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Gravitational Potential Energy

The gravitational potential energy of a body of mass m at a height h (h<<R, where R is the earth's radius) above the earth's surface could be obtained by simply finding the work done against the constant force (mg) in displacing the body through a distance h. Thus
PE = m g h
 
In this expression the zero of the potential energy is taken (for convenience) to be a point on the surface of the earth (h=0).
 
The choice of the reference point for the zero of PE is arbitrary and is chosen for convenience. It is immaterial where the reference point is chosen, for eventually we consider only the difference in potential energy. The increase in PE of a body when it is raised from the earth's surface (h=0) to a height h.
 
Consider a body of mass m at a distance (r > R) from the centre of the earth of mass M, R being the radius of the earth. The gravitational force between them is
 
It is evident that F is zero when r lends to infinity, i.e. no force is required to hold the body at an infinite distance from the earth. Hence, in this case, it is convenient to choose the reference point for zero of PE to be a point at infinity.
 
The gravitational potential energy of a mass m at a distance r from another mass M is defined as the amount of work done in bringing the mass m from infinity to a distance r.
 
The work done in moving m by a small amount dr along the line joining the centres of m and M is
dW = F dr
 
The total work done in moving m from infinity to a distance r is then given by
 
Thus  
 
The minus sign indicates that the potential energy is negative at any finite distance r. The reason is that we have arbitrarily chosen the zero of PE to correspond to infinite separation between the masses m and M. With this choice of the zero potential energy, PE is always negative and increases to zero at infinity.
 
The decrease in PE as the mass m moves in from infinity inwards the mass M implies that the work done by the gravitational force of mass M on mass m is positive. since the gravitational force is conservative, holds no matter what path the mass m follows in moving in from infinity to a position r.

We can compute the amount of work done against the gravitational force of mass M to move the mass m from a distance r1 (position A) to a distance r2 (position B) as shown in figure.

This work is given by


 

which represents the increase in PE as m moves from A to B. we can apply this general expression to the special case of a mass near the surface of the earth.

Potential energy near the surface of the earth


Suppose a body of mass m is raised from the surface of the earth to a height h. In this case, setting r1=R and r2 = (R
+ h),

we have (R is the earth's radius)

 

For a body near the surface of the earth, h<<R and (R + h) in the denominator can be replaced by R. Thus we have

PE = mgh

 

Note: The work done W12 measures the change in potential energy when the mass m is moved from r1 to r2 which may also be written as

Change in PE =

  1. If r1 < r2, W12 will be positive. This means that an outside agent has to do positive work (force and displacement in the same direction) in separating the two masses against their mutual gravitational attraction.
  2. On the other hand, if r1>r2, W12 will be negative. This means that the work has to be done by the gravitational force in bringing the two masses closer. When the two masses are released from a given separation, they accelerate towards each other because of mutual attraction.
The potential energy decreases resulting in an increase in kinetic energy. Thus when a stone is dropped, it accelerates downwards towards the earth at the rate of 9.8 ms-2. The earth must also accelerate upwards towards the stone. This acceleration of the earth is very small due to its enormous mass compared to that of the stone

Gravitational Potential and Gravitational field

Gravitational Potential
Just as we define electrical potential at a point in the electric field around a charge, we define gravitational potential at a point in the gravitational field around a mass.
 
The gravitational potential V, at a point in the gravitational field of a mass, is defined as the work done in moving a unit mass from infinity to that point. Thus if we set m=1 in Potential V due to mass M at a distance r from it, i.e.

 
The minus sign indicates that the potential at infinity (which is taken to be zero by convention) is higher than the potential close to mass M.
 
Gravitational Field  
Newton's law of universal gravitation enables us to calculate the field at any point set up by an array of point masses (or spheres). Force fields are usually represented graphically by sets of field lines and equipotentials.
 
A field line is a continuous line that is tangent to the field at every point along its length. For a single point mass, a field line is a straight line extending from the point mass to infinity.
 
For a set of masses or extended objects, field lines are curves that extend to infinity without crossing (except, as we will see, when they are constrained by symmetry). Equi potentials are perpendicular to the field at every point along their length. For a single point mass an equipotential is a circle, and for a set of point masses it is a closed curve.
The sample program plots field lines and equipotentials in the vicinity of identical spheres placed at the vertices of an equilateral triangle. We can start the curves anywhere we wish, but it is usual when dealing with spheres to start field lines at points equally spaced around each sphere, and plot out to the boundary in the direction opposite to the field.
We can calculate the field at a point, draw a line segment in the field direction, calculate the field at the next point, and continue the process as long as we wish, but we do not get a very accurate plot by this procedure. It is not easy to see if a field line is drawn accurately, but an equipotential that is not drawn accurately shows up clearly because it fails to close.
g = F/m.




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