Constants of motion

Bernoulli’s principle in a spray gun

July 9th, 2022jeefirst0 Comments

JEE Advanced 2014 Paper 2, Questions 13 and 14

A spray gun is shown in the figure where a piston pushes air out of a nozzle. A thin tube of uniform cross section is connected to the nozzle. The other end of the tube is in a small liquid container. As the piston pushes air through the nozzle, the liquid from the container rises into the nozzle and is sprayed out. For the spray gun shown, the radii of the piston and the nozzle are $20 {\rm~mm}$ and $1 {\rm~mm}$ , respectively. The upper end of the container is open to the atmosphere.

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Q.1…

Heat transferred along two paths

April 10th, 2022jeefirst0 Comments

JEE Advanced 2014 Paper 1, Question 20

A thermodynamic system is taken from an initial state $i$ with internal energy $U_{i} = 100 \, {\rm J}$ to the final state $f$ along two different paths $i a f$ and $i b f$ , as schematically shown in the figure. The work done by the system along the paths $a f, i b$ and $b f$ are $W_{a f} = 200 \, {\rm J}, W_{i b} = 50 \, {\rm J}$ and $W_{b f} = 100 \, {\rm J}$ respectively. The heat supplied to the system along the path $i a f, i b$ and $b f$ are $Q_{i a f}, Q_{i b}$ and $Q_{b f}$ respectively. If the internal energy of the system in the state $b$ is $U_{b} = 200 \, {\rm J}$ and $Q_{i a f} = 500 \, {\rm J}$ , the ratio $Q_{b f}/Q_{i b}$ is

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Solution

The problem gives us partial information about internal energies, heat transferred, and work done at various points in the PV diagram. We …

Coriolis effect and angular momentum

February 20th, 2022jeefirst0 Comments

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 …

Mass on a semicircular block

February 1st, 2022jeefirst0 Comments

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.

Solution:

(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 …

Sliding on a block with a circular cut

January 30th, 2022jeefirst0 Comments

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?

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The position

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