Home
Class 12
PHYSICS
A small particle of mass m moves in such...

A small particle of mass m moves in such a way that the
potential energy `U = ar^2`, where a is constant and r is the distance of the
particle from the origin. Assuming Bhor model of quantization of angular
momentum and circular orbits, find the radius of nth allowed orbit.

Text Solution

Verified by Experts

The force at a distance `r is f = - dU// dr = - 2 ur`.
Suppose`r` be the redius of nth orbit . Then the necessary centripetal force is provided by the above force, then.
`(mv^(2))/( r) = 2 ur` (i)
Further the equatization of angular momentum gives
`mvr = (nh)/(2 pi)` (ii)
Solving Eqs. (i) and (ii) for `r` we get `r = ((n^(2) h^(2))/(8 um pi^(2))) ^1//4` (iii)
Promotional Banner

Topper's Solved these Questions

  • ATOMIC PHYSICS

    CENGAGE PHYSICS ENGLISH|Exercise Exercise 4.2|12 Videos

  • ATOMIC PHYSICS

    CENGAGE PHYSICS ENGLISH|Exercise Subject|17 Videos

  • ATOMIC PHYSICS

    CENGAGE PHYSICS ENGLISH|Exercise Solved Examples|23 Videos

  • ALTERNATING CURRENT

    CENGAGE PHYSICS ENGLISH|Exercise Single Correct|10 Videos

  • CAPACITOR AND CAPACITANCE

    CENGAGE PHYSICS ENGLISH|Exercise Integer|5 Videos

Similar Questions

Explore conceptually related problems

A small particle of mass m move in such a way the potential energy (U = (1)/(2) m^(2) omega^(2) r^(2)) when a is a constant and r is the distance of the particle from the origin Assuming Bohr's model of quantization of angular momentum and circular orbits , show that radius of the nth allowed orbit is proportional to in

A small particle of mass m move in such a way the potential energy U = (1)/(2) m^(2) omega^(2) r^(2) when a is a constant and r is the distance of the particle from the origin Assuming Bohr's model of quantization of angular momentum and circular orbits , show that radius of the nth allowed orbit is proportional to in

A small particle of mass m , moves in such a way that the potential energy U = ar^(3) , where a is position constant and r is the distance of the particle from the origin. Assuming Rutherford's model of circular orbits, then relation between K.E and P.E of the particle is :

A small piece of mass m moves in such a way the P.E. = -(1)/(2)mkr^(2) . Where k is a constant and r is the distance of the particle from origin. Assuming Bohr's model of quantization of angular momentum and circular orbit, r is directly proportional to : (a) n^(2) (b)n (c) sqrt(n) (d)none of these

The potential energy of a particle in a conservative field is U =a/r^3-b/r^2, where a and b are positive constants and r is the distance of particle from the centre of field. For equilibrium, the value of r is

suppose an electron is attracted towards the origin by a force k//r , where k is a constant and r is the distance of the electron form the origin. By applying bohr model to this system, the radius of n^(th) orbit of the electron is found to be r_n and

For a hypothetical hydrogen like atom, the potential energy of the system is given by U(r)=(-Ke^(2))/(r^(3)) , where r is the distance between the two particles. If Bohr's model of quantization of angular momentum is applicable then velocity of particle is given by:

Find the workdone to move an earth satellite of mass m from a circular orbit of radius 2R to one of radius 3R.

A particle of mass m moves in a one dimensional potential energy U(x)=-ax^2+bx^4 , where a and b are positive constant. The angular frequency of small oscillation about the minima of the potential energy is equal to

A particle of mass m is moving in a horizontal circle of radius r, under a centripetal force equal to (-K//r^(2)) , where k is a constant. The total energy of the particle is -