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15 votes

How can forces be added?

It is self-evident that successive displacements add by the tail-to-head arrow method. Velocities, being rates of change of displacement, add in the same way, as must momenta, and therefore forces, ...
  • 29.1k
8 votes

How can forces be added?

Here is one way of looking at it that might help you. To add vector A to vector B in the diagrams you provided, take each vector and draw its horizontal and vertical component like in this image. It ...
6 votes

How can the position representation make sense with compatibility of addition? (Dirac Notation)

You're right that you're being tripped up by notation. $x$ and $y$ are $\mathbb R$-valued labels for the (generalized) eigenvectors of the position operator, i.e. $|x\rangle$ is a vector such that $\...
  • 55.6k
5 votes
Accepted

What is the difference between |$\frac{ d\vec{r}}{dt}$| and $\frac{d|\vec{r}|}{dt}$?

The time derivative of the absolute value of the velocity $$\dfrac{\mathrm d|\mathbf{v}|}{\mathrm dt}$$ doesn't take into account change of direction of the velocity, i.e. acceleration in directions ...
  • 2,233
4 votes
Accepted

Issue with canonical proof of Schwartz inequality

The basic idea is that a vector lets you split the space into a direct sum of a one-dimensional subspace it spans and an orthogonal complement. The decomposition you ask about is exactly this ...
  • 30.9k
4 votes

Why does Taylor take the derivative or only $r \bf{\hat r}$ for the velocity/derivative of the position vector?

The information of angle $\theta$ is entirely "encoded" in the position unit vector $\mathbf{\hat{r}}$, even though it isn't encoded in the radial variable $r$. You can see that if you ...
  • 322
4 votes
Accepted

Prove that Gauss's Law Holds Under Translations

Let $\textbf{g}=\textbf{x}-a$ If $E(\textbf{x} - p) = (E_1(\textbf{x} - p), E_2(\textbf{x} - p), E_3(\textbf{x} - p))$ then $\nabla \cdot E(\textbf{x} - p) = \partial_x E_1(\textbf{x} - p)+ \partial_y ...
3 votes

How can forces be added?

The idea of vectors addition, subtraction, resolution, etc. is found everywhere in life. Using basic trigonometry, one could show that some force $F$ that makes an angle $\theta$ with the horizontal ...
3 votes

What is $k$ in this form of the wave equation?

This is a static field, so not really a wave. This however: $$\vec{E}=E_0\vec{n}\,\cos(\omega t-\vec{k}.\vec{r})$$ is a wave. $\vec{k}$ is called the wavevector: its norm is the wave number its ...
  • 4,501
3 votes

How can forces be added?

Let's rearrange this last situation as you are regarding force instead of displacement. The above diagram actually more likely represent displacement. The forces seems to be on seperate points. This ...
3 votes

Does acceleration acting on velocity only change direction or magnitude both?

Well, I am not sure about your physics background so I attempt to explain the issue conceptually without the use of calculus or vectors. For starters, let us clear up what you have done in your "...
3 votes

How do you show that the temporal part of an object's 4-velocity is decreasing as that object approaches the speed of light?

In special relativity, with the most negative convention and $u^{\rho}$ the four velocity, we have: $$u^{0}=\gamma c=\frac{c}{\sqrt{1-\frac{\vec{v}^2}{c^2}}}$$ From here we see that if $\vec{v}^2$ ...
3 votes
Accepted

How do I calculate $x^\nu$ using the below definition?

You can't just divide by the metric like that because there's an implicit summation due to the Einstein Summation convention. $x_\mu = \eta_{\mu\nu} x^\nu = \sum_\nu \eta_{\mu\nu} x^\nu$ where $\eta_{...
3 votes

Does an inertial frame have to be orthogonal?

You can set up whatever co-ordinate axes you like in an inertial frame of reference (or, indeed, in a non-inertial frame of reference) - orthogonal, non-orthogonal, cylindrical, spherical etc. What ...
  • 36.8k
3 votes
Accepted

Why does Taylor take the derivative or only $r \bf{\hat r}$ for the velocity/derivative of the position vector?

