Trying to solve a harmonic motion problem

In summary, the equation for the velocity of the mass is arccos(0.8) + pi/2, which gives the answer of .232 sec.
  • #1
1,041
1
"simple" harmonic motion

Trying to solve a harmonic motion problem (mass oscillating on a spring), given:
amplitude=.100 m
k=8.00 N/m
mass=.5 kg
to solve for time elapsed for the mass to travel from x=0.00 to x=.080 m.
It's easy to solve for w (omega) = sqrt(8/.5) = 4, and using the equation for shm from my physics textbook I got this equation:
.08 = .1 cos(4t-pi/2)
and then
t = ((arccos 0.8) + pi/2)/4 = .554 sec. which unfortunately is wrong. This turns out to be the time it takes for the mass to go past .08 m, past its maximum displacement of .1 m, and BACK to .08 m.

Now, this equation is equivalent:
.08 = .1 sin(4t)
Check it...the graphs are exactly the same. But this one solves to
t = .232 sec. which is the answer given by the book, and tracing the graph confirms that this is the correct answer.

So, there must be a way to solve the first equation [.08 = .1 cos(4t-pi/2)] to get the correct answer of .232 sec., but I can't see it.

What am I missing?
 
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  • #2
When using inverse trig operations, be aware that there is more than one answer, for example: -

sinx = 0
x = arcsin(0)
x = n(pi) where n is an integer.

When applying such maths to physics, it is necessary to choose which answer you require, which may, or may not be obvious.

In your question, you assumed that the only solution to arccos(0.8) is 0.6435... when in fact, -0.6435 is an equally valid solution. (This is a consequence of cos(x) = cos(-x)). Substituting this into your equation gives the answer provided by the book.

If you are uncomfortable with having to choose which answer is correct without using educated guesswork or drawing graphs, use the equation for the velocity of the object to find out whether each solution corresponds to a positive or negative velocity.
 
  • #3
Thanks Claude, that was very helpful.
 

1. What is harmonic motion?

Harmonic motion is a type of periodic motion in which an object oscillates back and forth around an equilibrium point. It can be described by a sinusoidal function and is commonly seen in systems such as pendulums and springs.

2. How do I solve a harmonic motion problem?

To solve a harmonic motion problem, you will need to use the equations of motion for harmonic oscillators. These include the displacement equation, velocity equation, and acceleration equation. You will also need to identify the initial conditions and any known values in the problem.

3. What is the difference between simple harmonic motion and damped harmonic motion?

Simple harmonic motion is idealized and assumes that there is no external resistance or damping acting on the system. Damped harmonic motion, on the other hand, includes the effects of external resistance or damping, which causes the amplitude of the oscillations to decrease over time.

4. How does the spring constant affect harmonic motion?

The spring constant, also known as the force constant, determines the strength of the restoring force acting on the system. In harmonic motion, a higher spring constant results in a higher frequency and shorter period of oscillations, while a lower spring constant results in a lower frequency and longer period.

5. Can I use harmonic motion to model real-life systems?

Yes, harmonic motion is commonly used to model real-life systems such as pendulums, musical instruments, and even the vibrations of atoms and molecules. However, it assumes ideal conditions and may not fully represent the behavior of complex or nonlinear systems.

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