Save all ants from Monster Cube !

Classical Mechanics Level 5

A solid Cube of side Length L and Mass M is moving on horizontal smooth table with velocity ${ V }_{ o }$ . ( Side View is shown in figure )

An ridge ( Considered it as a Small stone ) is fixed at the corner of table.( Red coloured,as shown )

And If below the table , there are infinitely many ants are eating their food !

Then Find The Maximum velocity of Cube so that No ant get Hurt !

If Maximum Velocity can be expressed as :

${ V }_{ o,max }\quad =\quad \sqrt { \cfrac { a(\sqrt { b } -c) }{ d } gL }$ .

Then Find value of : " a + b + c + d " ? ?

Details And Assumptions

$\bullet$ gcd(a,d) = 1 & gcd(a,c) = 1 & gcd (c,d)=1 & 'b' is square free integer.

$\bullet$ Ant's are infinite in numbers and present at everywhere on the ground .

$\bullet$ Assume If Cube Falls then ant's get Hurt .

Source : This is modified Form of an Similar Problem, which i got from my friend , we together make made this situation .

2 solutions

Abhishek Sharma
Nov 29, 2014

Conserving the angular momentum about ridge,

$\frac { MV_{ o }L }{ 2 } =\frac { 2M{ L }^{ 2 } }{ 3 } \omega$

Here $\omega$ is the final angular velocity.

To save the ants the cube must not fall down, therefore it rotates till its diagonal is vertical,

Conserving energy,

$\frac { 1 }{ 2 } \frac { 2M{ L }^{ 2 } }{ 3 } { \omega }^{ 2 }=Mg\frac { L }{ 2 } (\sqrt { 2 } -1)$

Solving the two equation we obtain,

${ V }_{ 0 }=\sqrt { \frac { 8(\sqrt { 2 } -1) }{ 3 } } gL$

same way!!!!! nice solution.... and a great question

A Former Brilliant Member - 3 years, 5 months ago

A Former Brilliant Member
Dec 31, 2016

first conserve angular momentum then energy as some energy will be dissipated as this is no pure elastic collision ! $\ddot \smile$