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If U(xyz)= (xy+yz+zx), then find force act on particle at point (1m, -1m, 2m).
Asked by patra04011965 | 08 Oct, 2019, 19:29: PM
answered-by-expert Expert Answer
Potential energy function, U(x,y,z) = ( x y + y z + z x )  ...............(1)
 
Force = - ∂U/∂r = - ( ∂U/∂x ) begin mathsize 14px style i with hat on top end style- ( ∂U/∂y ) begin mathsize 14px style j with hat on top end style- ( ∂U/∂z ) begin mathsize 14px style k with hat on top end style .....................(2)
where begin mathsize 14px style i with hat on top end style , begin mathsize 14px style j with hat on top end style and begin mathsize 14px style k with hat on top end style are unit vectors along x, y and z direction.
∂U/∂x = y + z  ; ∂U/∂y =  z + x ; ∂U/∂z = x + y
 
Hence force from eqn.(2),   Force = - (y+z) begin mathsize 14px style i with hat on top end style  - (z+x) begin mathsize 14px style j with hat on top end style - (x+y) begin mathsize 14px style k with hat on top end style ....................(3)
Force at ( 1 m , -1 m , 2 m ) is given by,  - (-1+2) begin mathsize 14px style i with hat on top end style  - ( 2 + 1) begin mathsize 14px style j with hat on top end style - (1 -1) begin mathsize 14px style k with hat on top end style =  - begin mathsize 14px style i with hat on top end style  - 3 begin mathsize 14px style j with hat on top end style  ....................(4)
magnitude of force = ( 12 + 32 )1/2 = 3.16 N
 
direction of force is such that force vector makes 72º angle with -ve x-axis
Answered by Thiyagarajan K | 08 Oct, 2019, 23:21: PM

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