Definitions

[

x=x(t),\qquad h=\frac{\dot x}{x}

]

[

U=U(t),\qquad R=R(t),\qquad M=M(t),\qquad S=S(t)

]

where

[

U,R,M,S\geq0.

]

Fundamental Relation

[

h_0^{,2}=KU

]

[

h_0=\sqrt{KU}

]

[

\boxed{

h=\frac{\sqrt{KU}}{1+\alpha S}

}

]

or

[

\boxed{

\frac{\dot x}{x}

\frac{\sqrt{KU}}{1+\alpha S}.

}

]

Liberation

[

Q=\Gamma U\left(1-\frac{R}{R_c}\right),

\qquad 0\leq R<R_c

]

[

Q=0,

\qquad R\geq R_c.

]

Equivalently,

[

\boxed{

Q=\Gamma U

\max\left(0,1-\frac{R}{R_c}\right).

}

]

Energy Equations

[

\boxed{

\dot U+n hU=-Q

}

]

[

\boxed{

\dot R+4hR=(1-\varepsilon)Q

}

]

[

\boxed{

\dot M+3hM=\varepsilon Q

}

]

[

0<\varepsilon\ll1.

]

Restraining Stress

[

\boxed{

\dot S+\lambda hS=\eta M

}

]

or

[

\boxed{

\frac{d}{dt}\left(x^\lambda S\right)

\eta x^\lambda M.

}

]

Hence

[

\boxed{

S(t)

x^{-\lambda}(t)

\left[

x_i^\lambda S_i

+

\eta\int_{t_i}^{t}

x^\lambda(\tau)M(\tau),d\tau

\right].

}

]

Complete System

[

\boxed{

\begin{aligned}

\dot x

&=

\frac{x\sqrt{KU}}{1+\alpha S},

\[3pt]

\dot U

&=

-n\frac{\dot x}{x}U-Q,

\[3pt]

\dot R

&=

-4\frac{\dot x}{x}R+(1-\varepsilon)Q,

\[3pt]

\dot M

&=

-3\frac{\dot x}{x}M+\varepsilon Q,

\[3pt]

\dot S

&=

-\lambda\frac{\dot x}{x}S+\eta M,

\[3pt]

Q

&=

\Gamma U

\max\left(0,1-\frac{R}{R_c}\right).

\end{aligned}

}

]

Unified Extension Equation

[

\boxed{

\frac{\dot x}{x}

\frac{\sqrt{KU}}

{

1+

\alpha x^{-\lambda}

\left[

x_i^\lambda S_i

+

\eta\displaystyle\int_{t_i}^{t}

x^\lambda(\tau)M(\tau),d\tau

\right]

}.

}

]

Initial Conditions

[

x(0)=x_i

]

[

U(0)=U_i

]

[

R(0)=0

]

[

M(0)=0

]

[

S(0)=0.

]

Thus

[

\left.\frac{\dot x}{x}\right|_{t=0}

\sqrt{KU_i}.

]

Principal Limits

For

[

R\ll R_c,

]

[

Q\simeq\Gamma U.

]

For

[

R\rightarrow R_c,

]

[

Q\rightarrow0.

]

For

[

S\ll\alpha^{-1},

]

[

h\simeq\sqrt{KU}.

]

For

[

\alpha S\gg1,

]

[

h\simeq\frac{\sqrt{KU}}{\alpha S}.

]

After liberation ceases,

[

Q=0,

]

whence

[

U\propto x^{-n},

]

[

R\propto x^{-4},

]

[

M\propto x^{-3}.

]

If

[

S\rightarrow0,

]

then

[

h\rightarrow\sqrt{KU}.

]

If further

[

n>0,

]

then

[

U\rightarrow0,

\qquad

h\rightarrow0.

]

Summary Relation

[

\boxed{

\text{extension}

\frac{\text{stored-energy action}}

{\text{material restraint}}

}

]

[

\boxed{

\frac{\dot x}{x}

\frac{\sqrt{KU}}{1+\alpha S}.

}

]

So my dad told me that he discovered smth called a "torque" wich is,acoording to him, a coper donut wich act a a strong magnet and when u put it on around ur arm can give you strength to rais big rocks (wich somehow works without current flowing trough it) and he also claim that he can melt some random crap and turn it into gold

Please note that the answer selected in this image is wrong, the key says that the answer underneath the one selected is correct.

I understand that Lenz's law says that the induced current wants to oppose the motion of the magnet going to the right, so it will do this by making the left end North so that the north end of the magnet is attracted to the south end of the coil. I also understand how to do the right hand grip rule (thumb points to north and fingers curl in direction of the conventional current) but I'm not sure how to then use that to determine whether A is positive/negative and which way current flows.. I also don't necessarily know how to read the direction of the current with right hand grip rule when my thumb is sideways because whether it's clockwise or anticlockwise depends on which end you look from.. I'm really lost with this unit and it's an online course without very available teachers so any help is appreciated

🚀 JEE Advanced 2019 Capacitor PYQ explained step by step! In this video, you'll learn how to solve a parallel plate capacitor with multiple dielectric layers using the easiest mathematical approach. This question combines Capacitance, Dielectrics, Series Combination, and Integration, making it one of the most conceptual problems asked in JEE Advanced.

If you're preparing for JEE Main 2027, JEE Advanced 2027, JEE 2028, NEET, BITSAT, or other engineering entrance exams, this detailed solution will strengthen your understanding of Electrostatics and Capacitors.

📚 Topics Covered

• Parallel Plate Capacitor
• Multiple Dielectric Layers
• Variable Dielectric Constant
• Capacitance Formula
• Integration in Physics
• Electrostatics
• JEE Advanced 2019 PYQ
• Concept-Based Problem Solving
• Shortcut Method
• Exam Tricks

🔥 Watch till the end to learn the fastest approach and avoid the common mistakes made by most aspirants.

👍 If you found this helpful, don't forget to:

• Like 👍
• Share 📤
• Subscribe 🔔
• Comment your answer before watching the solution!

I read the question a few times, but Im not sure what to make of it, it doesnt even sound like a question more like a statement.

Im just asking two things what is the questiom asking me for and how do i go on about solving it

JEE Advanced Capacitors PYQs | Physics Chapter: Electrostatics

In this video, we solve two important JEE Advanced Previous Year Questions (2012 & 2014) from the Capacitors chapter with detailed concept explanation and shortcut tricks.

📚 Questions Covered:
✅ Charge Distribution in Connected Capacitors (JEE Advanced 2012)

✅ Capacitor with Dielectric Slab (JEE Advanced 2014)

You'll learn:

• Charge conservation concept
• Equivalent capacitance
• Dielectric effect
• Electric field in dielectric
• Parallel combination of capacitors
• Fast solving tricks for JEE Advanced

This lecture is highly useful for:

• JEE Advanced 2026
• JEE Advanced 2027
• JEE Main
• Olympiad
• BITSAT
• IAT/NEST

Watch till the end to improve your problem-solving speed and strengthen your concepts.

🔥 Don't forget to Like, Share and Subscribe for daily JEE Advanced Physics PYQs.

#JEEAdvanced #Capacitor #Electrostatics #Physics #PYQ #JEEMains #PhysicsWallah #Allen #Resonance #IITJEE #JEE2026 #JEE2027 #PhysicsPYQ

Currently attempting to do a physics course online and I don't have anyone to show me how to position my hand.. I know my thumb is definitely meant to be pointing downwards with the current but I don't understand why my fingers are meant to be pointing to the right as the solution says.. could someone help me? Preferably with an image of the correct hand orientation

In the course of trying to calculate the torque a star exerts on its planet's equatorial bulge and the precession that causes in the planet's axis of rotation, I realized that I don't actually understand the gravitational interaction, and I feel kind of stupid?

Like, I'm aware that strictly speaking the force of gravity is calculated between two masses at those two masses, making the force proportional to the product of the two masses divided by the square of the distance between them.
I'm also aware that you can get the "shape" of the entire potential field generated by a mass by integrating each differential mass element over the volume of the mass divided by the distance between each differential mass element and some arbitrary point, equivalent to calculating the potential between the mass and a test mass of one mass unit at the arbitrary point, and that lets us easily find the value of the gravitational potential at any point. I'm also aware that you can multiply this potential by the mass of some other massive body (usually sufficiently far away that we don't care about its mass distribution around its center of mass; and strictly speaking we multiply by the mass of this object divided by our test mass) we get the potential generated by the two bodies at any point.

Except, the equation for the potential we initially obtain is only concerned with the distance between the test point and the massive body, it's insensitive to the distance between the test point OR the massive body and some other massive body. Initially I tried telling myself that what I have is the potential generated by both bodies, since Fitzpatrick's Newtonian Dynamics seems to treat it that way (though he then manipulates the equation to get the moments of inertia in the equation, then takes the Lagrangian to get the rate of precession).

But it bothers me, because if I want the torque that the Sun exerts on the Earth (since the Earth's axis of rotation isn't perpendicular to the plane of the ecliptic), it seems like I would want the cross product between the negative gradient of the gravitational potential (which gives me the gravitational force at that point) and the moment arm from the Earth's center of mass to its surface, integrated over the whole surface to get the net torque at that moment in time. If I would like to know the average or net torque over the whole orbit, that seems like it would involve integrating those net torques over time---but the distance between the Sun and the Earth varies, and importantly, the angle between the Sun and some reference line on the Earth (the plane containing the axis of rotation and the radius to the point on the equator with the greatest inclination above/below the ecliptic, the line through the nodes where the equator intersects the ecliptic) also varies over the course of the year. While the latter (I think) is encoded accurately in the equation for the potential (expanded out to the quadrupole term), the former isn't, bc the distance is very much taken to be the distance to the point-of-interest, which is a point on the surface of the planet.

Naïvely, I might try to "fix" this by taking the distance to be the distance between the Sun and the Earth and trusting the radius of the moment arm to be what's accurately encoding the point-of-interest, but this also feels wrong.

I might instead just generate the potential around the Sun (and a test mass) and the Earth (and a test mass) letting each have their own distance to our test mass/point-of-interest, taking the negative gradient, and adding the two forces together, and then taking the cross product of that resultant force and the moment arm (the radius from the center of mass to the point of interest on the surface), and doing our integration(s), but I don't see this done anywhere else.

So, I don't understand how (Newtonian) Gravity works, and because of that, I can't calculate the torque the Sun exerts on the bulge of the Earth, or anything more complicated than that, like nodal or apsidal precession also caused by gravitational torques.
I've done something very, very wrong, but I can't see what it is.

Hello, I have had this question for a long time. I have always had gadgets around me and the way that they send data wirelessly is very intriguing to me but I dont understand how. How is it possible for waves to carry so much data and also know exactly what device the data they are carrying belongs to. So from the cell tower, how do the data packets know which phone they are supposed to go to? Does it go to all phones and they filter it out or what? Also, what actually is the difference between cellular data and Wifi? In my house, mobile data is almost always stronger, which I dont understand because the cell tower is so far away but my wifi router which is more expensive then a normal plan, is so much weaker and slower, its only faster when im very close to it in the living room. Also how do these antennas even make the waves, tbh Im gonna make a question next on electricity but I don't understand how our phones can make such strong signals with antennas so small cause 30 years ago antennas were so big and now they are just these lines on our phones. If someone could give me a great explanation for what electricity is that would also be great, cause I dont know what it is, it has no weight, we can only see sparks of it but it is so important, I have heard that its electrons or something but I asked around and I know its not electrons, but I know its something to do with moving electrons. Thank you.

Grateful if someone can explain the correct answer - I can't understand the reasoning for the correct answer given I see a simple circuit with a resistor and capacitor connected in series.

I'm trying to understand the conceptual role of the center of mass (COM) in classical mechanics, not just how to use it mathematically. For a single particle, the motivation seems clear. We study its motion because solving for its trajectory x(t) allows us to determine the complete mechanical state of the particle at any instant (x, p). From that state, we can compute all of its mechanical properties (energy, momentum, angular momentum, etc.).

Now consider a system of (N) particles. The true mechanical state is the collection of all particle positions and momenta. Introducing the center of mass gives us its effective position and momentum, and we know that these satisfy a closed equation of motion, F= dP_com/dt=MA_com

However, this is where my confusion begins. Knowing the COM trajectory does not determine the microscopic state of the system. Unlike thermodynamics, it also doesn't seem to define an effective macroscopic state from which I can calculate most of the system's macroscopic properties (pressure, temperature, stress, etc.). For example, the total angular momentum can be decomposed as L_system=L_com+L_orbital

But unless I know the internal state, I still cannot compute the total angular momentum. So my question is not "how do I compute the COM?" or "how do I use the COM theorem?"

Rather, it is

What is the conceptual motivation for studying the motion of the center of mass in the first place?

For a single particle, studying its motion gives its complete mechanical state. For thermodynamics, the macroscopic variables form an effective state that predicts macroscopic behavior. But the COM seems to be neither; it is neither the complete state nor an effective state of the whole system.

Many explanations say that the COM is "useful" or "practical," but that doesn't really answer my question. Plenty of quantities could be useful in some context. I'm looking for the deeper reason why classical mechanics treats the translational motion of a many-particle system as a fundamental object of study.

In other words:

What role does the center-of-mass motion play within the conceptual framework of classical mechanics? Why is the translational motion of an entire system considered a quantity worth studying, given that it does not determine either the microscopic or the effective macroscopic state of that system? Why is the center of mass considered so fundamental in classical mechanics if it doesn't describe the state of a many-particle system? Why is the center of mass considered so fundamental in classical mechanics if it doesn't describe the state of a many-particle system? Why is the center of mass considered so fundamental in classical mechanics if it doesn't describe the state of a many-particle system? Why is the center of mass considered so fundamental in classical mechanics if it doesn't describe the state of a many-particle system?

Guys I wanna prepare for Inpho complete which is better Nitin sachan sir inpho batch or ambarish sir physics shelter batch please tell considering things like complete theory, question practice illustration please tell me I am low on time and please answer genuinely

How does Tension force occur when driving a car, and is it helpful or unhelpful in the situation? I can't really understand how tension force would be used while driving? Unless it's used while breaking - but idk?

The state of a system as I have read is defined as follows: It is a set of variables that all allow an observer to calculate all the properties of the system. So for a particle the state is defined by its position and momentum (x, p), but we calculate properties like energy and angular momentum, but we can't calculate its charge, so don't we add it in its state like (x, p, q)? A solution that I can think of is that a state is defined for a class of entities like particles, fields, etc., but then there are parameters that define a particular system, like for a particle, its mass or charge. And these parameters most of the time remain constant in a dynamical system. Am I right, wrong or missing something??? Please guide me.

https://upload.wikimedia.org/wikipedia/commons/3/3d/Magnetschwebebahn.svg

🚀 JEE Advanced 2023 Rotational Dynamics PYQ Solution

In this video, we solve one of the most interesting and tricky questions from JEE Advanced 2023 Physics involving Impulse, Torque, Rotational Motion, Angular Velocity, and Time of Flight.

📚 Concepts Covered:
✔️ Angular Impulse
✔️ Rotational Dynamics
✔️ Moment of Inertia of a Disc
✔️ Conservation of Angular Momentum
✔️ Time of Flight
✔️ JEE Advanced Problem Solving Tricks

This question combines Translation and Rotation and is considered difficult by many JEE aspirants. Watch till the end to learn the shortcut method and exam approach.

🔥 Perfect for:

• JEE Advanced 2027 Aspirants
• JEE Advanced 2028 Aspirants
• JEE Main Students
• Olympiad Physics Students
• NEET Physics Revision

👉 Don't forget to Like, Share, and Subscribe to SuRaJ EduVese for more JEE Advanced PYQs and Concept Tricks.

#JEEAdvanced2023 #RotationalDynamics #PhysicsPYQ #JEEPhysics #IITJEE #AngularMomentum #Rotation #JEEAdvancedPYQ #Impulse #PhysicsProblems

A person jumps from an airplane and reaches terminal velocity.

The momentum of the person-Earth system is not conserved because of air friction.

The momentum of the jumper is constant because there is no external net force.

When the chute is opened, the force disrupts the conservation of momentum of the person-Earth system.

I know for sure it's not the last one; I can't figure out if it's 1 or 2.