New answers tagged homework-and-exercises
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Exponential of the sum of two non-commuting operators where their commutator is proportional to one of them
Found it. The correct expression is:
$$ \exp(A+B)=\exp(A)\exp\left(\frac{-2}{k}B \left(1-e^{\frac{k}{2}}\right)\right)$$
Notice the missing minus sign in the last exponential. This property is used ...
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Help me understand power and torque
Power and torque transmitted on a shaft or to a wheel are not unrelated variables:
$\frac{P}{T}=f$
Here $f$ is the revolution frequency. Power, torque and frequency are expressed in W, Nm and Hz ...
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7
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Calculating Momentum Change?
Sounds to me, you are not familiar with vectors. So I highly suggest to learn about vectors quickly, which is a simple field of mathematics.
Let's have a look in some more detail. x-axis points to the ...
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How does Work done by external force gets COMPLETELY converted to potential energy when there is friction?
When friction is present between two bodies, energy is dissipated as heat if one body is moving relative to the other. But if the two bodies are static relative to one another then no energy is ...
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6
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Calculating Momentum Change?
I'd recommend that you think first about a simpler case: you throw a ball straight at a wall (in the x-direction, let's say) and it bounces back with equal and opposite velocity. The wall has to exert ...
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6
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Calculating Momentum Change?
One way to look at this is the following: for any system, you can relate the force the object experiences to its change in momentum by
$$
\Delta \vec{p} = \int \vec{F} \, dt
$$
In particular, if the ...
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How can you derive the formula for the apparent weight at any latitude on the Earth?
For the apparent reduction in weight, you need to consider the acceleration vector "anti-parrallel" to $g$.
Centrifugal acceleration $a = ω^2r = ω^2R\cosλ $
Acceleration "anti-...
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Clarifying Bra-Ket Notation: Orthonormal Bases
The trace is a number, and the sum of the diagonal elements
so take the diagonal elements $\langle e_j\vert A\vert e_j\rangle$ and sum over them:
\begin{align}
\hbox{Tr}(A)&=\sum_{j}\langle e_j\...
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Clarifying Bra-Ket Notation: Orthonormal Bases
$$\require{cancel}\text{tr}(A) = \sum_{k=1}^n \langle k|A| k \rangle = \sum_{k=1}^n \frac{1}{n} \langle k \left| \sum_{r,q} (-1)^{r+q} |r \rangle \langle q | \right| |k \rangle = \frac{1}{n} \sum_{k,r,...
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Does a simple pendulum have some radial acceleration at its extreme positions where its speed becomes zero?
No, the nature of the string or rope of the pendulum means constant length, and this translates to zero radial acceleration.
Suppose that wasn't the case and you had the coordinates of the mass ...
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3
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Clarifying Bra-Ket Notation: Orthonormal Bases
$\underline{\text{Hint:}}$ you can use the formula
$$\text{tr}A=\sum_{p=1}^n\langle p|A|p\rangle$$
which essentially picks up all the diagonal elements of the operator's $A$ matrix form. Substituting $...
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Does a simple pendulum have some radial acceleration at its extreme positions where its speed becomes zero?
Yes you are right - although you may have made a sin/cos error or defined angles differently than me. Since the velocity/angular velocity is zero, the tension in the string doesn't need to pull ...
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Magnitude of force and point of application to keep semi-cylindrical gate in a fluid at equilibrium
You correctly conclude that the liquid provides no torque about $O$, and neither can the pivot, so $F$ must be zero. Otherwise, it would cause a net torque. This answers your second question. The ...
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How do I convert vectors between rotated cartesian coordinate systems?
For this specific problem involving aircraft relative to Earth, you have chosen a decent frame: North East Down (NED), which is a local tangent plane: https://en.wikipedia.org/wiki/...
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How do I convert vectors between rotated cartesian coordinate systems?
This problem can be solved in a general way via a basis change, where one basis is related to the other via a 90° rotation. More about this can be found on the following wiki page: https://en....
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Why only does the tangential component of gravitational force on an object in elliptical orbit do work?
. . . . . decompose the gravitational force on the orbiting object in an elliptical orbit into components normal and tangential [to the velocity of the body] . . . . .
Work is done when there is a ...
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Coulomb's law problem (add 1 point charge to 2 other point charges such that they are in equilibrium)
For $x\lt a$, the sign of the second term is wrong.
(It might be better to put the origin at the negative charge, so ${\rm sgn}(x)$ determines the direction of the attractive force.)
An unstable ...
