The potential energy for a conservative force field is given by U(x, y) = cos(x + y). Find the force on a particle at the point (0, pi/4).

-1/sqrt(2) * (i-hat + j-hat)
1/sqrt(2) * (i-hat + j-hat)
(1/2) * i-hat + (sqrt(3)/2) * j-hat
(1/2) * i-hat - (sqrt(3)/2) * j-hat

Correct answer: 1/sqrt(2) * (i-hat + j-hat)

Solution

F = -grad(U). dU/dx = -sin(x+y), dU/dy = -sin(x+y). So Fₓ = -(-sin(x+y)) = sin(x+y), F_y = sin(x+y). At (0, pi/4): sin(0 + pi/4) = sin(pi/4) = 1/sqrt(2). F = (1/sqrt(2)) i-hat + (1/sqrt(2)) j-hat = (1/sqrt(2))(i-hat + j-hat).

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