What fraction of its kinetic energy is rotational?

AI Thread Summary
The discussion revolves around calculating the angular velocity and the fraction of kinetic energy that is rotational for a sphere rolling down an incline. The gravitational potential energy is equated to the sum of linear and rotational kinetic energy, leading to the use of formulas for rotational kinetic energy and the moment of inertia of a sphere. The participant successfully determines the angular velocity but seeks assistance with calculating the fraction of kinetic energy that is rotational. The conversation emphasizes the application of conservation of mechanical energy in pure rolling motion. Overall, the focus is on understanding the relationship between linear and rotational dynamics in this scenario.
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An 8.90-cm-diameter, 310 g sphere is released from rest at the top of a 2.00-m-long, 17 degree incline. It rolls, without slipping, to the bottom.

a) What is the sphere's angular velocity at the bottom of the incline?
b) What fraction of its kinetic energy is rotational?

If someone could help me out, it'd be great...I'm not exactly sure how to tackle this problem...
 
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To start off...

Gravitational Potential Energy = Linear Kinetic Energy + Rotational Kinetic Energy

...Rotational Kinetic Energy = 1/2 I w^2

w(its called omega) = v/r

I of Sphere = (2/5)mr^2...

..thusly...Rotational Kinetic Energy = (1/2)[(2/5)mr^2][v/r]^2...which simplifies to..
...(1/10)mv^2

Work from there...
 
anaylizing both movement rotational and linear we should consider a kinetic energy of the sum of both the linear of its center of mass and the rotational. Apply Conservation of Mechanical Energy because it's pure rolling motion (no slipping).
 
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I kind of get what you're saying but I'm still sort of lost?
 
Actually now I got part A...I just need part B...
Thanks for the help by the way
 
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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