The gravitational potential in a certain region of space is given by the expression V = -x²y³ + 3z² , where V is in J/kg and x, y, z are in meters. What is the magnitude of the gravitational field intensity at the point (1, 1, -1) m ?

Step-by-step solution

The gravitational field intensity E is related to the gravitational potential V by the relation: E = - ( V x i + V y j + V z k ) Given V = -x²y³ + 3z² , we find the partial derivatives: V x = -2xy³ V y = -3x²y² V z = 6z Evaluating these at the point (1, 1, -1) : V x = -2(1)(1)³ = -2 V y = -3(1)²(1)² = -3 V z = 6(-1) = -6 Substituting these values into the expression for E : E = -(-2 i - 3 j - 6 k ) = 2 i + 3 j + 6 k The magnitude of the gravitational field intensity is: | E | = 2² + 3² + 6² = 4 + 9 + 36 = 49 = 7 N/