A piston performs reciprocal motion in a straight line which can be modeled as SHM. Its velocity is 16 m/s when the displacement is 80 mm from its midpoint, and 4 m/s when the displacement is 160 mm from mid-position. Determine the frequency and amplitude of the motion and the acceleration when displacement is 120 mm from mid position

The velocity is linked to the displacement as by the formula

v = \omega \sqrt{x_o\ ^2 - x^2}

Where v is the velocity at any point x from the mid-position,
omega the angular velocity,
xo the amplitude
x the displacement from the mid-position.

Acceleration is then given by:
a = -\omega^2 x

I still don't understand this could you please explain what I have to do abit more thanks

Using the equations that Unk gave you:


v = \omega \sqrt{x_0^2 - x^2}


where x_0 is the amplitude of the SHM, and \omega is the rotational veocity in radians/second.

You have been given two data points:

v = 16 m/s when x = 0.08m;
v = 4 m/s when x = 0.16m

Note that I changed the dimensions to be consistenty in meters. Now substitute those values into the above equation, to give you two equations in two unknowns:


16m/s = \omega \sqrt{x_0^2-0.08m^2} \\
4m/s = \omega \sqrt{x_0^2 - 0.16m^2}


Solve this for \omega and x_0. Then the frequency of the system is: f = \frac {\omega}{2 \pi}
and the acceleraton when x = 0.12m is: a= -\omega^2 \cdot 0.12m
in meters per second squared.

if I divide both these equations by each other dose that illiminate w and the squaring but I still can't figure out x

is x 62.5

if i divide both these equations by each other dose that illiminate w and the squaring but i still can't figure out x


You're on the right track:


\frac {16} 4 = \frac {\omega \sqrt{ x_0^2 - 0.08^2} } {\omega \sqrt{ x_0^2 - 0.16^2} } \\
\\
4 = \sqrt {\frac {x_0^2 - 0.08^2}{x_0^2 - 0.16^2}}


Square both sides:


16 = \frac {x_0^2 - 0.08^2}{x_0^2 - 0.16^2}


Multiply through by the denominatoir:


16 (x_0^2 - 0.16^2) = x_0^2 - 0.08^2}


Rearrange to get x_0 by itself:


16x_0^2 - x_0^2 = 16 \times 0.16^2 - 0.08^2 \\

x_0^2 = \frac {16 \times 0.16^2 - 0.08^2} {15} \\

x_0 = \sqrt {\frac {16 \times 0.16^2 - 0.08^2} {15}}

is x 62.5

62.5 what? Millimeters? And do you mean x_0? Obviously this is not right since you are given data about what happens when x = 80mm or x = 160 mm. Clearly x_0 must be greater than 160 mm.

from that equations I got 0.04233 so is x 423.3 mm

from that equations i got 0.04233 so is x 423.3 mm

Nope. I think when you did the calculation you may not have reaized that the denominator is inside the square root. In other words you took the square root of the numerator and then divided by 15, as opposed to dividing the numerator by 15 and then taking the square root.

Also - if the answer had turned out to be 0.04233m that would be 42.3mm, not 423.3 mm. