homework and exercises - Variable refractive index question

A ray of light travelling in air is incident at grazing incidence on a slab with variable refractive index, $n (y) = [k y^{3/2}+ 1]^{1/2}$ where $k = 1 m^{-3/2}$ and follows path as shown in the figure. What is the angle of refraction when the ray comes out.$(A) 60^°$$(B) 53^°$$(C) 30^°$$(D)$ No deviation .................………My approach:-Let the angle of emergence br $r$At origin i. e $y=0$ $n_1=1$ and when $y=1$ $n_2=\sqrt2$Using Snell's law$$n_1 \sin90^° = n_2 sinr $$On solving this I get $r=45^°$ but the sad part is that my answer doesn't ma...Read more

homework and exercises - Maximum displacement of a projectile with $x$ initial velocity and drag resistance

So, as a personal project, I imagined a scenario where a projectile with an initial velocity $x$ is affected by air resistance (drag) but not gravity, and I wanted to find out the distance it would travel until it stops. I know that drag force is determined by: $F_D= (1/2) \rho v^2 C_d A$, which I simplified to $F_D= bv^2$ by grouping all the constants. And I know that $\text{work = force} \times \text{displacement}$. So in theory I should be able to find the displacement of the projectile when work done by drag is equal to the kinetic energy ...Read more

homework and exercises - Obtaining a general equation for velocity (in 2D projectile motion)

I'm trying to obtain a general equation for the instantaneous velocity of a projectile moving on a Cartesian plane.I began with the equation for a projectile's trajectory (air resistance neglected):$$y = x(tanθ) - \frac {gx^2}{(u^2)(cosθ)^2}$$where $u$ is the projection velocity, and $θ$ is the projection angle.I then sought to differentiate the above-mentioned equation with respect to time. This yielded:$$y' = x'(tanθ) - \frac {2gxx'}{(u^2)(cosθ)^2}$$Where $'$ stands for a differential with respect to time.Now, re-writing the equation:$$v_y = ...Read more

homework and exercises - How do I find the magnitude and angle of the velocity?

I'm confused about the following problem. A ball is shot from the ground into the air. At a height of $9.1\text{ m}$, its velocity is $v = (7.6\hat{\imath}+ 6.1\hat{\jmath})\text{m/s}$, with $\hat{\imath}$ horizontal and $\hat{\jmath}$ upward. To what maximum height does the ball rise? What total horizontal distance does the ball travel? What is the magnitude and angle (below the horizontal) of the ball’s velocity just before it hits the ground? The given answers are: (1) 11m, (2) 22.76m, (3) 16.57m/s, 62.69We know $S=ut + 1/2 at^2$. ...Read more

homework and exercises - Air Drag in orbit

Artificial satellites don't orbit the earth forever. Eventually the Earth's atmosphere, thin as it may be up there, will bring them down. But did you know the linear speed of a satellite in a near circular orbit will increase because of the air drag? The satellite will experience an acceleration forward along its path, and the accelerations's magnitude will be the same as if the air drag were turned around and were pushing the satellite along. How can that be? Hello I don't really understand this question and hope someone can help me solve it....Read more

homework and exercises - Lagrange Undetermined Multipler

Q1) Write down the Lagrangian of the system in terms of y(t)Q2) Obtain the Eqn of motionQ3)Using Lagrange Multiplier method find the forces of constraints1)We have a constraint such that $$f=y-r\theta=0 $$And the lagrangian is $$L=1/2m[\dot{y}^2+\frac{R^2\dot{\theta}^2}{2}]+mgy$$from here I have to get rid of the $\theta$ by using constraint.Then I get$$L=1/2m[\dot{y}^2+\frac{\dot{y}^2}{2}]+mgy$$$$\ddot{y}=2g/3$$ where g is gravity of earth.Now the question asks to find the constraints so I dont use the constraint in the lagrangian and I just w...Read more

homework and exercises - Hoop rolling inside a circular hole

A hoop of radius $b$ and mass $m$ rolls without slipping within a stationary circular hole of radius $a > b$ and is subject to gravity. Use the generalized coordinates the rotationangle $\phi$ of the hoop and the angular position of the hoop’s center $\theta$. We have the rolling without slipping constraint$$b\phi - a\theta=0.$$The Lagrangian of the system is$$L=\frac{1}{2}m(a-b)^2\dot{\theta}^2+\frac{1}{2}mb^2\dot{\phi}^2+mg(a-b)\cos\theta.$$The Euler-Lagrange equations with Lagrange multiplier are$$m(a-b)^2\ddot{\theta}+mg(a-b)\sin\theta=\...Read more

homework and exercises - How to calculate spherical coordinate components of dipole field?

I understand well enough how to calculate the radial and tangential components in spherical coordinates at a point due to a magnetic dipole field using the magnetic potential gradient ($\overrightarrow{B}=\nabla\times \overrightarrow{W}$). Is it possible to calculate radial ($B_r=\frac{2\mu_om\cos\theta}{4\pi r^3}$) and tangential ($B_\theta=\frac{\mu_om\sin\theta}{4\pi r^3}$) components in spherical coordinates without using the magnetic potential? I.e. I would like to know if it is possible to demonstrate these components in spherical coordin...Read more

homework and exercises - Is there a better choice of coordinates for a bead on a rotating helical wire?

A bead of mass $m$ is threaded around a smooth spiral wire and slides downwards without friction due to gravity. The $z$-axis points upwards vertically. Suppose the spiral wire is rotated about the $z$-axis with a fixed angular velocity $\Omega$. Determine the Lagrangian and the equation of motion.This is related to a previous problem in which the wire shape is given as$$z = k\psi, \hspace{3mm} x = a\cos\psi, \hspace{3mm} y = a\sin\psi$$where $a$ and $k$ are both positive.My attempt at a solution: We still have $z = k\psi$, but now $x = a\cos(\...Read more

homework and exercises - Lagrangian formalism (demonstration)

My question is about the multiplicity of the Lagrangian to a Physics system.I pretend to demonstrate the following proposition: For a system with $n$ degrees of freedom, written by the Lagrangian $L$, we have that: $$L'=L+\frac{d F(q_1,...,q_n,t)}{d t}$$ also satisfies Lagrange's equations, where $F$ is any arbitrary function, but differentiable.I saw the resolution to this problem and I found that another interesting proposition was needed to complete the demonstration, and it was: $$\frac{\partial \dot{F}}{\partial \dot{q}}=\frac{\partial ...Read more

homework and exercises - Bead on a smooth rotating rod

Suppose a bead ($m$) is free to move on a thin rod in the otherwise empty space. The rod is made to rotate at constant angular speed $\omega$. Lets assume the initial position of the bead is $(r_o, \theta_o)$. To solve this problem I can write down the Lagrangian of the system. $$L=\frac{1}{2}m(\dot{r}^2+r^2\dot{\theta}^2)$$Hence the equation of motion will be, (note $\dot{\theta}=\omega\rightarrow$ constant)$$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{r}}\right)=\frac{\partial L}{\partial r} \implies \ddot{r} = r\dot{\theta}^2 = r\omega...Read more

homework and exercises - Is the Hamiltonian conserved or not?

The question is the very last sentence at the end of this post. In this post, I'll first show that the Hamiltonian is conserved since it does not have explicit dependence on time and then show that the Hamiltonian is not conserved since when directly calculate, the derivative is found not to vanish.A bead is threaded on a friction-less vertical wire loop of radius $R$. The loop is spinning w.r.t. a fixed axis shown in the figure at a constant angular speed $\omega$. The Lagrangian is given by $$L=\frac{1}{2}\dot\theta^2R^2+\frac{1}{2}R^2\sin^2\...Read more

homework and exercises - Help with Chrstoffel symbols for geometric mechanics problem?

I am working through the book Geometric Control of Mechanical Systems by Bullo and Lewis https://www.amazon.com/gp/product/0387221956/ and I am stuck on a problem, E4-18. The problem was evidently at one point the subject of a research paper, https://pdfs.semanticscholar.org/387d/4bb1c336aa0da87ab1d3a59f53532a2c74d2.pdf . I am trying to reproduce what the authors did in that paper so that I may solve the Bullo and Lewis problem. Taking the paper's authors' kinetic energy function as correct, including their "kinetic energy Riemannian metric", I...Read more

homework and exercises - Distance travelled by an object slowed only by drag?

Let's say I throw an object horizontally off a cliff with a fixed height, and I know the time it takes to fall. I wanted to know how far it travels, but it has an acceleration opposite the direction of initial velocity due to air resistance. Therefore, I integrated velocity with respect to time; in this case, velocity as a function of time was equal to v0 - at. Okay, what's acceleration (in this case)? I looked into air resistance, and it turns out it's dependent on area, some coefficient, air density, and... instantaneous velocity... As a high...Read more

homework and exercises - Find the force needed to accelerate body to a certain velocity for a certain time with respect to drag force

So, the problem is straightworward when we suggest that air resistance force is constant:$$ \vec F = \frac {\vec V_1 - \vec V_0} {t} m / b $$$$ \vec V_0, \vec V_1 - \text {initial and final velocities respectively},\\ t - \text {time, during which velocity will become its final value}, \\b - \text {some constant for drag force} $$ The problem arises when using quadratic air resistance: $$m a = -c v^2$$ I don't have any clue for solving this and asking for help.What I must do is to accelerate plane to the certain velocity in a certain period ...Read more