Potential Problems

Potential Problems

Potential energy of a body very close to the Earth's surface is defined as the work done to bring the body of mass m, from ground to a certain height h. It is calculated using work = force x distance, and is given by E = mgh.

A body can also be very far from the Earth's surface, or in the field of a sphere (or small mass). The potential energy of the body at a point P, is then defined as the work done to bring the body from infinity to P. How do we calculate this work?

The problem is that the force F changes with distance d from the sphere. We cannot simply use work W = Fd. This only works if the force is the same. The way to do it is to divide the distance into many small intervals. Within each interval d, the force is approximately constant. We find the work done in each interval using W = Fd, and add up the work done in all intervals.

In order to sum over the large number of small intervals, we use integration. The force is given by Newton's law of gravitation, F = GMm/r², where M is the mass of the sphere producing the field, m is the mass of a small body in the field, and r the distance from centre of the sphere to the small body. The interval of distance is a small interval of r.

Carrying out the integration gives the potential energy as U = -GMm/r.

The minus sign happens because the gravitational force is attractive. So when we bring the body m from infinity, work is done by the force instead of by us.

The potential Φ is the potential energy per unit mass. So Φ = U/m. Subsituting U from above, we get Φ = -GM/r.

Exercise. The gravitational potential energy of a body at point A is -10 J, and that at point B is -15 J. Find the work done to bring the body from B to A. Answer. The work done to bring the body from B to A, is equal to the work done to bring it from B to infinity and back to A. From B to infinity, the work is negative of -15 J. From infinity to A is -10 J So work done from B to A is (-10) - (-15) = 5 J.

Exercise. Use U = -GMm/r to find the potential energy on the Earth's surface for a body of mass 1 kg. Find the potential energy 1 m above the Earth's surface. Hence find the work done to bring the body from the surface to 1 m above. Compare with the result using mgh. Given that the radius of the Earth is 6400 km, and the mass of the Earth is 6 x 10²⁴ kg. Comment.
Answer. Let m be the mass of the body, M the mass of the Earth, and R the radius of the Earth. The potential energy on the surface is U₁ = -GMm/R. Putting in the values, we find U₁ = -62784000.00. The potential energy 1 m above the surface is U₂ = -GMm/(R+1). Putting in the values, we find U₂ = -62783990.19. The work done from surface up 1 m is U₂ - U₁ = 9.81 J. Using mgh, we find 1 x 9.81 x 1 = 9.81 J, the same. This shows that near the surface of the Earth mgh is good enough.

Note that practical problems often involve using the potential energy formula to find the work done to bring a body from one place to another, like in the examples above.

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