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The reason why a gyroscope does behave in this strange way is that if you try to rotate it's axis in some direction, the "endpoints" of this axis have to be pushed perpendicular to what our first intuition would say. In order to verify why the axis starts rotating in this strange way, let's make some simplifications: the gyroscope consists of two identical ...

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Any rigid body in motion can be described as rotating about in instantaneous axis of rotation (IAR) and translating along the same axis at the same time. Example/Proof A rigid body in moving and at time instant a point A riding on the rigid body has position vector $\vec{r}_A$ and instantaneous linear velocity $\vec{v}_A$ at A. The whole body is rotating ...

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1- Is it right to say "the motion of the robot can be described as a transitional motion of center of mass plus a rotational motion about that point?" Pick a point on (or off) your robot; pick any point. The motion can always be described in terms of the translational motion of that point plus a rotational motion about that point In general, the ...

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It seems to me what you're asking is pretty simple. You say you can control the angular velocity of each wheel. That, times the wheel radius, give you the forward velocity of each wheel on the ground. That tells you the robot's forward speed (the average of the forward speeds of the wheels), and it tells you the rate at which the robot is turning (the ...

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As one of the comments mentions, it is simpler to consider a linear case. Dropping a body of mass $m$ on one moving with mass $M$ and velocity $v$ is essentially considered the instantaneous transformation $M \to M + m$. Momentum must be conserved in the collision, but the mass of the system effectively increases, producing a smaller kinetic energy:  ...

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Energy is conserved, but if you ignore some kinds of energy then it will look like it isn't conserved. You can imagine a really big disk with some radial pointing two by fours attached at the one o'clock, two o'clock etcetera positions then attach springs to each two by four with the spring pointing in the clockwise/counter-clockwise directions. Add a nice ...

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Does a body always rotate purely about its center of mass? Well, that depends. The first assumption you need is that the body is rigid. Violate this assumption and all bets are off the table because you can't even necessarily classify all motions as "rotations": for example if a long thin board starts twisting sinusoidally into/out-of a helix shape, ...

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Say the projectile was thrown with a velocity $v$ at an angle $\theta$ with respect to the horizontal. We ignore all friction effects (air drag, side winds). Define a coordinate system with a vertical $y$-axis, a horizontal $x$-axis and the point of origin $O$ the point from which the projectile starts its trajectory. The trajectory can now be decomposed in ...

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Faced by the same question and a background that includes courses in vector calculus, I have sought a simpler answer. My answer is much that same as to why one can easily balance on a typical bicycle. Bicycles are constructed so that the point where the extension of the front fork pivot would hit the ground is in front of the point where the front tire ...

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