# Tag Info

Accepted

### 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 ...

### 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 ...
• 5,145

### 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 ...
• 294
Accepted

### 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 ...
• 40.8k
Accepted

### 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 ...
• 31.5k

### 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 ...
• 43.7k
1 vote

### 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-...
• 299

### 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\...
• 42.2k

• 43.7k

### 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 ...
• 28.1k

### 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 ...

• 46.6k

### 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. ...
• 227

### 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 ...
• 40.8k
Accepted

### 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 ...
• 4,763

### 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$\omega$ , ...
• 212

### 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 ...
• 35.7k

### 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 ...
• 48
Accepted

### 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 ...
• 40.8k