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$\bullet$ $\bullet$ A block with mass $M$ rests on a frictionless surface and is connected to a horizontal spring of force constant $k,$ the other end of which is attached to a wall (Figure 11.41 ). A second block with mass $m$ rests on top of the first block. The coefficient of static friction between the blocks is $\mu_{s}$ . Find the maximum amplitude of oscillation such that the top block will not slip on the bottom block.

$\frac{\mu_{\operatorname{gg}}(M+m)}{15}$

Physics 101 Mechanics

Chapter 11

Elasticity and Periodic Motion

Equilibrium and Elasticity

Periodic Motion

Cornell University

Rutgers, The State University of New Jersey

University of Washington

Hope College

Lectures

04:12

In physics, potential energy is the energy possessed by a body by virtue of its position relative to others, stresses within itself, electric charge, and other factors. The unit for energy in the International System of Units is the joule (J). One joule can be defined as the work required to produce one newton of force, or one newton times one metre. Potential energy is the energy of an object. It is the energy by virtue of an object's position relative to other objects. Potential energy is associated with restoring forces such as a spring or the force of gravity. The action of stretching the spring or lifting the mass is performed by a force which works against the force field of the potential. The potential energy of an object is the energy it possesses due to its position relative to other objects. It is said to be stored in the field. For example, a book lying on a table has a large amount of potential energy (it is said to be at a high potential energy) relative to the ground, which has a much lower potential energy. The book will gain potential energy if it is lifted off the table and held above the ground. The same book has less potential energy when on the ground than it did while on the table. If the book is dropped from a height, it gains kinetic energy, but loses a larger amount of potential energy, as it is now at a lower potential energy than before it was dropped.

02:18

In physics, an oscillation is the repetitive variation, typically in time, of some measure about a central value or between two or more different states. The oscillation may be periodic or aperiodic.

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So in this problem, the maximum acceleration of the lower block can't exceed the maximum acceleration that can be given to the other block by the friction force. And so for him, for the smaller block sitting on top, the friction force which is static in this case, is equals to the coefficient of static friction times the normal force and the normal force. Due to a balance of forces perpendicular to the direction the ramp is equal to mg. And so now some of the forces along the direction of the ramp for the small mass on top is equal to the mass of the small mass times acceleration. This implies that Mu S O M G, which is here, is equal to mass times acceleration. And this implies that the acceleration is equal to the coefficient of static friction, times gravity. And so now let's set this aside And let's look at hooks law for the combined system, folks, Law says that F is equal to negative. K X within the force of their combined system is equal to the mass times acceleration command system. Solving this for a gives A is equal to negative ke eggs over him, but the now I'm going to figure out what the maximum acceleration is, and the way you do that is to plug me in the amplitude for the ex, since that is where the acceleration is maximum. I also admitted the negative sign, since it only tells us the direction which I don't care about and writing the maximum acceleration. But this is the combined system. I was applying Newton's second law here for the combined system, which means that little him here is actually little AM plus Bigham. Since these air the two masses of the blocks, it's the massive combined system. Now we're going to set the maximum acceleration to be this acceleration that we found here and we're going to solve for the amplitude. Whenever I do that, I give that the amplitude is equal to the coefficient of static friction times Gravity Time's a little in plus Bigham over the spring constant, and that is the amplitude that this system we'll move at

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