New answers tagged

1

Abrasive is intended to increase friction. You want your brake pads to have (and retain) a certain shape, so you use some structure material that resists well to changes in temperature. Not sure what performance material is supposed to be. In order to limit the costs, in places where no structural integrity is threatened, and no friction exists, you can use ...


0

The motion of N atoms in three dimensions (x,y,z) produces 3N degree of freedom. Every molecule also has whole body rotation (as the atoms are now bonded together) about each of the 3 axes and translational motion along each axis making 6 motions altogether. If the molecule is linear, rotation about the principal symmetry axis in not measurable so there ...


2

Often $X$ is a coadjoint orbit of a Lie group. These have a natural symplectic structure; see https://en.wikipedia.org/wiki/Symplectic_reduction


1

From the equations $\dot q = p/m$ and $\dot p=V'(q)$ and the definition $Q=D(q)$ you can derive by differentiation $\dot Q=D'(q)p$ and $\ddot Q=D''(q)pp-D'(q)V'(q)$. The second equation produces an ODE for $Q$ if you can express $p$ in terms of $\dot Q$ up to terms in the null space of $D''(q)$ (i.e., translation and rotation degrees of freedom). This should ...


0

Friction force always opposes relative motion between two surfaces. In most cases (like yours), one of those surfaces is fixed. So, you should recognize that how does the other surface tend to move. For this purpose, you should consider that what force or torque want to move the body. Then, you can determine the correct direction of friction. While ...


0

The book you are looking for is called "A decade of SIN plus 16" it has questions and solutions from first twenty six years of the Sir Isaac Newton (SIN) contest run by the University of Waterloo. check the Where can I find more practice exams question here


3

Your reasoning is essentially correct, apart from the last paragraph. To conclude, note that Newton's equation: $$\ddot {\mathbf r}(t) = \mathbf f(\mathbf r (t)),$$ with initial condition $\mathbf x (0)=(x_0,0,0)$, $\dot {\mathbf x} (0)=(0,0,0)$ can be solved by puttin $y(t)=z(t)\equiv 0$, thus reducing to a one dimensional problem: $$\ddot x (t)=f(x(t)),$$ ...


2

Mass removal due to wear relates to sliding distance. This gives to wear rate relates to sliding speed. However, wear rate is not only only a function of sliding speed. Surface hardness also plays a role. AL-SI, according to this paper, will be hardened in the beginning session. With the surface hardness increased, the wear rate decreases. When it can no ...


0

I shouldn't give a direct answer since this is an exercise question. Since the System (Man+seat) is rotating on a parallel to the ground plane. That means that there is no motion on the vertical axis-y. That means that the total force in this axis is zero. $$ \sum \vec F_y = \vec 0$$ You have to find what are the forces on this axis. It can easily be done ...


3

It seems relevant to mention the importance of distinguishing between explicit, implicit, and total time-dependence. The Lagrangian $L=L(q,v,t)$ depends implicitly on time via the position $q$ and the velocity $v$. The total time derivative of the Lagrangian $L=L(q,v,t)$ is $$\underbrace{\frac{dL}{dt}}_{\text{total $t$-derivative}}~=~\underbrace{\frac{\...


1

The Lagrangian only depends on the potential energy and the kinetic energy. What the statement you quoted means is that if both the potential and kinetic energies are constant w.r.t. time, then so is the Lagrangian. This makes a lot of sense. Usually, we have: $$\mathscr{L}=K(x,t)-P(x,t)$$ Where $K$ and $P$ are the kinetic and potential energies. But if ...


1

We must all keep in mind that for average atmospheric pressure, and assuming Zhang could pull a hard vacuum with his abdomen (which is probably not feasible, but serves to provide us with a bound), the maximum (negative) pressure he could achieve is only about 14.7 psia. Given 36 tonnes, you can back calculate what the diameter of the bowl would have to be. ...


3

What should be the minimum value of $v_0$ in order to hit the monkey while it's in air? Minimum value for $v_0$ is when arrow hits to the monkey just before it (monkey) reaches to the ground. Or, minimum value for $v_0$ is when arrow's range is equal to horizontal distance between hunter and monkey. So you need to find the time of monkey's fall. Or, you ...


1

The exact mechanism you describe for how suction cups work is how the rice bowl work. Instead of the bowl being flexible, though, it's his body (skin and muscles) that are providing the change in volume necessary for the suction. So, instead of the suction cup creating the volume change, it's the surface the suction cup (the bowl) is attached to, Mr. Zhang'...


0

Classical Mechanics: Systems of Particles and Hamiltonian Dynamics by Walter Greiner. This is a very good book for the same reasons that all the books belonging to the series of books written by Greiner are good. They are clear, they do not shy away from mathematics (they are written for people who want to pursue theoretical physics) and they have many ...


0

Capacitors connected in series $$\frac{1}{C_{\mathrm{total}}} = \frac{1}{C_1} + \frac{1}{C_2}$$ (Where $C$ is the capacitance) Capacitors in parallel $$ C_{\mathrm{total}} = C_1 + C_2$$ As you can see, capacitors are total opposite of resistors when connected in series or parallel More capacitance or $C$ means the capacitor can store more charge. As ...


2

The forces acting on the box will be those by 'gravity', the normal reaction force by the table and the friction force by the table. The force of gravity has two components (passing through the center of inertial mass owing to the equivalence principle) along the surface of the table and normal to it. The friction acts along the surface and the normal ...


4

Angular momentum, or a measure of rotation, in very large astrophysical or cosmological bodies and energy can become relativistic, and must be treated using General Relativity. In General Relativity (GR) angular momentum is not too different from momentum and energy. Different quantities, the last two are contained in the stress energy momentum tensor and ...


7

This question leads to some subtleties. There are at least two distinct notions of "revolution" that could be meaningful in physics. Namely, "to revolve" can mean: To have angular momentum; To transform by a particular kind of Euclidean isometry (a rotation) (or, to be broader and more technical, a representation of that Euclidean isometry). As far as we ...


0

There are other shapes of galaxies. In particular, look at ellipticals.


7

Without touching on electromagnetism, I'd like to bring up this construction from mechanics (it's in the Feynman lectures). Consider two equal particles approaching each other with equal speed. A----> <----B You can argue from first principles that if they stick together they will not be moving afterwards -- any argument you could make ...


0

I think that a good point to start are decent introductory books about continuum mechanics in general to get an idea about how the framework works in general and how systems are described, how different quantities are transported. For this I myself really liked as a starting point: Liu, Continuum Mechanics, Berlin, Heidelberg: Springer, 2002 It is by far ...


2

Newton's second law, force f is $$f=m\frac{d^2 x}{d t^2}$$ x is position vector of the particle. $$f=-\frac{d v}{dx}$$v is the potential energy. $$m\frac{d^2 x}{d t^2}=-\frac{d v}{dx}$$ Multiply both sides with $\dot x$ $$\frac{m}{2} \frac{d\dot x^2}{dt}=-\frac{dv}{dt}$$ $$ \frac{d}{dt}(\frac{1}{2}m\dot x^2+v)=0$$ i.e., $$\frac{dE}{dt}=0$$Energy is ...


2

The flaw is your assumption that In this [accelerating] frame we don't see any force so the first law of dynamics is respected. In the accelerating reference frame you do see evidence of a force, even though you don't see the effect you are expecting (acceleration of the object). Like an observer standing on the surface of the Earth, acted on by the ...


10

As the wiki article you quote states, momentum is defined as the product of the velocity times the mass of an object. Classical mechanics developed theoretically on the lines explained by WetSavanna in the other answer, the conservation of momentum and energy being cornerstones of the theory. Classical mechanics is a very successful theory, and ...


45

Momentum / energy are the conserved Noether charges that correspond, by dint of Noether's Theorem to the invariance of the Lagrangian description of a system with respect to translation. Whenever a physical system's Lagrangian is invariant under a continuous transformation (e.g. shift of spatial / temporal origin, rotation of co-ordinates), there must be a ...


5

Newton's third law tells us that the momentum imparted on one body is equal and opposite to the momentum imparted on another if they interact. We then have $$ \Delta \vec p_1~=~-\Delta\vec p_2. $$ The change in momentum is $\Delta \vec p_i~=~m\vec a_i\Delta t$, $i~=~1,~2$. The change in momentum is with Newton's second law due to a force so that $$ \vec F_1~...


4

The flaw is that you've failed to do an experiment which will tell you whether the frame is inertial or not. If you do such an experiment -- for instance take a test mass, initially at rest with respect to the frame, release it, and see if it remains at rest -- you will immediately discover that the frame is not inertial.


1

I believe the analysis of the Norton Dome is flawed (as many philosophers thought experiments). The ball does not stay at rest and start to move spontaneously in the absence of any force. If there were no forces it will stay there forever. The reason it starts to move is some small perturbations. They could be either external (random variations in pressure ...


0

Your formula itself is wrong. $$ \frac{d\vec{r}}{dt} = \vec{v} = \omega \times \vec{r}$$ Angular Velocity ($\omega$) is an axial vector whereas velocity is a polar vector. Axial vectors are vectors whose directions oriented in a non-intuitive way. For example, when you rotate a disc by a certain angle ($d\phi$), the $d\phi$ vector lies along the axis of ...


0

Those definitions are either for the mean velocity and the mean acceleration respectively, or for the instantaneous speed for non-accelerated motion (the first). To find the instantaneous velocity, acceleration or actual position over time in the general case you need to solve the equations of motion, which will result in different "formulas" than those ...


0

If you already know the path to be taken, along with the time stamp for each position; or having measured the actual path, and the corresponding times, then in either case you have all of the information for the kinematics. The more usual problem is that you start with less information, but you know the dynamics: then Newton's Laws of Motion can be used ...


4

The first form is true whenever the Forces are derivable from a scalar, i.e when $Q_i=-\frac{\partial V}{\partial q_i}$ The second equation however is true even when none of the forces can be derived from a scalar The third is true when some of the forces are derivable from a scalar and some are not, i.e. $L$ contains potential of the conservative forces ...


