Q3 of 30 Page 134

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.

The kinetic energy of a particle of mass m moving with a velocity v is given as

K=


Since m is always positive, K is also always positive.


The total energy (E) of a particle is equal to the sum of Kinetic energy (K) and Potential energy (PE).


⇒ E = K + PE


⇒ K = E – PE


a) For x>a, PE(=V0)>E.


This means that the kinetic energy is negative. Hence, the particle cannot exist in the region x>a.


b) For x<a and x>b, PE(=V0)>E.


This means that the kinetic energy is negative. Hence, the particle cannot exist in the region x<a and x>b.


c) In the region a<x<b, the kinetic energy is positive. The potential energy is -V1. So, K = E-(-V1) = E + V1.


In order for K to be positive, E> -V1. So, the minimum total energy of the particle is -V1.


d) For the regions -b/2<x<-a/2 and a/2<x<b/2, the potential energy of the particle is greater than the total energy which suggests that the kinetic energy is negative in this region. Hence, the particle cannot exist in this region.


The lowest potential energy is -V1. So, K = E-(-V1) = E+V1.


For K to be positive, E should be greater than -V1. So, minimum total energy is -V1.


More from this chapter

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1

The sign of work done by a force on a body is important to understand. State carefully if the following quantities are positive or negative:

A. work done by a man in lifting a bucket out of a well by means of a rope tied to the bucket.


B. work done by gravitational force in the above case,


C. work done by friction on a body sliding down an inclined plane,


D. work done by an applied force on a body moving on a rough horizontal plane with uniform velocity,


E. work done by the resistive force of air on a vibrating pendulum in bringing it to rest.

 2

A body of mass 2 kg initially at rest moves under the action of an applied horizontal force of 7 N on a table with coefficient of kinetic friction = 0.1. Compute the

A. work done by the applied force in 10 s,


B. work done by friction in 10 s,


C. work done by the net force on the body in 10 s,


D. change in kinetic energy of the body in 10 s, and interpret your results.

 4

The potential energy function for a particle executing linear simple harmonic motion is given by V(x) = kx2/2, where k is the force constant of the oscillator. For k = 0.5 N m-1, the graph of V(x) versus x is shown in Fig. 6.12. Show that a particle of total energy 1 J moving under this potential must ‘turn back’ when it reaches x = � 2 m.

 5

Answer the following:

A. The casing of a rocket in flight burns up due to friction. At whose expense is the heat energy required for burning obtained? The rocket or the atmosphere?


B. Comets move around the sun in highly elliptical orbits. The gravitational force on the comet due to the sun is not normal to the comet’s velocity in general. Yet the work done by the gravitational force over every complete orbit of the comet is zero. Why?


C. An artificial satellite orbiting the earth in very thin atmosphere loses its energy gradually due to dissipation against atmospheric resistance, however small. Why then does its speed increase progressively as it comes closer and closer to the earth?


D. In Fig. 6.13(i) the man walks 2 m carrying a mass of 15 kg on his hands. In Fig. 6.13(ii), he walks the same distance pulling the rope behind him. The rope goes over a pulley, and a mass of 15 kg hangs at its other end. In which case is the work done greater?