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4

I'll give you a couple of ways to think about this. First, geometrically, the circle you are thinking about drawing should contain the entire circular path of the car. If we're assuming that the car is remaining at a constant "elevation" on the banked surface, then the center of that circle has to be at the same elevation: otherwise you'd be drawing a cone ...


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Without loss of generality, we assume that a particle is moving in a circle of radius $R$ centered at the origin and lying in the $x$-$y$ plane. Using cylindrical coordinates $(\rho, \phi, z)$, the angular velocity of the motion is $\boldsymbol\omega = \omega \hat{\mathbf z}$. The velocity is tangent to the circle and given by $\mathbf v = ...


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At the instantaneous moment shown in the diagram, we can write: $$2R\alpha_{ring}=a_{disc}$$ as both are in pure rolling. This also tells us that the point on the ring where the thread is attached has an acceleration $=2R\alpha_{ring}=2a_{ring}$ so we find that: $$a_{disc}=2a_{ring}$$ Note that when the string moves to another position this will not be true, ...


3

Gravity doesn't have a horizontal component. The component of gravity normal to the plane in your diagram can be said to have a horizontal component, sure (and a vertical component of magitude $mg\cos^{2}\theta$). But there is also a component of gravity parallel to the plane of magnitude $mg\sin{\theta}$. That component can be resolved into a vertical and ...


3

The EV efficiency goes down very quickly with power output as $I^2R$ losses are mounting in the motor, the wiring, the power electronics and the batteries. Since an acceleration phase can only be so long (10-20s) and the car will not be used with constant velocity changes (unless it is designed for racing), the entire design will be optimized in such a way ...


3

See moment of inertia is analogous to mass. Moment of inertia can be thought of as a physical "property" of the object similar to that of mass. And as we know that mass does not depend on any force or gravitational field or any other external effect, so does moment of inertia. Hope this answers your question.


3

Motion is a very diffuse concept :) you have to add a frame of reference to make it meaningfull. In the frame of reference of the surrounding water the force definitely tries to stop the particle. So if you have a stone rolled along the ground by a swift stream, the force goes in the direction of motion (in the usual, external, frame of reference), since ...


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Rotation of a 3-vector We'll find an expression for the rotation of a vector $\mathbf{r}=(x_1,x_2,x_3)$ around an axis with unit vector $\mathbf{n}=(n_1,n_2,n_3)$ through an angle $\theta$, as shown in Figure . The vector $\mathbf{r}$ is analysed in two components \begin{equation} \mathbf{r}=\mathbf{r}_\|+\mathbf{r}_\bot \tag{01} \end{equation} ...


2

Remember that the variation of the angular momentum equals the external torque. If there are no external torque (as in your case), the angular momentum is conserved.


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The principle of conservation of angular momentum says that angular momentum remains conserved unless an external torque acts on it. The net torque on a body is defined as: $$\vec{\tau\,}=\dfrac{\mathrm d\vec{L\,}}{\mathrm dt}$$ We can clearly see from this definition that since external torque on the body is zero, the angular momentum is going to remain ...


2

To my understanding, work is done on object A when object B is applying a force on object A, causing object A to displace. Work is done whenever a force displaces an object. Since energy is the ability to do work, what work does a moving object do, due to its kinetic energy? A moving object might not do any work at all. Imagine an empty ...


2

In your scenario, angular momentum $m v r$ is preserved (because your pulling force is radial, with no tangential component). So if you reduce $r$ by half, $v$ must double, and since $\omega = v/r$, it increases by a factor of four. Note this means in a small amount of time that the area swept out by the string is proportional to $v$ and $r$. Since they ...


2

There isn't a "centripetal force" vector. As the car goes around the banked curve, the normal force on the car increases relative to what it would be on an un-banked straight road. The vertical component of the normal force supports the weight of the car, and the horizontal component of the normal force provides the centripetal force necessary to cause the ...


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Any time you have an equal and opposite force acting on two bodies, the effect is proportional to $\frac{1}{m}$ for each one. The combined effect is proportional to $$\text{(effect)} = \left( \frac{1}{m_1} + \frac{1}{m_2} \right) \text{(action)}$$ When inverted to find which action has a desired effect we get the reduced mass $$ \text{(action)} = \left( ...


