An object rolling along a hemispherical bowl

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The discussion focuses on a physics problem involving a solid sphere rolling inside a hemispherical bowl. The potential energy lost by the sphere when it reaches the bottom can be calculated using the formula PE = mgh, where the height is determined by the angle of release. To find the translational and angular velocities at the bottom, the total energy conservation principle must be applied, equating initial potential energy to the sum of translational and rotational kinetic energies. A relationship between angular and linear velocities is necessary due to the rolling without slipping condition. The conversation emphasizes the importance of correctly applying energy conservation and the relationships between different forms of energy.
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Homework Statement



A uniform solid sphere (moment of inertia = 2/5 mr^2) of mass 1.5kg and radius r = 0.473m, is placed on the inside surface of a hemispherical bowl of radius R = 2.77m. The sphere is released from rest at an angle of 66.9 degrees from the vertical and rolls without slipping.
a) How much potential energy has the sphere lost when it reaches the bottom of the bowl?
b) What is the translational velocity of the sphere when it reaches the bottom of the bowl?
c) What is the angular velocity of the sphere when it reaches the bottom of the bowl?

Homework Equations



KE = 1/2 Iw^2. PE = mgh.

The Attempt at a Solution



For part a, do I just do (1.5kg)(9.8)(2.77sin66.9)?
After I find PE, I will just equate it with 1/2 mv^2 to find the translational velocity and 1/2 Iw^2 to find the angular velocity?

Any help is appreciated. Thank you!
 
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Momentum09 said:
For part a, do I just do (1.5kg)(9.8)(2.77sin66.9)?
After I find PE, I will just equate it with 1/2 mv^2 to find the translational velocity and 1/2 Iw^2 to find the angular velocity?

Any help is appreciated. Thank you!
part a looks fine. part b is not fine. the total energy initially is PE + KE_trans + KE_rot. this must equal the total final energy PE_f + KE_trans_f+KE_rot_f. Part a is the value of PE and PE_f=0.

thus set PE = KE_trans_f + KE_rot_f

not equal to each individually.
 
And find a relation between the angular velocity and the linear velocity, since it 'rolls without slipping'.
 
I got it. Thank you so much!
 
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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