Assignment V
(Work and Energy)
Q1) A particle on the x-y plane is acted on by a variable force given by

F(x, y)  A (xy 2 î  x 2 y ĵ )
It is taken from the point (0, 0) to the point (1, 1) under the action of the above force along the following
different paths. Calculate the work done on the particle in each case.
i) The parabolic path: y  x 2 ii) The parabolic path: x  y 2 ii) The straight line: y  x iv) The line from
(0, 0) to (1, 0) along the x-axis, followed by the line from (1, 0) to (1, 1) along the y-axis v) The line from
(0, 0) to (0, ½)) along the y-axis followed by the straight line from (0, ½) to (1, 1). (Ans: All of them give ½)
Is the force likely to be conservative or non-conservative?
Q2) Consider the variable force of (Q1) above. The particle is taken from (0, 0) to the point (1, 1) along an
arbitrary path: y  f ( x ) . By using your knowledge of basic integration, show that the work done does
not depend upon the exact form of the function f ( x ) . Connect this result to that of Q1. Can you say with
certainty that the force is conservative now?
Q3) Mass m whirls on a frictionless table, held to a circular motion by a string which passes through a hole
in the table. The string is slowly pulled through the hole vertically down at a constant rate so that the
radius of the circle changes from  1 to  2 .
a) Write down the equations of motion of the particle in polar co-ordinates, and using them, find the force
required to pull the string as a function of the radius of the circle.
Ans:
m 1412 m 4222

r3
r3
b) Calculate the work done by this force in pulling the string. Ans:
1
m  2222  1212 
2
c) Show that the work done is equal to the change of kinetic energy of the mass.
Q4) Problem no. 4.21 of the text book (kleppner and Kolenkow).
P  v 0 (gy  v 02 ) ;
dE 1 3
 v 0  gv 0 y
dt 2
Q5) Problem no. 4.23 of the text book.
Ans: 9h
Q6) Problem no. 4.24 of the text book. Ans: 1/6
Q7) Problem no. 4.26 of the text book. Ans: 1/3
*************
Ans: F( y)  (gy  v 02 ) ;