Time Dilation: Showing Relation b/w Intervals

AI Thread Summary
The discussion focuses on deriving the relationship between time intervals for a stationary clock and a moving clock on a rotating disk using time dilation in special relativity. The user seeks to demonstrate that the difference in time intervals is proportional to the square of the radius and angular speed, expressed as (△ t'-△ t)/(△ t) ~ r²w²/2c². A concern is raised about obtaining a minus sign in the calculation, which seems to confuse the user. Participants emphasize the importance of substituting the linear speed, v = rω, into the time dilation formula to clarify the derivation. The conversation underscores the application of special relativity principles in a rotating reference frame.
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For a disk that rotates with angular speed w, there is a clock at the center(at rest), adn one at distance r, with speed rw,. I need to show that frome time dilation in special relativity, time inervals (△ t) for the clock at rest, and (△ t') for the moving clock are related by
(△ t'-△ t)/(delta t) ~ r²w²/2c²

I got a minus sign in front, by using the time dilation.
 
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I don't see what the difficulty is? You know the formula for time dilation using linear speed v, don't you?
You say that the speed is rω. Plug that into the formula.
 
Thread 'Variable mass system : water sprayed into a moving container'

Thread 'Variable mass system : water sprayed into a moving container'

Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
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