How to find the at rest position of a particle when trig functions

In summary, the given function s(t) is equal to cos(pi*t/8), and after taking the derivative, the velocity function v(t) is equal to -pi/4 * sin(pi*t/4). To find when the particle is at rest, we set v(t) equal to 0 and use the formula for the period of sine and cosine to find the values of t where sin(pi*t/4) is equal to 0. These values are t=0, 4, and 8.
  • #1
TitoSmooth
158
6
so my given: s(t)=cos(pie8*t/4)

took the derivative= velocity function

then, v(t)= -pie/4 *sin(pie*t/4)

When is the particle at rest? v(t)=0

now, 0= -pie/4 *sin(pie*t/4)


im lost here. I know it's very simple I am just over thinking. What do I do from here?

thanks
 
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  • #2
figured it out. had to remind myself that the period of sin and cosin is -2pie->2pie.

then use the formula forget what its called. but 2pie/b

2pie/pie/4 = 8. then when divide the interval into 4 sub intervals and we have 0, 2, 4, 6, 8.

remembering that sin is negative at 0, pie, 2pie. so velocity is zero at t=0,4,8
 
  • #3
$$-\frac{\pi}{4} \sin \frac{\pi t}{4} = 0$$
is true if and only if ##\sin(\pi t/4) = 0##. For what values of ##x## is ##\sin(x) = 0##?
 

FAQ: How to find the at rest position of a particle when trig functions

How do I determine the at rest position of a particle using trig functions?

To find the at rest position of a particle using trig functions, you will need to use the equation x(t) = A sin(ωt + φ), where A is the amplitude, ω is the angular frequency, and φ is the phase shift. The at rest position can be found by setting the velocity (dx/dt) equal to zero and solving for t. Once you have the value of t, plug it into the equation x(t) to find the position of the particle at rest.

Can the at rest position of a particle be found using other mathematical methods?

Yes, the at rest position of a particle can also be found using calculus. By taking the derivative of the position function, you can find the velocity function and set it equal to zero to find the at rest position. However, using trig functions can be a simpler and more intuitive way to find the at rest position.

What information do I need to know in order to use trig functions to find the at rest position?

To use trig functions to find the at rest position, you will need to know the amplitude, angular frequency, and phase shift of the particle's motion. These can be determined from the given position function or by analyzing the physical properties of the particle's motion.

Can the at rest position of a particle change over time?

No, the at rest position of a particle remains constant over time. This is because at rest position is defined as the point where the particle has zero velocity, therefore it cannot change as long as the particle remains at rest.

Are there any limitations to using trig functions to find the at rest position?

Trig functions may not be suitable for all types of particle motion. They are most commonly used for simple harmonic motion, where the particle's motion follows a sinusoidal pattern. If the particle's motion is more complex, other mathematical methods may be more appropriate for finding the at rest position.

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