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39

The following fact lies at the heart of this and many similar issues with sizes of things: Not all physical quantities scale with the same power of linear size. Some quantities, like mass, go as the cube of your scaling - double every dimension of an animal, and it will weigh eight times as much. Other quantities only go as the square of the scaling. ...

33

The units for torque, as you stated, are Newton-meters. Although this is algebraically the same units as Joules, Joules are generally not appropriate units for torque. Why not? The simple answer is because $$W = \vec F \cdot \vec d$$ where $W$ is the work done, $\vec F$ is the force, $\vec d$ is the displacement, and $\cdot$ indicates the dot product. ...

12

If you want to prove that $\vec{L}=\vec{r}\times \vec{p}$ is constant with respect to time for a particle in a central force field $\vec F = \phi(r) \vec r$, just show that the angular momentum doesn't change with time, i.e. $\frac{d}{dt}\vec{L}=0$. Using the product rule we get two terms: $\frac{d}{dt}\vec{L}=\frac{d}{dt}(\vec{r}\times \vec{p}) = ... 11 Take the right pedal as an example. It uses a right-hand thread, so turning the pedal spindle clockwise (CW) relative to the crank will screw the spindle in, counter-clockwise (CCW) will unscrew it. Say you put the bike in a repair stand, grab the right pedal and gently simulate the motion of someone riding the bike (always keeping the pedal platform ... 11 Newton's first law of motion for a point particle states that a particle at rest will stay at rest and a particle in motion will stay in motion unless acted on by an unbalanced force. In other words, if the net force on the particle is zero, then the velocity of the particle will stay constant. Newton's first law of motion for a system of particles states ... 11 This apparent paradox is actually not a paradox at all. Infact, it is because of Newton's second law, that we can say that the object will rotate exactly about it's centre of mass. This is true for any body on which net external force is 0, but net torque is present. In Newtonian mechanics, centre of mass serves to simplify calculations, for exactly the ... 10 You can grow arbitrarily large as long as you are essentially flat. For example, one fungus covers several thousand acres; there's a grove of clonal aspen trees that may have higher mass. Scaling in three dimensions is much harder, though. The pressure on the bottom is proportional to the height--eventually that pressure is too great for tissue to ... 9 The moment of a vectorfield$\vec{v}$at a position$\vec{r}$is equal to $$\vec{r}\times\vec{v}.$$ So torque is simply a special case where the vectorfield we look at is the force field,$\vec{v} = \vec{F}$. Another way of saying this is that torque is the moment of force. 9 The reason we distinguish the two is that torque is vector quantity, where as energy is a scalar quantity. So while we give the magnitude of torque the same units as energy, there is in fact additional information that tells us the direction the torque is applied. UPDATE: As dmckee has pointed out in the comments, to be perfectly corrected torque is a ... 7 There is indeed a term involving the time derivative of the changing coupling between the masses. First, let's derive the equation for a single mass. $$L = \frac{1}{2} I\, \dot{\theta}^2 - V(\theta)$$ $$\frac{\partial L}{\partial \dot{\theta}} = I\, \dot{\theta}$$ $$\frac{\partial L}{\partial \theta} = -\frac{dV}{d\theta} = \tau$$ $$\tau = \frac{d}{dt} ... 7 Yes, electric motors can always deliver the required torque. What really matters is the efficiency of the conversion. Over a long range of speeds, the motors may convert 90 percent of the energy to its mechanical form. You know, it's just some electromagnetic fields that may become arbitrarily stronger - depending on how much energy is sent into them - and ... 7 When studying angular things - torque, angular velocity, angular momentum, etc. - physicists do a clever thing to avoid having to describe curves. You see, you might be tempted to draw a curved arrow for a torque, indicating that you are twisting something around in a circular-ish way. But then when you try to add two such arrows together, all of a sudden ... 7 Since F = \phi(r) \vec r , you can find the torque around the origin. Torque \tau = \vec F \times \vec r = \phi (r) \vec r \times \vec r But \vec r \times \vec r is zero, so the torque around the origin is also zero. Since torque is just rate of change of angular momentum \frac{ d\vec L}{dt}, the angular momentum doesn't change, which is what ... 7 Since torque is defined as the rate of change of angular momentum, the more fundamental question would be whether angular momentum is a vector in SR. The answer is no, because there is no vector cross product in four dimensions. Angular momentum is a rank-2 tensor. 6 You are correct in your assertion that pairs of charged point particles can interact magnetically in ways that seemingly violate Newton's 3rd law, and therefore also seem to violate the conservation of both linear and angular momentum. This is a fundamental result and it is the decisive (thought) experiment which forces us to change our viewpoint on ... 