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A ball moves ...

Question

$A$ ball moves over a fixed track as shown in the figure. From $A$ to $B$ the ball rolls without slipping. Surface BC is frictionless. $K_{A}, K_{B}$ and $K_{c}$ are kinetic energies of the ball at $A, B$ and $C$ , respectively. Then

A

h_{A} > h_{C}; K_{B} > K_{C}

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B

h_{A} > h_{C}; K_{C} > K_{A}

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C

h_{A} = h_{C}; K_{B} = K_{C}

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D

h_{A} < h_{C}; K_{B} > K_{C}

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Solution

The correct options are
A $h_{A} > h_{C}; K_{C} > K_{A}$
B $h_{A} > h_{C}; K_{B} > K_{C}$
D $h_{A} < h_{C}; K_{B} > K_{C}$
$E_{A} = m g h_{A} + K_{A}$
$E_{B} = K_{B}$
$E_{c} = m g h_{c} + K_{C}$
Using conservation of energy
$E_{A} = E_{B} = E_{c}$
$K_{B} > K_{C}$
$K_{B} > K_{A}$
$M g (h_{A} - h_{c}) + (K_{A} - K_{c}) = 0$
$\Rightarrow h_{A} - h_{c} = \frac{K_{C} - K_{A}}{M g}$

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