New answers tagged

1

The above isn't drawn correctly. And it is far too complicated in my opinion. I think I was able to solve this using high school physics: This bugged me a long time. I am surprised no one ever posted an answer on the internet. The length of his rope was 60.8 meters, assuming a drop above the target building of about 12 meters. The difference in height ...


1

If you've got an older web browser kicking around that still runs Java applets you should check out Paul Falstad's Loaded string simulation. You can add harmonics to your heart's content.


2

Basically Oscars answers says it all, but I just want to add a few more things. When a string is plucked its motion need to follow the wave equation $$ \frac{d^2}{dt^2}y(x,t) - c^2 \frac{d^2}{dx^2}y(x,t) = 0 $$ with Dirichlet boundary conditions (the ends of the string are fixed). $c$ is the speed of sound of the string's medium. The function $y_n(x,t) = \...


2

If a string has multiple waves expressed in it, this is done by adding the waves individually. Each frequency in the harmonic series can be expressed by a wave, a guitar string is the sum of these waves in different proportions. The resulting wave is significantly different than the others. See below for the sum of the first three frequencies in the ...


8

Carrying out the Fourier transform, I get a slightly different result for the frequency spectrum than 'knzouh'. I used $u$ instead of $y$ and $c$ instead of $v$, so the PDE becomes: $$u_{tt}=c^2u_{xx}-Au_{xxxx}$$ Fourier transforming the equation: $$F\{u_{tt}\}=F\{c^2u_{xx}\}-F\{Au_{xxxx}\}$$ Transforming $x$ to $k$: $$\hat{u}(k,t)=\int_{-\infty}^{+\infty}u(...


20

In plain English, there is stiffness at the ends of the strings where they are fixed in place, which makes the string's frequency of vibration slightly higher (sharper)—effectively shortening the length of the string slightly, for all practical purposes. And the resistance to bending is dependent on the frequency. It behaves more “stiffly” with regard to ...


156

This effect is known as inharmonicity, and it is important for precision piano tuning. Ideally, waves on a string satisfy the wave equation $$v^2 \frac{\partial^2 y}{\partial x^2} = \frac{\partial^2 y}{\partial t^2}.$$ The left-hand side is from the tension in the string acting as a restoring force. The solutions are of the form $\sin(kx - \omega t)$, ...


0

Somewhere in between $T_1$ and $T_2$ -- but exactly what depends on details about how the friction between the string and the pulley varies, how the string stretches under tension and the pulley deforms while being accelerated by the string ... All of these are things that cannot be deduced from an idealized picture such as this. In one extreme, if the ...


1

Tension $T$ is the reactive force in the string when you pull at both ends with force $F$. If the string has stopped stretching or accelerating then $T=F$. The action-reaction pair of forces in Newton's 3rd Law are two equal but opposite forces with the same cause which act mutually on two different bodies : object A exerts force $F$ on object B, and B ...


1

To simplify this answer assume that the string is made up of a line of molecules so each molecule bar the end ones have only two nearest neighbours. When there is no tension force in a string then on average the molecules which make up the string are at their equilibrium spacing and have no net force acting on them. Imaging that you apply a force $F$ on ...


1

If the rope is in static equilibrium then $T=F$. If $T\neq F$ then that section of the rope (where tension is $T$) must be accelerating, which may happen if the rope is slack or if it is extensible.


1

For a pulley that has mass and moment of inertia, there must be a net torque on the pulley for the pulley to demonstrate an angular acceleration. Assuming that mass "M" is greater than mass "m", a net torque necessarily requires that the counterclockwise torque from mass "M", given by the equation Torque1 = T1(R), is larger than the clockwise torque from ...


1

Yes. Tension can vary if external forces are acting between the ends of the string - such as gravity (if the string has mass) and friction where the string makes contact with other objects (such as the pulley). For example, suppose you attach one end A of a uniform massless string to a support and the other end C to a vertically hanging mass M. This ...



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