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Question

A ball of mass $$5kg$$ is suspended from a chord of length $$l=0.5m$$ is attached to a cart $$A$$ of mass $$2.5kg$$, which slides freely on a frictionless horizontal track as shown in figure. The ball given an initial horizontal velocity $$1m/s$$ relative to the $$ground$$ while the cart is at rest.
Find
(a) The velocity of the ball when string makes $$minimum$$ angle with vertical
(b) The maximum vertical distance through which the ball will rise.
694406_95f1d8f624284a3cbe47970e3b9a0699.png


Solution

(a) At time of horizontal velocity of ball is same of cart velocity .
By linear momentum conservation 
$$5x_{ 1 }=5xv+2.5v\\ \cfrac { 5 }{ 7.5 } =v\\ v=\cfrac { 2 }{ 3 } m/s$$
(b) By energy conservation theorem 
$$\cfrac { 1 }{ 2 } 5(1)^{ 2 }=\cfrac { 1 }{ 2 } 5\times { v }^{ 2 }+\cfrac { 1 }{ 2 } 2.5\times { v }^{ 2 }+5g(h)\\ 5=\cfrac { 20 }{ 9 } +\cfrac { 10 }{ 9 } +10gh\\ 5-\cfrac { 30 }{ 9 } =10\times 10h\\ \cfrac { 15 }{ 9 } =100h\\ h=\cfrac { 5 }{ 300 } m\\ =\cfrac { 5 }{ 3 } cm$$

961077_694406_ans_5f40dd55c53e4a43907eaa0d5744b884.png

Physics

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