Coriolis effect and angular momentum

Imagine a mass m moving on the surface of a rotating sphere. For instance, the mass could be parcel of air moving away from a high pressure region in the Earth’s atmosphere. It experiences a Coriolis force which, in the example shown in the figure below, pushes it from its original trajectory (orange) to move eastward (blue). Why does this happen, and how do we understand it intuitively?

Rendered by

Formally, the Coriolis force on m is given by

(1)   \begin{equation*}   {\bf F}_{\rm Coriolis} = - 2 m {\bf \Omega} \times {\bf v}_{\rm rot} ,  \end{equation*}

where {\bf \Omega} is the angular velocity of the rotating frame (Earth), and {\bf v}_{\rm rot} is the velocity of m as seen by an observer on the Earth’s …

Continue Reading

Missing energy in a rope and a capacitor

Consider a uniform rope of mass density \lambda coiled on a smooth horizontal table. One end is pulled straight up with a constant speed v_0 as shown.

  1. Find the force exerted on the end of the rope as function of the height y.
  2. Compare the power delivered to the rope with the rate of change of the rope’s mechanical energy.

(This is a problem from chapter 5 of Kleppner and Kolenkow)

Rendered by

To find the force exerted at the top end, note that if we were to pull up a fixed mass with constant velocity v_0, the total force on the mass should …

Continue Reading

Mass on a semicircular block

A heavy particle of mass m is placed at the top of a semicircular block of radius R. Find the height at which the particle falls off, assuming (i) the block is fixed to the ground, and (ii) the block has a mass M and is free to move. Assume all surfaces are frictionless.

Rendered by

Related problem: Sliding on a block with a circular cut.


(i) We first consider the case where the block is fixed to the ground. As the mass slides down the block, there are three forces acting on it: the weight mg, the centrifugal force m R \dot{\theta}^2, and …

Continue Reading

Sliding on a block with a circular cut

JEE Advanced 2017 Paper 1, Question 2

A block of mass M has a circular cut with a frictionless surface as shown. The block rests on the horizontal frictionless surface of a fixed table. Initially the right edge of the block is at x=0, in a co-ordinate system fixed to the table. A point mass m is released from rest at the topmost point of the path as shown and it slides down. When the mass loses contact with the block, its position is x and the velocity is v. At that instant, which of the following options is/are correct?

Rendered by

  1. The position
Continue Reading

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