Given in Figure, 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.

Asked by Abhisek | 1 year ago | 82

The total energy is given by E = K.E. + P.E.

K.E. = E – P.E.

Kinetic energy can never be negative. The particle cannot exist in the region, where K.E. would become negative.

**(a)** For the region x = 0 and x = a, potential energy is zero. So, kinetic energy is positive. For, x > a, the potential energy has a value greater than E. So, kinetic energy becomes zero. Thus the particle will not exist in the region x > a.

The minimum total energy that the particle can have in this case is zero.

**(b)** For the entire x-axis, P.E. > E, the kinetic energy of the object would be negative. Thus the particle will not exist in this region.

**(c)** Here x = 0 to x = a and x > b, the P.E. is greater than E, so the kinetic energy is negative. The object cannot exist in this region.

**(d)** For x = -b/2 to x =-a/2 and x = a/2 to x = b/2 . Kinetic energy is positive and the P.E. < E. The particle is present in this region.

A person trying to lose weight (dieter) lifts a 10 kg mass, one thousand times, to a height of 0.5 m each time. Assume that the potential energy lost each time she lowers the mass is dissipated.

**(a)** How much work does she do against the gravitational force?

**(b)** Fat supplies 3.8 × 10^{7}J of energy per kilogram which is converted to mechanical energy with a 20% efficiency rate. How much fat will the dieter use up?

The windmill sweeps a circle of area A with their blades. If the velocity of the wind is perpendicular to the circle, find the air passing through it in time t and also the kinetic energy of the air. 25 % of the wind energy is converted into electrical energy and v = 36 km/h, A = 30 m^{2} and the density of the air is 1.2 kg m^{-3}. What is the electrical power produced?

A body of mass 0.5 kg travels in a straight line with velocity \( v =ax^\dfrac{3}{2} \) where\( a = 5 m^\dfrac{-1}{2}s^{–1}\) What is the work done by the net force during its displacement from x = 0 to x = 2 m?

A trolley of mass 300 kg carrying a sandbag of 25 kg is moving uniformly with a speed of 27 km/h on a frictionless track. After a while, the sand starts leaking out of a hole on the floor of the trolley at the rate of0.05 kg s^{–1}. What is the speed of the trolley after the entire sandbag is empty?

The bob of a pendulum is released from a horizontal position. If the length of the pendulum is 1.5 m, what is the speed with which the bob arrives at the lowermost point, given that it dissipated 5% of its initial energy against air resistance?