1

I think you are confusing static and dynamic or are trying to separate the two in a way that isn't helpful and leads to 'over-thinking' of a simple problem. Let's look at the force diagram again: In the point $P$ a downward (but as you stated not necessarily vertical) force $\vec{F}$ acts on the end of the string. We decompose it into two components: $\...


1

This might further clarify (or confuse) things: The force that an object exerts on the string is equal in magnitude to the force that the string exerts on the object. The force that an object exerts on the string (or similarly the force that the string exerts on the object) is not necessarily equal in magnitude to the component of a force acting on the ...


1

Your statement is quite a complex one, involving several clauses. It is not clear which clause is the focus of your question. You state that the string is vertical and the applied force is downward, so the force $F$ has no component perpendicular to the string, only in the direction of the string. You do not explain what you mean by the "constraint ...


2

You would be right, if the string is like a real string in that it's flexible (that is, can neither exert transverse force nor bear transverse loads). I have to point out here that you haven't described a system (classically, you couldn't draw a static free body diagram for the string)--to be static, the entire force $\vec{F}$ would have to be parallel to $\...


4

Experimental results can deviate from ideal. Outcomes depend sensitively on small differences in mass and alignment, and the extent to which kinetic energy is conserved. Alignment of non-identical balls is much more difficult. Considerable effort and expense may be required to achieve reliable results. The following sites agree with this view : https://...


0

One different aspect to this in case we are really talking about a living planet. Because gravity really kicks in at that scale, anything the size of a planet is so smoothly round that a polished billard ball feels ashamed for its own imperfection. So if this being is really planet-sized, it better be of really low density ...


3

The notion arises fairly naturally when one is thinking about how a constant cross section beam reacts to forces and bending moments. You imagine a beam to be made up of elastic fibers. When it is bent in a plane, nearest the center of curvature, the fibers must "squash", so they are in compressive strain. On the side outermost from the curvature center, ...


0

One of the most common uses for the area moment of inertia is to analyze the bending of beams. A beam which has a large area moment of inertia is harder to bend by applying a given load than a beam with a small area moment of inertia. You can see this by taking a long strip of wood or other material, like a ruler, where the cross section has a small ...


2

Replace the salt with tea leaves, and you'll find the answer... with wikipedia. The tea leaf paradox is exactly what you describe: denser than water particles accumulate in the vortex center. This is an indirect effect of the pressure gradient cited by philip_0008. As you stir, the liquid accumulates at the periphery so that the extra height generates a ...


0

I've actually tried it when I was younger, intentionally stirring water circularly very fast in a large basin, such as to make a large 'vortex', and noticed that everything that I put in goes toward the center. The reason seems to be that: From wikipedia: "The fluid motion in a vortex creates a dynamic pressure (in addition to any hydrostatic pressure) that ...


0

One candidate for a force that will oppose the particle's inertia is the so-called inertial lift that is created in the neighbourhood of the wall: grossly speaking, the flow created around the particle interacts with the wall and pushes the particle away from it. You can see this paper: http://www.pnas.org/content/104/48/18892 ...but it will not dominate ...


11

Thanks to Michael Seifert's answer, I found a paper he referenced: Satellite Relocation by Tether Deployment by G. A. Landis and F. J. Hrach, 1989. By extending a tether radially, a satellite can increase or decrease its orbital speed (pictures below copied from the paper): The principle then can be used to pump an eccentric orbit: Similarly a planet ...


1

Fun question! Try this very simple answer (Newtonian as you asked).. If a planet changes its shape from a round ball into a twine spool like shape (i.e. with a thicker center and elongated thinner ends) and assuming the elongation is done exactly along the radial line to the star (Sun), then for simplicity sake, the amount of mass that gets closer to the ...


6

You can always use the process of tidal acceleration/deceleration. In nature this process might be very slow, such as for the system Earth/Moon. However, you can alway speed it up, by artificially increasing the frequency of the shape oscillations. In a natural system tidal acceleration will stop when the two objects are in tidal locking (both object ...


6

I'm not sure if it helps you with your students, but maybe gives you some background: I guess the underlying reason for orthogonal basis vectors is that you are implicitly using a euclidean metric that will just have diagonal values. These would e.g. be $$g_\mathrm{\mu\nu, ~euclidean}=I=\pmatrix{1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1} \...


14

A different mechanism: On a long timescale, by increasing the surface area exposed to the sun (flattening the planet), the radiation pressure would increase, boosting to a higher orbit. Changing the albedo would be a more effective means to the same end but could allow assymetric force as well Either way it would be simpler in a tidally-locked planet. ...


67

If you allow for non-Newtonian gravity (i.e., general relativity), then an extended body can "swim" through spacetime using cyclic deformations. See the 2003 paper "Swimming in Spacetime: Motion by Cyclic Changes in Body Shape" (Science, vol. 299, p. 1865) and the 2007 paper "Extended-body effects in cosmological spacetimes" (Classical and Quantum Gravity, ...



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