# Thread: Displacement of a particle

1. ## Displacement of a particle

Ok I'm guessing this is simple harmonic motion?
I don't understand how these values relate to the displacement at all, can anyone help?

The displacement, d, in cm, of a particle is given by
d = 3 sin(40 π t) – 4 cos( 40 π t ) t ≥ 0
· Find expressions for the velocity and acceleration of the particle
· Calculate the velocity and acceleration when t = 2
· Determine when the particle is first at rest
· Calculate the displacement and acceleration when the particle is at rest

(π = pi)

Thanks!

2. ## Re: Displacement of a particle

Find the derivative to get the velocity, and the second derivative to find the acceleration with respect to time.

3. ## Re: Displacement of a particle

Are you sure?

4. ## Re: Displacement of a particle

Originally Posted by srh
Are you sure?
Yes.
$\displaystyle v=d'~\&~a=v'=d''$

6. ## Re: Displacement of a particle

Then when is the particle at rest?

7. ## Re: Displacement of a particle

$\displaystyle d = 3 sin(40 nt) - 4 cos( 40 nt )$

$\displaystyle d^{'} = 120ncos(40nt) + 160nsin(40nt)$

Particle at rest when velocity =0.

$\displaystyle 0 = 120ncos(40nt) + 160nsin(40nt)$

$\displaystyle -120ncos(40nt) = 160nsin(40nt)$

$\displaystyle -\frac{4}{3}tan(40nt) = 1$

$\displaystyle tan(40nt) = -\frac{3}{4}$

and then solve for t....

8. ## Re: Displacement of a particle

Thanks for your time but that doesn't help me in the slightest I'm afraid.

Can this be explained in a more simplistic way?

9. ## Re: Displacement of a particle

Also the π symbol is supposed to be the pi symbol

10. ## Re: Displacement of a particle

Originally Posted by srh
Also the π symbol is supposed to be the pi symbol
Use the reply with quote tab. On the tool bar you see $\displaystyle ~~\Sigma~~$ that is the LaTeX rap.
[TEX]\pi [/TEX] gives $\displaystyle \pi$.

11. ## Re: Displacement of a particle

The displacement, d, in cm, of a particle is given by
d = 3 sin(40 π t) – 4 cos( 40 π t ) t ≥ 0
· Find expressions for the velocity and acceleration of the particle
So we know how to get velocity and acceleration i.e.

$\displaystyle velocity = 120n cos(40nt) + 160n sin(40nt)$ --> which is the derivative of displacement
$\displaystyle acceleration = -4800 n^2 sin(40nt) + 6400n^2 cos(40nt)$ --> which is the derivative of the velocity.

Calculate the velocity and acceleration when t = 2
now we substitute t=2 into the above:

$\displaystyle velocity = 120(\pi) cos(40\pi(2)) + 160\pi sin(40\pi(2)) = 120\pi$
$\displaystyle acceleration = -4800 \pi^2 sin(40\pi(2)) + 6400\pi^2 cos(40\pi(2)) = 6400\pi^2$

Say the part you don't understand

12. ## Re: Displacement of a particle

Any of it really :S

13. ## Re: Displacement of a particle

Originally Posted by srh
Thanks for your time but that doesn't help me in the slightest I'm afraid.

Can this be explained in a more simplistic way?
For this problem to make any sense to you at all, you have to know some Calculus. Are you taking a Calculus course? If not, what course is this for?

14. ## Re: Displacement of a particle

I'm studying mechanical engineering, I've got only a basic grasp of calculus from what I've been able to teach myself online.

15. ## Re: Displacement of a particle

These are equations that describe the motion.

Let's take a simple example

displacement = t for $\displaystyle t\geq0$

For the 1st second: displacement = 1 cm
For the 2nd second: displacement = 2 cm and so on....

One can easily that the velocity is 1cm per second.
So how do we do this mathematically,
we use the formula $\displaystyle velocity = \frac{change\hspace{2mm} in\hspace{2mm} displacement}{change\hspace{2mm} in \hspace{2mm}time}$ = derivative of displacement
from now on I will use s to refer to displacement...

velocity = $\displaystyle \frac{ds}{dt}$(t)= 1 cm per second.

Now to calculate the acceleration:
You the same as above but now you take the derivative of the velocity, so we get

acceleration = $\displaystyle \frac{dv}{dt}$ (1)= 0.

You can verify this by seeing that the velocity is constant.

Second example

$\displaystyle displacement = t^2$ for $\displaystyle t\geq0$

For the 1st second: displacement = 1 cm
For the 2nd second: displacement = 4 cm and so on....

we use the formula $\displaystyle velocity = \frac{change \hspace{2mm}in\hspace{2mm} displacement}{change \hspace{2mm}in \hspace{2mm}time}$ = derivative of displacement
from now on I will use s to refer to displacement...

velocity = $\displaystyle \frac{ds}{dt}$ $\displaystyle (t^2)$= 2t cm per second.

so if t=1, v= 2(1) = 2; if t=2, v= 2(2) = 4

Now to calculate the acceleration:

acceleration = $\displaystyle \frac{dv}{dt}$(2t)= 2cm per second squared.

Now apply the same principles to your equations...

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