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3
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Exponential of the sum of two non-commuting operators where their commutator is proportional to one of them
Hint: If $$[A,B]~=~aB, \qquad a~\in~ \mathbb{C},$$ then one may show that the BCH formula collapses to $${\rm BCH}(A,B)~=~A+\frac{a}{1-e^{-a}}B$$
when the denominator is non-zero.
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Quantum fidelity commutativity proof
Define square roots $R:=\sqrt{\rho}$ and $S:=\sqrt{\sigma}$. Both $R,S$ are semi-positive definite bounded operators. Define the bounded operator $T:= SR$. Notice that $T^{\dagger}=RS$.
The ...
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How does narrowing and splitting/branching of a pipe affect speed of liquid flow?
I know that the volumetric flow rate is supposed to be the same in any cross-sectional area, and that, in order to maintain this, the speed must change. When diameter halves, area is quartered, this ...
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Center of Mass calculation in configuration of $3$ pennies inscribing equilateral triangle
You know that the centres of the pennies form a triangle with sides of length $2R$. If the vertices of this triangle are A, B and C and the midpoint of side BC is O, then AOB is a right angled ...
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Commutator under unitary transformation
$$U^\dagger [A,B] U U^\dagger = [U^\dagger A U U^\dagger , U^\dagger B U U^\dagger]$$
$$ U^\dagger [A,B] =[U^\dagger(U A U^\dagger), U^\dagger( U B U^\dagger)] $$
$$ [A,B] = [A', B'] $$
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Compression of spring when an object of given mass is placed on it
I'm looking at your work an I was looking at other work an I ended up back here
Your formula was right if trying to find initial compression of a spring
its just its
x=sqrt((2mgh)/k)
its really the ...
2
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How can you derive the formula for the apparent weight at any latitude on the Earth?
What you're neglecting is that the gravitational force, the normal force, and the centripetal acceleration are vectors. This means that you must have
$$
m\vec{g} + \vec{F}_N = (m \omega^2 R_e \cos \...
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Potential outside two infinite sheets with hole
So the potential between two infinite sheets is linear: constant electric field in the obvious (by symmetry) direction. Instead of cutting a hole in the sheets, consider adding 2 disk of equal charge ...
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Derive Linearized Einstein's equation from Lagrangian approach
Below is a derivation of the linearized Einstein's equation from the given Lagrangian.
Since $h^{\mu\nu}$ does not appear in the Lagrangian, we have $\frac{\delta L}{\delta h^{\alpha\beta}} = 0 $. So ...
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How can I calculate the solid angle of a planet?
Looking at a distant ($d$) hemisphere with irradiance $I$, each surface element emits according to Lambert's cosine-law:
Integrate that over the hemisphere with:
$$ dA = R^2d\Omega = R^2\sin\theta d\...
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3
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Non-relativistic limit of relativistic energy
Taylor series to first order:
$$ \frac 1 {1-\epsilon} \approx 1 + \epsilon $$
$$ \sqrt{1 + \epsilon} \approx 1 + \frac 1 2 \epsilon $$
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Diagonalizing the tridiagonal matrix for finding the normal mode
The matrix eigenvalue problem leads a recurrence equations with constant coefficients
$$
x_{n+1}-2 x_n +x_{n-1}= \Omega^2 x_n, \quad x_{N+1}= x_1
$$
for the periodic case, and
$$
x_{n+1}-2 x_n +x_{n-...
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Why isn't the work minus the potential energy when bringing a charge in from infinity?
Simple Answer:
Since x=2a is closer to the +ve charge, it must be of higher potential. so, you (external force) must do work to bring a +ve charge from infinity overcoming the repulsion of field. ...
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Gas pressure forces in metal gas bottles looks too big for my layman's mind. Are my numbers wrong?
The pressures inside a gas canister are high, but not ridiculously high - according to Wikipedia a steel-hulled submarine can withstand pressures of up to 580 psi.
The key thing to note is that the ...
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Diagonalizing the tridiagonal matrix for finding the normal mode
You can try to solve the eigenvalue problem directly, the calculations are not too hard. You just solve the second order induction and match the boundary conditions.
A faster method is to revert to ...
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Extensions of bars under different loads?
You can go by formula of Young's modulus (as material are same so young's modulus will be same)
$\gamma$ = $\frac{Stress}{Strain}$
Stress = $\frac{F}{A}$ (Here force is given as $\omega$ , 2$\omega$ , ...
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Extensions of bars under different loads?
