The gravitational field in a region is given by . Find the work done in shifting a particle of mass 3 kg from origin (0, 0) to a point (3, 7).

The problem involves calculating the work done in moving a particle in a gravitational field. The work done is given by the integral of the force over the distance moved. In this case, the force is given by the product of the mass of the particle and the gravitational field.

The gravitational field is not specified in the problem, but let's denote it by g. The force on the particle is then F = m*g, where m is the mass of the particle.

The work done in moving the particle from the origin to the point (3, 7) is given by the integral of F over this path. In Cartesian coordinates, this is

W = ∫ F dx = ∫ m*g dx from (0,0) to (3,7).

This is a line integral, which can be computed as follows:

W = m * ∫ g dx from (0,0) to (3,7).

The limits of integration are the initial and final positions of the particle. The integral of g dx is the change in the gravitational potential energy of the particle, which is given by mgh, where h is the height of the particle above the origin.

In this case, the height of the particle changes from 0 to 7, so

W = m * g * 7 = 3 kg * g * 7.

The value of g is not given in the problem, but if we assume that the gravitational field is that of the Earth, then g = 9.8 m/s^2, and

W = 3 kg * 9.8 m/s^2 * 7 = 205.8 Joules.

So the work done in shifting the particle from the origin to the point (3, 7) is 205.8 Joules.