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#.

Answer 1
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^@#
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Answer 2

To determine the magnitude and direction of the acceleration of the block at the point (0.24 m, 0.52 m), you would need additional information such as the forces acting on the block, its mass, and potentially its velocity or acceleration components at that point. Without this information, it's not possible to calculate the magnitude and direction of the acceleration accurately.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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