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Damped Harmonic Motion on a Spring

by Johnny0290
Tags: damped, harmonic motion
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Johnny0290
#1
Apr29-10, 11:29 PM
P: 9
1. The problem statement, all variables and given/known data

In this problem we will investigate a particular example of damped harmonic motion. A block of mass m rests on a horizontal table and is attached to a spring of force constant k. The coefficient of friction between the block and the table is mu. For this problem we will assume that the coefficients of kinetic and static friction are equal. Let the equilibrium position of the mass be x = 0. The mass is moved to the position x = +A, stretching the spring, and then released.

2.1 Apply Newton's 2nd law to the block to obtain an equation for its acceleration for the first half cycle of its motion, i.e. the part of its motion where it moves from x = +A to x < 0 and (momentarily) stops. Show that the resulting equation can be written d2x'/dt2 = -omega^2 * x', where x' = x - x0 and x0 = mu*m*g/k. Write the expression for position of the block, x(t), for the first half cycle (be sure to express omega, the angular frequency, in terms of the constants given in the problem statement). What is the smallest value of x that the mass reaches at the end of this first half cycle?


2.2 Repeat the above for the second half cycle, i.e. wherein the block moves from its maximum negative position to its (new) maximum positive position. First show that the differential equation for the block's acceleration can be written d2x''/dt2 = -omega^2*x'' where this time x'' = x + x0. Next, match the amplitude for the beginning of this half cycle with the amplitude at the end of the last one. Write the expression for the position of the block, x(t), for the second half cycle.


2.3 Make a graph of the motion of the block for the first 5 half cycles of the motion in the case where A = 10.5*x0. Plot the position of the block normalized to x0 as a function of the fractional period, T = 2*Pi/omega (i.e. plot x(t)/x0 vs t/T). Attach the graph to your email.


2.4 Something interesting happens at the end of the 5th half cycle - what changes in the physical situation and what is the motion after this half cycle?

2. Relevant equations

F=ma
F=-kx
F_friction=mu*m*g

3. The attempt at a solution

I was able to get answers for 2.1 and 2.2

Acceleration
F_spring-F_friction=m*a
-k*x+mu*m*g=m*a
a=(-kx+mu*mg)/m
This works for 2.1 and 2.2 and I am able to simplify it to -omega^2*x' and -omega^2*x''

Position
x(t)=x'*cos*(omega*t) for 2.1
x(t)=x''*cos*(omega*t) for 2.2

x' and x'' seemed like the amplitudes at each of those intervals so I plugged them into the position equation for general harmonic motion.

Now I'm working on 2.3 and confused about graphing this.
I'm not sure why the question wants x(t)/x0 vs t/T.
Won't t/T give me a unit-less x-axis since time and radians cancel out?
The concept of x' and x'' confuses me. Does anyone care to explain?

2.4
I haven't graphed it but I'm guessing the graph starts to go flat or something like that as the mass begins to reach equilibrium.
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