The gravitational field in a region is given by the equation E = (5i + 12j) N/kg. If a particle of mass 2 kg is moved from the origin to the point (12m, 5m) in this region, the change in gravitational potential energy is

Explanation

Change in position vector:The particle moves from the origin (0, 0) to the point (12m, 5m).The change in position vector (Δr) is: Δr = (12i + 5j) mWork done by the gravitational field:The change in gravitational potential energy (ΔPE) is equal to the negative of the work done by the gravitational field (W) on the particle: ΔPE = -WRelationship between work and force:The work done by a constant force (F) acting on an object undergoing a displacement (Δr) is: W = F ⋅ ΔrGravitational force:The gravitational force acting on the particle is the product of its mass (m) and the gravitational field (E): F = mEApplying the concepts:Substitute the expressions for F and Δr into the work equation: W = (mE) ⋅ (12i + 5j) mSubstitute the given values for mass (m = 2 kg) and the gravitational field (E = 5i + 12j N/kg): W = (2 kg) ⋅ ((5i + 12j) N/kg) ⋅ (12i + 5j) mPerform the vector dot product: W = 2 kg * (60 N + 60 N) = -240 J (since the negative sign indicates work done by the field)Therefore, the change in gravitational potential energy (ΔPE) is -240 J, which corresponds to option b.