Calculating KE in rotational motion

AI Thread Summary
To calculate the total kinetic energy of a rolling solid disk, both translational and rotational kinetic energy must be considered. The formula for kinetic energy combines translational KE, given by (1/2)mv^2, and rotational KE, represented as (1/2)Iw^2, where I is the moment of inertia and w is the angular velocity. The moment of inertia for a solid disk is I = (1/2)Mr^2, and the relationship between translational speed and angular speed is crucial for accurate calculations. Ignoring rotational KE leads to an incomplete understanding of the disk's motion. Therefore, the total kinetic energy must account for both forms of energy to provide an accurate result.
MinaHany
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Homework Statement


A 20kg solid disk (I=1/2Mr^2) rolls on a horizontal surface at the rate of 4.0m/s
Calculate its total kinetic energy


The Attempt at a Solution


I think that simply equating the KE to (0.5)(m)(v^2) would be a wrong solution because then I would not use the moment of inertia given in the question, although I don't know why it is wrong.


Thank you.
 
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MinaHany said:
I think that simply equating the KE to (0.5)(m)(v^2) would be a wrong solution because then I would not use the moment of inertia given in the question, although I don't know why it is wrong.
It's wrong because it ignores the rotational KE. The rolling disk has both translational and rotational KE. What's the formula for rotational KE? Hint: How does translational speed relate to angular speed? (Assume it rolls without slipping.)
 
Use:

KE = KEtranslational + KErotational = (1/2)mvcm2 + (1/2)Iw2

vcm : velocity at the center of mass[sorry Doc Al, I didn't see you post]
 
Last edited:
Thank you Doc AL and lewando..
Your replies made the idea clear for me.
 
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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