The axial stiffness of a rod of length $\ell$ and cross section $A = \pi d^2/4$ is
$$ k = \frac{E A}{\ell} = \left(\frac{\pi E}{4 } \right) \left( \frac{d^2}{\ell } \right)$$
where $E$ is Young's ...
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Extensions of bars under different loads?
F/A = young's modulus * extension/L
Since the material is same the young's modulus for all materials is same
rearrange for extension and input values for all 3 cases and compare.
(I'd recommend never ...
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5
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A case of normal force working in the “wrong direction”
By assuming that the acceleration of the painter and the scaffold are both equal to $a$, you are implicitly assuming that the painter and the scaffold are connected together e.g. the painter has tied ...
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3
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Quantum fidelity commutativity proof
If the operators $\rho$ and $\sigma$ act in a finite-dimensional space, then you can easily see that all nonzero eigenvalues of the operators $\sqrt{\rho}\sigma\!\sqrt{\rho}$ and $\sqrt{\sigma}\rho\!\...
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Question about the deduction of Hamiltonian of Schwinger model
$$
H_{\rm interaction}=\frac 12 \int_{-\infty}^\infty e^2 (\partial_x^{-1} j_0(x))^2 dx\\ =\frac 12
\int_{-\infty}^\infty \int_{-\infty}^\infty dx dx' \delta(x-x') (\partial_x^{-1} j_0(x)) (\...
- 46.6k
5
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Purity of the bosonic Gaussian state $e^{\lambda (b_1^\dagger b_2^\dagger - b_1b_2)} |0\rangle$
This problem has the symmetry $1 \leftrightarrow 2$ and the operator in the exponent creates and annihilates only $12$ pairs . Therefore it admits a straightforward solution. Let's consider the state ...
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Two constraints of $\bar\psi$ from equations of motion for Free Dirac Field Lagrangian
(I haven’t checked the equations of motion you have written, I am assuming they are right)
In quantum field theory fields $(\phi)$are operators but positions $x$ are not. So $\partial_\mu$ is also not ...
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Proving that commutator algebra of Dirac matrices are isomorphic to that of matrix generators of lorentz group
The idea here is based on the concept of a Lie algebra and a representation of a Lie algebra. So the Lorentz group is a Lie group (if you do not know what this is exactly don't worry about it) i.e. a ...
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Integral Form of Conservation or Energy and Momentum in General Relativity
We assume that $T^{\mu\nu}(t, {\bf x})$ vanishes at large spatial distance ${\bf x}$ .
You want
$$
\partial_t \int_{{\mathbb R}^3} T^{tj}(t, {\bf x})d^3x.
$$
As $\partial_\mu T^{\mu\nu}=0$, this is ...
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$n$-dimensional irreducible representation of $SU(2)$
You must surely know that for a $n=2j+1$
$$
J_z|j,m\rangle = m|j,m\rangle, \quad -j\le m\le j,\\
J_{\pm} |j,m\rangle= \sqrt{j(j+1)- m(m\pm 1)}|j,m\pm 1\rangle,
$$
so there are your $n$-by-$n$ matrices....
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Been confused about string constrained motion for a while
Your teacher is correct. The bead's actual velocity is horizontal. It makes sense to break that into components parallel and perpendicular to the string.
The beads actual velocity is not along the ...
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A 1 kg steel ball is dropped and another is thrown downward with a velocity of 5 ms-1, which will have greater acceleration?
Well, you haven't mentioned if the balls are in free fall (idealized condition). If that's so, both balls will have the same acceleration ~ acceleration due to gravity, g.
Now you may ask: Why?
Well, ...
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In what case does the parameter of surface tension vary?
Surface tension acts tangentially to the surface of water, So when you immerse a nail in the water, water from both sides are being in effect
Hence 2*L is used in the formula
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Energy spectrum in Klein-Gordon equation in general relativity
No, the spectrum is different and in general depends on the global properties of the pseudo-Riemannian manifold. Questions you should ask yourself to better understand this:
What exactly is the ...
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What is the ratio of the turns in the transformer when it is impedance-matching?
See a transformer does not change the phase of a transformed quantity(i.e.:-V, I) therefore the respective difference between their phases remains same:-phi1=phi2.;)
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Tension of rope. Different Answers?
As others have pointed out, there cannot be unequal forces on the two ends of a massless rope - the lack of mass (or inertia) means that any force is instantly transferred to all parts of the rope.
In ...
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How to exactly determine the position and sign in vector quantity like displacement?
I assume that you have chosen a unit vector, $\hat {\mathbf w}$ that points along the ruler towards the wall. In that case you should have found the resultant displacement to be +32 cm $\hat {\mathbf ...
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