Physics, 12.03.2020 06:10 iiChantxx

A cylinder with moment of inertia I1 rotates with angular speed ω0 about a frictionless vertical axle. A second cylinder, with moment of inertia I , initially not rotating, drops onto the first cylinder. Since the surfaces are rough, the two eventually reach the same angular speed ωf.

a. Calculate ωf. (Use I1 for I1, I2 for I2, and omega_i for ωi, as necessary.)

b. Show that the kinetic energy of the system decreases in this interaction by calculating the ratio of the final to initial rotational energy. Express your answer in terms of ωi.

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Answer from: pr173418

a) The final angular speed of the system is $\omega_{f} = \frac{I_{1}}{I_{1}+I_{2}}\cdot \omega_{i}$ radians per second.

b) Since r < 1 , then the kinetic energy of the system decreases due to the interaction.

a) The coupling between the two cylinders can be modelled as a entirely inelastic collision, such phenomenon is modelled after the principle of angular momentum conservation, that is:

$I_{1}\cdot \omega_{i} = (I_{1}+I_{2})\cdot \omega_{f}$ (1)

Where:

$I_{1}$ - Moment of inertia of the moving cylinder, in kilogram-square meters. $I_{2}$ - Moment of inertia of the resting cylinder, in kilogram-square meters. $\omega_{i}$ - Initial angular speed of the moving cylinder, in radians per second. $\omega_{f}$ - Final angular speed of the resulting system, in radians per second.

Then the final angular speed of the system is:

$\omega_{f} = \frac{I_{1}}{I_{1}+I_{2}}\cdot \omega_{i}$

The final angular speed of the system is $\omega_{f} = \frac{I_{1}}{I_{1}+I_{2}}\cdot \omega_{i}$ radians per second.

b) The system experiment a decrease in total energy if the final kinetic energy to intial kinetic energy ratio is:

$r = \frac{\frac{1}{2}\cdot (I_{1}+I_{2})\cdot \omega_{f}^{2} }{\frac{1}{2}\cdot I_{1}\cdot \omega_{i}^{2}}$

$r = \left[\frac{(I_{1}+I_{2})}{I_{1}}\right]\cdot \left(\frac{I_{1}}{I_{1}+I_{2}} \right)^{2}$

$r = \frac{I_{1}}{I_{1}+I_{2}}$

Since r < 1 , then the kinetic energy of the system decreases due to the interaction.

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Answer from: ellycleland16

Part(a): The final angular velocity is $\omega_{f} = \dfrac{I_{1}\omega_{i}}{(I_{1} + I_{2})}$

Part(b): The ratio of the rotational energies is $\dfrac{k_{f}}{k_{i}}& = \dfrac{I_{1}}{(I_{1} + I_{2})}$ ,showing the the energy of th system will decrease.

Explanation:

Part(a):

If '' be the moment of inertia of an object and ' $\omega$ ' be its angular velocity then the angular momentum '' of the object can be written as

$L = I \omega$

If ' $I_{1}$ ' and ' $I_{2}$ ' be the moment of inertia of the two cylinders and ' $\omega_{1}$ ' and ' $\omega_{2}$ ' be the initial angular velocity of the cylinders and ' $\omega_{1}{'}$ ' and ' $\omega_{2}^'}$ ' be their respective final angular velocity, then from conservation of angular momentum,

$I_{1} \omega_{1} + I_{2} \omega_{2} = I_{1} \omega_{1}^{'} + I_{2} \omega_{2}^{'}$

Given, $\omega_{1} = \omega_{i},~\omega_{2} = 0,~\omega_{1}^{'} = \omega_{2}^{'} = \omega_{f}$ . From the above expression

$&& I_{1} \omega_{i} = (I_{1} + I_{2}) \omega_{f}\\&or,& \omega_{f} = \dfrac{I_{1}\omega_{i}}{(I_{1} + I_{2})}$

Part(b):

Initial kinetic energy

$K_{i} = \dfrac{1}{2} I_{1} \omega_{i}^{2}$

and Final kinetic energy

$K_{f} = \dfrac{1}{2}(I_{1} + I_{2}) \omega_{f}^{2}$

Substituting the value of $\omega_{f}$ ,

$&& K_{f} = \dfrac{1}{2}(I_{1} + I_{2})\dfrac{I_{1}^{2}\omega_{i}^{2}}{(I_{1} + I_{2})^{2}} = \dfrac{1}{(I_{1} + I_{2})} \dfrac{1}{2}I_{1}\omega_{i}^{2} = \dfrac{1}{(I_{1} + I_{2})} K_{i}\\&\dfrac{k_{f}}{k_{i}}& = \dfrac{I_{1}}{(I_{1} + I_{2})}$

The above expression shows that the ebergy of the system will decrease.

Answer from: Quest

relativism, roughly put, is the view that truth and falsity, right and wrong, standards of reasoning, and procedures of justification are products of differing conventions and frameworks of assessment and that their authority is confined to the context giving rise to them. more precisely, “relativism” covers views which maintain that—at a high level of abstraction—at least some class of things have the properties they have (e.g., beautiful, morally good, epistemically justified) not simpliciter, but only relative to a given framework of assessment (e.g., local cultural norms, individual standards), and correspondingly, that the truth of claims attributing these properties holds only once the relevant framework of assessment is specified or supplied. relativists characteristically insist, furthermore, that if something is only relatively so, then there can be no framework-independent vantage point from which the matter of whether the thing in question is so can be established.

relativism has been, in its various guises, both one of the most popular and most reviled philosophical doctrines of our time. defenders see it as a harbinger of tolerance and the only ethical and epistemic stance worthy of the open-minded and tolerant. detractors dismiss it for its alleged incoherence and uncritical intellectual permissiveness. debates about relativism permeate the whole spectrum of philosophical sub-disciplines. from ethics to epistemology, science to religion, political theory to ontology, theories of meaning and even logic, philosophy has felt the need to respond to this heady and seemingly subversive idea. discussions of relativism often also invoke considerations relevant to the very nature and methodology of philosophy and to the division between the so-called “analytic and continental” camps in philosophy. and yet, despite a long history of debate going back to plato and an increasingly large body of writing, it is still difficult to come to an agreed definition of what, at its core, relativism is, and what philosophical import it has. this entry attempts to provide a broad account of the many ways in which “relativism” has been defined, explained, defended and criticized.

explanation:

Answer from: Quest

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A cylinder with moment of inertia I1 rotates with angular speed ω0 about a frictionless vertical axl...