center of mass."

When the skater pulls her center of mass a distance r2 from the pole, how can Darcy determine the skater's new angular speed, given r1, r2, and ω1 ?

"Use v = ωr to rewrite the conservation of angular momentum equation, mv1r1 = mv2r2, in terms of ω and r. Solve this for ω2. "

From these expressions for ICM, how can he describe how the mass distribution about the axis of rotation of an object relates to the moment of inertia?

"When the mass is farther from the axis of rotation, the moment of inertia is greater."

Suppose that the height of the incline is h = 15.4 m. Find the speed at the bottom for each of the following objects. In a race, which object would win?

*On the simulation (translational speed) solid sphere: 14 m/s spherical shell: 13.5 m/s hoop: 12.3 m/s cylinder: 14.2 m/s Winner: solid sphere

Chandra reminds Darcel that the rotational kinetic energy of a rolling object is equal to KR = 12ICMω2, where I is the moment of inertia of the object, the translational kinetic energy is equal to KT = 12MvCM2, and for a circular-shaped object that rolls without slipping, vCM = Rω. Which of the following is the conservation of energy equation that Darcel can write down for the object-Earth system?

1/2(ICM/R^2 + M)vCM^2 = Mgh

An ice skater of mass m = 60 kg coasts at a speed of v = 0.86 m/s past a pole. At the distance of closest approach, her center of mass is r1 = 0.35 m from the pole. At that point she grabs hold of the pole. (A) What is the skater's angular speed when she first grabs the pole? (B) What is the skater's angular speed after she now pulls her center of mass to a distance of r2 = 0.11 m from the pole?

2.46 and 24.9

Darcel challenges Chandra to use the conservation of mechanical energy equation, 12 ICMR2 + MvCM2 = Mgh, to write an expression for the speed of a rolling object at the bottom of an incline of vertical height, h. Which is the correct expression?

vCM= [2gh/1+(Icm/MR^2)]^1/2

Darcy considers the situation in which the skater, of mass m, is moving directly toward the pole. What does Darcy say the skater's angular momentum relative to the pole will be if the skater were skating at speed v at the instant when she is distance r from the pole?

zero