Plugging these in, we get two equations involving $\Delta p$, $v_ $, and $v_-$. Now, we know before and after the force acts, the dumbbell rolls without slipping. Similarly, the final angular momentum is $-I\omega_ $.) (note that the initial angular momentum is $-I\omega_-$, not $I\omega_-$, if we use the convention that positive angular momentum points out of the page. Since the force acts on the edge of the dumbbell, we know that $\Delta L=R\Delta p$. After this impulse, the dumbbell is again rolling without slipping. The net effect of this force is to create an impulse, $\Delta p=F\Delta t$. When the dumbbell hits the ground, there's going to be a frictional force on the dumbbell pointing to the right. Let $m$ be the mass of the dumbbell, and $I$ its moment of inertia. Let $r$ be the radius of the handle, and $R$ be the radius of the weights. Let $v_-$ and $v_ $ be the corresponding linear velocities. Let $\omega_-$ be the angular velocity before the dumbbell hits the ground, and let $\omega_ $ be the angular velocity after the dumbbell hits the ground. I am confused on the real answer, so I would really appreciate some concrete answers. This satisfied me until I heard rumors that the teacher (who enjoys being messing with students) said that the speed remains constant and was lying in class. However, the teacher later gave the explanation that the dumbbell would in fact increase in speed as when the handle was rotating, the weights would be rotating at the same speed and thus when it impacts, the weights which have larger circumference would have the same rotations/min as the handle (which has a smaller circumference) and thus as it is larger, with the same rotation rate, would increase speed. "If I have a dumbbell that is rolling down an incline with only its handle rolling (the weights are hanging off the sides of the incline), what would happen to it's speed when it tocuhes the ground and keeps rolling? Does it increase, decrease or stay constant?"Īt first I thought the answer was simple, it would decrease in speed as it would expend energy during its impact with the ground (I don't think this is a valid reason though). doi: 10.1137/0520100.As in the title, our Physics teacher gave us this brain teaser after learning introductory motion. "Transformations of the Jacobian Amplitude Function and its Calculation via the Arithmetic-Geometric Mean". "A comprehensive analytical solution of the nonlinear pendulum". The Pendulum: A Physics Case Study (PDF). Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences. "Sur une interprétation des valeurs imaginaires du temps en Mécanique". ^ "A Complete Solution to the Non-Linear Pendulum".(eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-5-5, MR 2723248 (2010), "Jacobian Elliptic Functions", in Olver, Frank W. The differential equation which represents the motion of a simple pendulum is The motion does not lose energy to friction or air resistance.the bob does not trace an ellipse but an arc. Motion occurs only in two dimensions, i.e.The rod or cord on which the bob swings is massless, inextensible and always remains taut.Since in this model there is no frictional energy loss, when given an initial displacement it will swing back and forth at a constant amplitude. This is a weight (or bob) on the end of a massless cord suspended from a pivot, without friction. Simplifying assumptions can be made, which in the case of a simple pendulum allow the equations of motion to be solved analytically for small-angle oscillations.Ī simple gravity pendulum is an idealized mathematical model of a real pendulum. The mathematics of pendulums are in general quite complicated. When released, the restoring force acting on the pendulum's mass causes it to oscillate about the equilibrium position, swinging it back and forth. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the equilibrium position. A pendulum is a body suspended from a fixed support so that it swings freely back and forth under the influence of gravity.
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