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=(1)/(2) m^2 omega^2 r^2,` where `omega` is a constant and r is the distance of the particle from the origin. Assuming Bohr model of quantization of angular momentum and circular orbits, show that radius of the nth allowed orbit is proportional to `sqrtn.`

Text Solution

Verified by Experts

`F=(delU)/(delr)={-(1)/(2)mb^(2)r}=mb^(2)r`....(i)
Quantization of angular momentum
`L=(nh)/(2pi)`……(ii)
Centipetal force `=(mv^(2))/(r )`
`F` will provide centripetal force, So, from (i) `F`
So, `mb^(2)r=(mv^(2))/(r )`
`b^(2)r^(2)=v^(2)`
`v=br`.....(iii)
`L=mvr=mbr^(2)`
From (ii)
`mbr^(2)=(nh)/(2pi)`
`r^(2)=(n)/(mb)(h)/(2pi)`
So, `r propsqrt(n)` Hence proved.
Doubtnut Promotions Banner Mobile Dark
|

Topper's Solved these Questions

  • ATOMIC PHYSICS

    RESONANCE ENGLISH|Exercise Exercise -3 part -I JEE (Advanced)|86 Videos

  • ALTERNATING CURRENT

    RESONANCE ENGLISH|Exercise HIGH LEVEL PROBLEMS|11 Videos

  • CAPACITANCE

    RESONANCE ENGLISH|Exercise High Level Problems|16 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

Knowledge Check

  • 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 -

    A
    `-k/r`
    B
    `-k/(2r)`
    C
    `k/(2r)`
    D
    `(2k)/r`

    • Similar Questions

    Explore conceptually related problems

    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 rodius of nth allowed orbit.

    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

    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 :

    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:

    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

    A particle of mass M is moving in a circle of fixed radius R in such a way that its centripetal acceleration at time t is given by n^2Rt^2 where n is a constant. The power delivered to the particle by the force acting on it, it :

    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

    Home
    Profile