Circular orbit in a harmonic potential

JEE Advanced 2018 Paper 1, Question 1

The potential energy of a particle of mass m at a distance r from a fixed point O is given by V(r)=k r^{2} / 2, where k is a positive constant of appropriate dimensions. This particle is moving in a circular orbit of radius R about the point O. If v is the speed of the particle and L is the magnitude of its angular momentum about O, which of the following statements is (are) true?

  1. v=\sqrt{\frac{k}{2 m}} R
  2. v=\sqrt{\frac{k}{m}} R
  3. L=\sqrt{m k} R^{2}
  4. L=\sqrt{\frac{m k}{2}} R^{2}


The force due to the given potential is

(1)   \begin{equation*}   {\bf F} = -\frac{\partial V}{\partial r} \hat{\bf r} = -k r \hat{\bf r} . \end{equation*}

The mass also experiences a centrifugal force due to its circular motion,

(2)   \begin{equation*}   {\bf F}_{\rm cent} = \frac{m v^2}{r} \hat{\bf r} \end{equation*}

For the …

Continue Reading

Mirror on a spring

JEE Advanced 2019 Paper 2, Question 12

A perfectly reflecting mirror of mass M mounted on a spring constitutes a spring-mass system of angular frequency \Omega such that \frac{4 \pi M \Omega}{h} = 10^{24} \, {\rm m}^{-2} with h as Planck’s constant. N photons of wavelength \lambda = 8 \pi \times 10^{-6} \, {\rm m} strike the mirror simultaneously at normal incidence such that the mirror gets displaced by 1 \mu{\rm m}.
If the value of N is x \times 10^{12}, what is the value of x?

[Consider the spring as massless]

Rendered by


The momentum carried by a single photon is given by the de Broglie relation p_{\rm ph} = h/\lambda. The total momentum carried by N photons is therefore N \times p_{\rm ph} = Nh/\lambda. All the photons hit the mirror simultaneously and are reflected …

Continue Reading

Pulleys and masses connected to a spring

JEE Advanced 2019 Paper 2, Question 2

A block of mass 2 M is attached to a massless spring with spring-constant k. This block is connected to two other blocks of masses M and 2 M using two massless pulleys and strings. The accelerations of the blocks are a_{1}, a_{2} and a_{3} as shown in the figure. The system is released from rest with the spring in its unstretched state. The maximum extension of the spring is x_{0}. Which of the following option(s) is/are correct?

[g is the acceleration due to gravity. Neglect friction]

Rendered by

  1. x_{0}=\frac{4 M g}{k}
  2. When spring achieves an extension of \frac{x_{0}}{2} for the first time, the speed of the
Continue Reading