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# Physics/Waves

Question
I am having difficulty figuring out what needs to be done. Any help will be greatly appreciated.

A block of unknown mass is attached to a spring with a spring constant of 10 N/m and undergoes
simple harmonic motion with an amplitude of 8.0 cm. When the block is 1/4 of the way between its
equilibrium position and the endpoint, its speed is measured to be 30.0 cm/s. Calculate
a) The mass of the block,
b) The period of the motion, and
c) The maximum speed of the block.

Hello Billy,

a. The general-case position equation of a simple harmonic oscillator is
x = A*cos(w*t+delta)
Let us say that the clock is started such that the phase shift, delta, may be zero. Therefore
x = A*cos(w*t)

The problem gives us the information that A = 0.08 m. It also tells us the speed when the block is at position x = A/4. We need the value of w*t at that moment.
A/4 = A*cos(w*t)
cos(w*t) = 0.25
And from that
w*t = arccos(0.25) = 75.5 degrees = 75.5 * pi radians/180
w*t = 1.32 radians

An equation for the velocity of the mass is obtained by taking the 1st derivative of the above equation for x. So
dx/dt = -A*w*sin(w*t)
When w*t = 1.32 radians, the block's speed is 0.30 m/s. So
0.30 m/s = -A*w*sin(1.32 radians) = -A*w*0.968
Since A = 0.08 m
0.30 m/s = -0.08 m*w*0.968
That minus sign is a complication. The mass passes through that point twice per cycle, with negative speed once and positive speed once. Let's choose the time with negative speed.
-0.30 m/s = -0.08 m*w*0.968
w = 0.30 s^-1 / (0.08*0.968) = 3.87 radians/s

A standard result that you will probably find in your book derived for simple harmonic motion is that
w^2 = k/m
(3.87 radians/s)^2 = (10 N/m)/m
m = (10 N/m)/(3.87 radians/s)^2 = 0.668 kg (substituting kg.m/s^2 for the N)

b. The period is given by
T = 2*pi/w = 1.62 s

c. The maximum speed is when the mass passes through the equilibrium point -- when x=0. Since
x = A*cos(w*t), that must happen when w*t = pi/2 (or 90 degrees).

So plugging that value of w*t into the velocity equation,
dx/dt = -A*w*sin(w*t) = -A*w = -0.08 m*3.87 radians/s = -0.31 m/s
Or when it passes through going the other direction, 0.31 m/s.

Note: please don't trust my math. Verify!
I hope this helps,
Steve
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#### Steve Johnson

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I would be delighted to help with questions up through the first year of college Physics. Particularly Electricity, Electronics and Newtonian Mechanics (motion, acceleration etc.). I decline questions on relativity and Atomic Physics. I also could discuss the Space Shuttle and space flight in general.

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