An ice skater spins with her arms extended and then pulls her arms in and spins faster. Which statement is correct?
A. Her kinetic energy of rotation increases because of the work she does to pull her arms in.
B. Her kinetic energy of rotation does not change because, by conservation of angular momentum, the fraction by which her angular velocity increases is the same as the fraction by which her rotational inertia decreases.
C. Her kinetic energy of rotation decreases because of the decrease in her rotational inertia; she does loses energy because she gradually gets tired.
I'm not sure of this one because, I know for sure that the angular rotation increases and the rotational inertia decreases, but does the kinetic energy changes or is it conserved?
A solid sphere rolls without slipping down an incline, starting from rest. At the same time, a box starts from rest at the same altitude and slides down the same incline, with negligible friction. Which object arrives at the bottom first?
A. Both arrive at the same time.
B. It is impossible to determine.
C. The box arrives first.
D. The solid sphere arrives first.
I think it's A.
Please help, I'm not sure of these.
first one: KE remains the same.
second: The inertia of the sphere causes it to absorb energy as rotational, the box all converts to translational. The box arrives first.
Thanks... now I understand, but when I choose B in the first one it says that it's wrong...
For the ice skater scenario, the correct statement is B. Her kinetic energy of rotation does not change because, by conservation of angular momentum, the fraction by which her angular velocity increases is the same as the fraction by which her rotational inertia decreases. This means that as the ice skater pulls her arms in, her rotational inertia decreases, allowing her to spin faster, but her kinetic energy of rotation remains constant.
For the scenario with the solid sphere and the box sliding down an incline, the correct answer is D. The solid sphere arrives first. This is due to the concept of rotational inertia. The solid sphere has a greater rotational inertia compared to the box sliding down the incline. As a result, the solid sphere takes longer to accelerate and roll down the incline, while the box slides down relatively faster.
For the first question about the ice skater, let's break down each option and see which one is correct:
A. Her kinetic energy of rotation increases because of the work she does to pull her arms in.
To determine if this statement is correct, we need to understand the concept of work and its relationship to kinetic energy. Work is defined as the product of force and displacement. In this case, as the ice skater pulls her arms in, she is applying a force to change the distribution of her mass closer to the axis of rotation. This force does work, and as a result, the rotational kinetic energy increases. So, option A is correct.
B. Her kinetic energy of rotation does not change because, by conservation of angular momentum, the fraction by which her angular velocity increases is the same as the fraction by which her rotational inertia decreases.
This statement refers to the conservation of angular momentum, which states that the total angular momentum remains constant unless acted upon by an external torque. However, this does not necessarily mean that the kinetic energy of rotation remains constant. As the skater pulls her arms in, her rotational inertia decreases, causing an increase in her angular velocity. This change in angular velocity results in a change in kinetic energy. So, option B is incorrect.
C. Her kinetic energy of rotation decreases because of the decrease in her rotational inertia; she loses energy because she gradually gets tired.
This statement suggests that the skater's kinetic energy of rotation decreases due to the decrease in rotational inertia. However, decreasing the rotational inertia by pulling her arms in actually results in an increase in kinetic energy. Additionally, the mention of the skater getting tired is unrelated to the change in kinetic energy. So, option C is incorrect.
Therefore, the correct answer is A. Her kinetic energy of rotation increases because of the work she does to pull her arms in.
Moving on to the second question about the solid sphere and the box going down an incline:
When a solid sphere rolls without slipping down an incline, it has both translational and rotational kinetic energy. On the other hand, a box sliding down the incline only has translational kinetic energy. The key here is that both objects start from rest at the same altitude.
Since the sphere has both translational and rotational kinetic energy, its total kinetic energy is shared between these two forms. As the rolling sphere starts to move, some of the initial potential energy is used to initiate both rotational and translational motion.
The box, on the other hand, only has translational kinetic energy. It doesn't need to transfer any energy to initiate rotational motion because it can freely slide without friction.
Due to the sharing of energy between rotational and translational motion, the sphere takes longer to reach the bottom compared to the box, which only has translational kinetic energy. Therefore, option C is correct, and the box arrives first.
I hope these explanations help clarify your understanding!