PHYS2001_Ch. 10

PHYS2001_Ch. 10 - Ch. 10 Simple Harmonic Motion 10.1 The...

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Unformatted text preview: Ch. 10 Simple Harmonic Motion 10.1 The Spring Force Probably all of you have some familiarity with a spring. If you push on a spring, it pushes back. If you pull on a spring it pulls back. And the harder you push or pull on the spring, the harder the spring pushes or pulls back. If youre careful not to pull too hard, then you will find the following to be true: x F Applied In words, the force you apply to the spring is directly proportional to the springs displacement. We can make this proportionality an equality by multiplying the right side by a constant: kx F Applied = The proportionality constant, k , is called the spring constant . Units? [Force]/[Displacement] = [N/m] Forces which are proportional to the displacement, like the spring force, are called elastic forces . The spring constant is a measure of the stiffness of the spring. The larger the spring constant, k , the stiffer the spring. F x Elastic region Slope = k Inelastic region If I pull or push on a spring with force F , then Newtons 3 rd Law tells me that the spring pushes or pulls back with an equal and opposite force. This force is called the Restoring Force . For an ideal spring the restoring force is given by: kx F- = If I pull on the ball with force F , the spring pulls back on the ball with restoring force F . F-F Hookes Law kx F- = The minus sign in Hookes Law indicates that the restoring force, F , always points in a direction opposite to the displacement of the spring. When the displacement is proportional to the restoring force, but in the opposite direction, like Hookes Law ( F = -kx ), the motion that results is called Simple Harmonic Motion (SHM). So what does SHM look like??? Lets consider a mass hanging vertically, connected to a spring. When I attach the mass to the unstretched spring, the weight of the mass pulls it down. The mass now hangs in a new equilibrium position. Notice how the forces cancel. F s W If I now pull down on the mass with some force F , the spring pulls back with an equal but opposite restoring force. And if I release the mass, then it executes SHM about its equilibrium position. -F F D The maximum distance the mass moves both above and below the equilibrium position is A , called the Amplitude of the Motion . So the mass is undergoing SHM. Lets make a plot of the y-position of the mass as a function of time. y t The position of the mass is periodic , or harmonic , in time, i.e. it repeats. Notice it oscillates between + A and A , the amplitude, and the motion repeats continuously in the absence of friction. +A-A Lets define a few quantities that we will use to characterize the SHM. y t +A-A Period : The time it takes to complete 1 cycle, T ....
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This note was uploaded on 09/22/2009 for the course PHYS 2001 taught by Professor Sprunger during the Spring '08 term at LSU.

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PHYS2001_Ch. 10 - Ch. 10 Simple Harmonic Motion 10.1 The...

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