2

To understand this, use the definition of force $\frac{d{\bf p}}{dt} = {\bf F}$, namely the force is equal to the rate of change of momentum. Something like a collision can be very complicated to model, but the average force is approximately given by ${\bf F}_{average} = \frac{\text{change in momentum}}{\text{time taken}}$. Typically, in a collision, the ...


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In general it changes although the reason is not exactly because its projections changes. For example. You start with a vector (let us say the electric field of a parallel plate capacitor) on the plane $xy$. Then you rotate the coordinate system by an angle. The components of the vector on the new coordinate system is changed. But the vector did not change ...


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What happens during folding is that the material undergoes plastic deformation. When a sheet of material is bent slightly that deformation is usually elastic, meaning it will return to its original shape when the deforming stress is withdrawn. But when the deformation is larger we enter the plastic zone: the material will no longer fully recover its ...


2

In the case of inorganic matter, such as metal sheets, folding/creasing produces a substantial bulk stress in the material which can modify the molecular structure in a large number of complex (and not fully understood) ways. For example, it can break bonds, cause amorphisation, and propagate dislocations. These same mechanisms are at play when you cut a ...


1

Since acceleration is a vector you can decompose it in the coordinate system you find convenient. If you define a cartesian coordinate system whose axis are along the normal to the plane and the plane itself you see there is a component of the acceleration $g\sin\theta$ along the plane. This is why the block accelerate in this direction. Notice that along ...


1

The normal force is not playing a role in this case because the force is perpendicular to the moving direction of the box. By "not playing a role" I mean, during this motion, $\ F_N$ is only be used to balance out $\ mgcos\theta $ so that the object won't be able to move "into the inclined plane". To find the force causing the block to accelerate down, we ...


1

what's the backward force to balance $F_{fs}$ so as to keep the car moving uniformly? Is that the Force produced by engines? The forwards force comes from the torque produced by the engine of the car and is transferred into the ground via static friction. The retarding force that keeps the car moving at the same speed is mostly air resistance (as well ...


1

I don't want to prove you that mass generates gravity. As ja72 already said there are hundreds of years of experience on that. Instead I just want to show you how weak and refutable are your arguments. My opinion is that it is the rotation that causes gravity, when we see a whirlpool in water it takes anything that comes near to it in to the center ...


1

If you measure a force (weight) for a given acceleration (gravity) in order to determine the mass of an object and you haven't started measuring then the mass is undefined. As soon as you apply an acceleration $a>0$ and you measure corresponding force $F>0$ you can determine the mass. Equations are useful only when they can be used to measure things, ...


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A physical quantity is a vector if it transforms in the same way as a position vector when the coordinate system undergoes a transformation.


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How do we formally define vectors in physics? An excerpt from chapter one, page 12 of "Mathematics of Classical and Quantum Physics" Originally, we introduced a vector as an ordered triple of numbers. The rule for expressing the components of a vector in one coordinate system in terms of its components in another system tells us that if we ...


1

The force of friction is defined as $F_f = \mu N$, where $N$ is the normal force. In the case of a flat surface free of external forces, you can use Newton's laws to determine that $N = mg$, where $m$ is the mass of the object. Notice that we have made no reference to the objects size, or area of contact. This is because in these examples we have ...


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Liquids are compressible but in lots of cases the fact that the volume of water decreases by about fifty parts per million for an increase in pressure of one atmosphere can be ignored. A similar approximation is also often made about the compressibility of solids.


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Short answer: It can be compressed, due to extreme pressure, but only slightly. The Wikipedia page on the Mariana Trench says that the seawater density is 4.96% greater there than at the surface. Also refer to Hydrostatic pressure - doesn't density vary with depth?. Further explanation (and assuming everything you currently know about water is that ...


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You will need to use two free body diagrams, one for each mass, one FBD will include the force of A on B and the other one the force of B on A. Both mass have the same acceleration (do they), so you end up with a little system of three equations and three unknowns.


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You should take moments about the points r1 and r2, which then you will get two seperate equations, which is what you have done, and that's fine. Take another look at both of your equations, r1 and r2 are forces and since you are taking moments about both those points, you need to multiply r1 and r2 by their distance. What you have essentially done is equate ...



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