Vector is a thing that has direction and length. $\hat{\boldsymbol{r}}$ is unit vector in the direction of $\boldsymbol{r}$ and encodes all the information about its direction, while r is its length. ...
  • 5,690
3 votes

Gravitational potential energy for more than two object

For an arbitrary number of particles, just sum that expression, $-Gm_im_j/|\vec r_i-\vec r_j|$, over all pairs $i$, $j$ of particles. This is easy to see if you imagine bringing each particle ...
  • 341
2 votes

Issue with canonical proof of Schwartz inequality

Let a unit vector $\;\mathbf{n}=(\rm n_1,n_2,n_3)\,, \Vert\mathbf{n}\Vert=1$. Any vector $\;\mathbf{r}\;$ could be decomposed in two components with respect to $\;\mathbf{n}\;$, see Figure-01 in the ...
  • 13.6k
2 votes

Where does pseudo force act at?

Summary Consider a system of particles, not necessarily a rigid body, as viewed in an inertial frame. The translational motion (the change in total linear momentum) can be evaluated assuming the ...
  • 7,036
2 votes

Does an inertial frame have to be orthogonal?

I think your question actually is: Should the basis vectors of a coordinate system be orthogonal? The answer is no, you can choose the basis vectors in any way you would like. An inertial frame ...
2 votes
Accepted

Geometric interpretation of tensor product

Well, in your example, you have a 3-vector w , each of whose components is multiplied by the same 2-vector v, to yield a 6-vector 𝑣⊗𝑤, which you may also think of as a 2×3 matrix, a sort of a dyadic ...
2 votes

Is torque possible without any pivot point?

You can calculate torque about any desired point; that point is not required to be pivoted. In $\vec{\tau} = \vec{r} \times \vec{F}$, vector $\vec{r}$ is the position vector of point where force $\vec{...
  • 230
2 votes

Why can't we take the larger angle in Cross Product?

Using the angle $2\pi - \phi$ would result in a vector pointing in the opposite direction as the standard cross product since $\sin(2\pi - \phi) = -\sin\phi.$ You could define another vector product ...
  • 21.8k
2 votes
Accepted

Is torque possible without any pivot point?

I'm assuming that the only forces actually applied to the bar are P and Q as I show in FIG 1 below. Then you are saying equivalent result force is P + Q = 15 located at point C. But you didn’t go far ...
  • 58.3k
2 votes

Gradient of scalar field

OP's eq. (1) follows from Poincare lemma: The co-vector/1-form $$\omega~:=~\frac{ X_a\mathrm{d}x^a}{\ X^2}\tag{A}$$ is closed $$\mathrm{d}\omega~=~0,\tag{B}$$ and hence locally exact, i.e. on the form ...
  • 175k
1 vote

Gradient of scalar field

I don't have that book, but if $\partial_a$, $\partial_b$ are Cartesian coordinates, the condition you provided are the coordinates of $\nabla \times \mathbf{F} = \mathbf{0}$, with $\mathbf{F} = \...
  • 2,233
1 vote

Coordinates-free, geometry-free, algebraic way to derive facts about circular motions

Let's call $\vec{OM}(t) = \mathbf{r}(t)$. Proof of Point 1. From 5. you know that $|\mathbf{r}(t)|$ is constant and thus the derivative of its square is equal to zero $0 = \dfrac{1}{2} \dfrac{d |\...
  • 2,233
1 vote

Is torque possible without any pivot point?

The acceleration of the center of mass (CM) is determined by the net external force. The net torque is the change in the angular momentum if you use the CM as the point about which to evaluate torque ...
  • 7,036
1 vote

How to draw vectors?

For one thing, if $F_x$ and $F_y$ are components of $\vec F$, they are usually drawn not to extend beyond the projection of $\vec F$ in that direction. E.g. the $F_x$ arrow in Fig 1 should be much ...
  • 3,685
1 vote

Two forces act on a body as shown

Length of the arrows proportional to the magnitude of the force, an then draw the force parallelogram or triangle.
  • 4,095
1 vote
Accepted

Circular motion equivalent in three dimensions

We can consider circular motion at constant angular velocity in an arbitrary number of dimensions $D$. In 2 dimensions, we have formulas like $v=\omega r$, $F=v^2/r=\omega^2 r$. We're dealing with ...
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