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Actually, spinning the pens bring a relative motion between cap and pen. When the surfaces of two objects are at rest with respect to each other static friction force acts between them and a kinetic friction force acts between then when they are in relative motion. Static frictional force $>$ kinetic frictional force. You may want to read about the ...


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Firstly I don't understand how if the ball is initially rotating how it gains rotational energy from the top of the plane to the bottom, since there is no friction to provide torque. Your understanding is mostly correct. We can choose any point on the object to measure rotation around; let us choose the center of the ball. Both the gravitational force ...


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No. Even if you include some additional things for angular momentum there are still many things not in Newton's laws of motion. For instance, the fact that mass doesn't change from one value on Tuesday to a different value on Wednesday. Newton talks about mass in the Principia but it isn't about motion or forces so it isn't part of his Laws of Motion. ...


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You are overcoming the static friction, which others have noted is generally greater than dynamic friction. But, the force needed at the point of contact is presumably the same when twisting versus when lifting, so why the difference in the behavior of the stand (lifting up, versus just sitting there)? The reason is because of the direction the force is ...


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The general equation is $$x(t) = x_0+ {\bf e}^{-\beta t} \left( A \sin \omega t + B \cos \omega t\right)$$ where the constants $A$ and $B$ depend on the initial conditions, and $\beta$ and $\omega$ on the mechanical properties of the system. Read http://hyperphysics.phy-astr.gsu.edu/hbase/oscda.html for more information For example if the initial ...


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It sounds like you are interested in symplectic reduction procedures. On of these methods is that of Routh's procedure to eliminate cyclic variables using a Legendre transform to a reduced-variable Hamiltonian called a Routhian. Forming a variational approach may be difficult for some reduction procedures, however we can view conserved quantities as ...


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While neither the Lagrangian $\mathcal{L}$ nor the action $S$ are invariant under boosts of the form $$\dot{q}(t) \to \dot{q}(t) + v, \quad v \in \mathbb{R},$$ the Euler-Lagrange equations are. The dynamics of the systems are unchanged for any transformation that preserves $\delta S = 0$, i.e. a transformation of the form $$ \mathcal{L}(q, \dot{q}, t) \to ...


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Why is it that two carbon atoms fired at each other will bounce off and not stick together? It is because as the atoms move close together their orbital electrons begin to repel more than their nuclei attract each other's electrons. The result is greater potential energy as they approach and this leads to the tendency to move apart much like compressing a ...



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