Given in Fig. 6.11 are examples of some potential energy functions in one dimension. The total energy of the particle is indicated by a cross on the ordinate axis. In each case, specify the regions, if any, in which the particle cannot be found for the given energy. Also, indicate the minimum total energy the particle must have in each case. Think of simple physical contexts for which these potential energy shapes are relevant.
a)
The graph of potential energy is given below.
The kinetic energy of a body cannot be negative because it is a positive quantity.
From the above graph it can be observed that V 0 > 0 for the region x > a .
The kinetic energy of the system is given as,
K . E = E − P . E
From the graph, it can be observed that the potential energy is greater than total energy for the given region so the kinetic energy will be negative and it is not possible. The particle will not exist in that region.
Thus, the particle will not exist in this region and the minimum total energy of the particle will be zero.
b)
The graph of potential energy is given below.
From the above graph, it can be observed that the potential energy is always ( − ∞ > x > + ∞ ) greater than the total energy. So, the kinetic energy will be negative and it is not possible for the particle to be exists in this region.
c)
The graph of potential energy is given as,
From the graph, it is clear that the potential energy is − V 1 for the region x > a and x < b and potential energy is V 0 for the region x < a and x > b .
The condition for the kinetic energy to be positive is satisfied only for the region x > a and x < b
K . E = E − P . E = E − ( − V 1 ) = E + V 1
Thus, the minimum total energy is − V 1 and the region is region x > a and x < b .
d)
The graph of potential energy is given as,
The condition for kinetic energy to be positive, the region − b 2 < x < b 2 and − a 2 < x < a 2 will be satisfied region.
The minimum potential energy is − V 1 .
The kinetic energy of the system is given as,
K . E = E − P . E = E − ( − V 1 ) = E + V 1
Thus, the minimum total energy is − V 1 and the region is − b 2 < x < b 2 and − a 2 < x < a 2 .
a)
The graph of potential energy is given below.
The kinetic energy of a body cannot be negative because it is a positive quantity.
From the above graph it can be observed that
The kinetic energy of the system is given as,
From the graph, it can be observed that the potential energy is greater than total energy for the given region so the kinetic energy will be negative and it is not possible. The particle will not exist in that region.
Thus, the particle will not exist in this region and the minimum total energy of the particle will be zero.
b)
The graph of potential energy is given below.
From the above graph, it can be observed that the potential energy is always
c)
The graph of potential energy is given as,
From the graph, it is clear that the potential energy is
The condition for the kinetic energy to be positive is satisfied only for the region
Thus, the minimum total energy is
d)
The graph of potential energy is given as,
The condition for kinetic energy to be positive, the region
The minimum potential energy is
The kinetic energy of the system is given as,
Thus, the minimum total energy is
