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?

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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 …

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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)

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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 …

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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.

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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 …

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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 …

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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]

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  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
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