Question

123. A ball of mass m falls symmetrically on asset of three hemispheres placed in contact with
each other on smooth horizontal ground from a height h. Mass of each hemisphere is m/2
and radius of each body is R.
////// ///
side view
Top view
Neglecting any rebounding of hemisphere .If sphere comes to rest after the simultaneous
collision with three hemispheres then ratio of KE of sphere before collision and KE of any
hemisphere after collision is P then find the value of P
Key: 9​

Answer 1

Answer:

the ratio of kinetic energy of falling sphere to each hemisphere is given as 9

Explanation:

As we know that the base of three hemisphere is connected such that their centers will make an equilateral triangle

Now the distance of centroid from its corner is given as

$r = \frac{2R}{\sqrt3}$

now we know that due to collision the impulse on the falling ball will stop it

so the direction of impulse from horizontal is given as

$cos\phi = \frac{2R/\sqrt3}{2R} = \frac{1}{\sqrt3}$

now net impulse in vertically upward direction on the falling ball is given as

$3J sin\phi = mv$

so we have

$J = \frac{mv}{3sin\phi}$

Now same impulse in opposite direction will act on each hemisphere

So here net horizontal impulse on each hemisphere is given as

$J cos\phi = \frac{m}{2}(v')$

$v' = \frac{2Jcos\phi}{m}$

$v' = \frac{2cos\phi}{m} (\frac{mv}{3sin\phi})$

$v' = \frac{2v}{3 tan\phi}$

now the ratio of kinetic energy of falling sphere to each hemisphere is given as

$R = \frac{1/2 mv^2}{1/2(m/2)(\frac{2v}{3tan\phi})^2}$

$R = \frac{9tan^2\phi}{2}$

Here we know that

$cos\phi = \frac{1}{\sqrt3}$

so we have

$tan\phi = \sqrt2$

so we have

$R = 9$

#Learn

Topic : impulse

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