What is the magnitude of the acceleration of the block when it is at the point #x= 0.24 m#, #y= 0.52m#? What is the direction of the acceleration of the block when it is at the point #x= 0.24m#, #y= 0.52m#? (See details).

A small block with mass
#0.0400 kg# is moving in the xy-plane. The net force on the block is described by the potential- energy function #U(x,y)= (5.90 J/m^2 )x^2-(3.65 J/m^3 )y^3#.

1 Answer
Feb 19, 2018

Since #xand y# are orthogonal to each other these can be treated independently. We also know that

#vecF=-gradU#

#:.x#-component of two dimensional force is

#F_x = -(delU)/(delx)#
#F_x = -del/(delx)[(5.90\ Jm^-2)x^2−(3.65\ Jm^-3)y^3]#
#F_x = -11.80x#

#x#-component of acceleration

#F_x=ma_x = -11.80x#
#0.0400a_x = -11.80x#
#=>a_x = -11.80/0.0400x#
#=>a_x = -295x#

At the desired point

#a_x = -295xx0.24#
#a_x = -70.8\ ms^-2#

Similarly #y#-component of force is

#F_y = -del/(dely)[(5.90\ Jm^-2)x^2−(3.65\ Jm^-3)y^3]#
#F_y = 10.95y^2#

#y#-component of acceleration

#F_y=ma_ = 10.95y^2#
#0.0400a_y = 10.95y^2#
#=>a_y = 10.95/0.0400y^2#
#=>a_y =27.375y^2 #

At the desired point

#a_y = 27.375xx(0.52)^2#
#a_y = 7.4022\ ms^-2#

Now #|veca| = sqrt[a_x^2 + a_y^2]#

#|veca| = sqrt[(-70.8)^2 + (7.4022)^2]#
#|veca| =71.2\ ms^-2#

If #theta# is the angle made by acceleration with #x#-axis at the desired point then

#tantheta = (a_y)/(a_x)#

Inserting calculated values

#tantheta = (7.4022)/(-70.8)#, (#2nd# quadrant)
#=>theta=174^@#