Calculating Gravity Using Angular Speed and Weight

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To calculate gravity using angular speed and weight, the equations for tension at the top and bottom of the swing are established. The initial approach of adding the equations to find Mg is incorrect due to neglecting the angle and energy conservation principles. Instead, kinetic energy at the bottom and potential energy at the top should be considered, as the ball's kinetic energy decreases while potential energy increases during the swing. By applying conservation of energy, the relationship between kinetic and potential energy can be used to derive the force of gravity. This method provides a more accurate calculation of gravity in the context of the swinging ball.
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Homework Statement


Suppose I tie a ball of mass M to a string of length L and I swing the ball on the string in a vertical fashion. If the speed at the top is V1 and the speed at the bottom is V2.
Would I have:

Let T = tension in string
(top) -T - Mg = -MV1^2/R
(bottom) T - Mg = MV2^2/R

So to find Mg (the force of gravity), I would add the 2 equations to get:

-2Mg = MV2^2/R - MV1^2/R.

But somehow that is wrong?
Thanks.
 
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When the ball is moving through the bottom with velocity V2 its kinetic energy = 1/2*m*v2^2. As it moves towards the top, its kE decreases but its PE increases. At the top, KE = 1/2*m*v1^2 and rise in PE = 2mgR.Apply conservation of energy to find mg.
 
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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