6 As in the comments, there's certainly something of a convention at work here and it's to do with the "co-incidence" that we live in three spatial dimensions. As in Greg's answer, torque is intimately linked with angular momentum through Euler's second law. That is, torque and angular momentum are about rotational motion. And rotations, in general, are ... 6 No ,the centre of mass would not accelerate,the object will rotate about the centre of mass. If the object is on a table(or anywhere here on the earth) and if you apply the forces as shown in the fig it may not be rotated about the centre of mass because of the frictional forces acting on the object,but in space where there is no friction the object will ... 6 The torque and power curves are of course related (power is related to torque times RPM) but the maximum torque and maximum power are achieved at different RPMs and tell us different things about the car, therefore both are specified. Best is of course to look at a diagram like the one below, showing the relationships of the BMW 335i twin-turbo engine as ... 6 Torque is the informal, practical man's way of calling this thing; the moment of force is the more quantitative, scientific term which is better at expressing the formula$$ \vec \tau = \vec r \times \vec F $$The position \vec r and the cross product, for this specific case, are responsible for the words "moment of" while \vec F is the force. A ... 6 The basic answer is that mass scales with the cube of linear dimension and strength of things like legs scales with the square of the linear dimension. Note that large animals have therefore evolved comparatively thicker legs than smaller ones. Linearly scale up a dog to elephant size, and its legs would snap. Even more extreme, think of scaling a ant to ... 6 Power is force multiplied by velocity. The engine power is actually (relatively) constant regardless of the gear. So when people say there is "more power" in a lower gear, it's the common misconception that "more powerful" is "more forceful" but that's only part of the equation. So if P is constant, then that means if you can combine a large force and a low ... 6$$ \text{bhp} \propto \text{torque} \times \text{engine speed}$$To start with a simple example consider linear motion. If we have some force,$F$, the as the force moves a distance$d$the work done is force times distance,$Fd$. The power is the work done divided by the time,$Fd/t$, but$d/t$is just the velocity (distance divided by time) so the power ... 6 OK, the direction of precession of a gyroscope. The first image shows a gimbal mounted gyroscope wheel. From outside to inside there is a yellow housing and a red housing. I define three axes: Roll axis - the gyroscope wheel spins around the roll axis. Pitch axis - motion of the red housing. As you can see, the gimbal mounting ensures the pitch axis ... 6 Some engineering texts use "moment" and "couple" to talk about forces that tend to rotate an assembly (what physicist mean when they say "torque", but the engineers sometimes have a slightly different meaning for that word). A roughly translation guide is... A "couple" is a pair of opposite forces whose points of action are not co-linear. A couple is ... 6 The horizantal beam on such scales is intentionally placed below the rotational axis. As long as the weights are in equilibrium the torque is equal on both sides. But as soon as the position changes e.g. tipping the left scale down, the torques differ because only the tangential part of the gravitational force vector in relation to the rotational axis ... 6 First of all, choose the reference frame co-moving with the paddle and assume that this reference frame is inertial. This is the key to all ball-and-wall problems. Of course, ignore gravity and air drag. Now we have a spinning ball incident on a stationary surface with friction. Let$\mathbf{V}$be the vector of the ball's velocity with respect to the ... 6 Ingo, when you consider the couple, you may put one of the "spouses" at the origin, so his torque is$P\times d_0$for$d_0=0$, so his torque vanishes. Meanwhile, she is located at a nonzero$d$so her contribution is$P\times d$and nonzero. Because his torque is zero, it doesn't matter whether you add him or not. The only difference between the whole ... 5 OP wrote(v1): So the torque should not be measured in N⋅m but rad⋅N⋅m. Would that then be completely consistent? No, that would not be consistent with the elementary definition of torque$\vec{\tau}=\vec{r} \times \vec{F}$as a cross-product between a position vector$\vec{r}$and a force vector$\vec{F}\$. An angle in radians is the ratio between the ...

5

Torque is force at a distance. Work is force through a distance. Same unit dimensions, different measurements.

5

Joule and Newton meter are two units that are algebraically identical; you might say they are two names for the same unit. This is not the only example: Ohms is a unit of resistance, while "ohms per square" is an algebraically identical unit of sheet resistance. Hertz is a unit of frequency, becquerel is a unit of frequency in the context of radioactivity. ...

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