Simple Oscillator: find x(t)

In summary, at time t = T/2, a simple oscillator with angular frequency \omega = \omega_0 will have a maximum amplitude U and a positive value. The textbook answer for finding x(t) is x(t) = -\frac {U}{\omega_0}sin(\omega_0t), where U = \omega_0A. There is some confusion about the negative sign in the answer, but the definition of the coordinate system is that the vertical axis represents displacement (x), velocity (v), and acceleration (a), while the horizontal axis represents time (t).
  • #1
cj
85
0
At time t = T/2 the speed of a simple oscillator of angular frequency

[tex]\omega = \omega_0[/tex]

has maximum amplitude U and positive value.

Find x(t).

----------
The textbook answer for this problem is:

[tex]x(t) = -\frac {U}{\omega_0}sin(\omega_0t)[/tex]

where

[tex]U = \omega_0A}[/tex]
----------
This is confounding me.

For a simple sin solution (I'm assuming zero phase shift), wouldn't x at t = T/4 have to be positive, i.e.:

[tex]x = \frac {U}{\omega_0}[/tex]

and NOT negative (as given by the alleged solution):

[tex]x = -\frac {U}{\omega_0}[/tex]

Comments?
 
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  • #2
What is the definition of the coordinate system?
 
  • #3
Vertical axis = displacement (x), velocity (v) and acceleration (a).

Horizontal axis = time, t
 
  • #4
I am trying to get you to think about the definition of what is meant by negative and positive. Have you given us the complete problem?
 
  • #5
Integral said:
I am trying to get you to think about the definition of what is meant by negative and positive. Have you given us the complete problem?

Yes, the posted question and textbook answer (which I guess could be wrong) is verbatim.

Maybe I don't understand the meaning of the negative sign.

I'm imagining a displacement-speed-acceleration relationship as given in the graph at the top of the following linked page:

Sample

The bottom line is that if x(t) is as given by the textbook, it would clearly have a negative value (relative to the equilibrium position at t = T/4. This seems incorrect since even junior high school students understand that at t = T/4 the value for a sine function is clearly a maximum and clearly positive.

Right?
 

1. What is a simple oscillator?

A simple oscillator is a physical system that exhibits regular periodic motion around an equilibrium point.

2. How do you find x(t) for a simple oscillator?

To find x(t), you need to solve the differential equation that describes the simple oscillator's motion. This equation is typically in the form of mx'' + bx' + kx = 0, where m is the mass, b is the damping coefficient, and k is the spring constant.

3. What is the significance of x(t) in a simple oscillator?

x(t) represents the displacement of the oscillator from its equilibrium point at any given time t. It is a crucial parameter in understanding the behavior of the oscillator.

4. Are all simple oscillators the same?

No, there are various types of simple oscillators, including pendulums, mass-spring systems, and electrical circuits. Each type has its unique characteristics and equations.

5. What factors affect the motion of a simple oscillator?

The motion of a simple oscillator is affected by several factors, including the mass of the object, the stiffness of the spring, the damping coefficient, and the initial conditions (e.g., initial displacement and velocity).

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