Work–energy on an elliptical rail (JEE Adv 2014)

physics

Problem

Consider an elliptically shaped rail

PQ

in the vertical plane with

OP = 3\,\text{m}

and

OQ = 4\,\text{m}

. A block of mass

1\,\text{kg}

is pulled along the rail from

P

Q

with a force of

18\,\text{N}

, which is always parallel to line

PQ

. Assuming no frictional losses, the kinetic energy of the block when it reaches

Q

(n\times 10)

Joules. Find

n

(take

g = 10\,\text{m s}^{-2}

). *Figure:* a quarter-ellipse arc in a vertical plane.

O

is a right-angle corner at lower-left; a horizontal segment

OP = 3\,\text{m}

runs right to

P

(ground level, bottom-right); a vertical segment

OQ = 4\,\text{m}

runs up to

Q

(top-left). The rail bulges right, joining

P

Q

; the

18\,\text{N}

force is parallel to the straight chord

PQ

. Thus

P=(3,0)

Q=(0,4)

, and chord

PQ=\sqrt{3^2+4^2}=5\,\text{m}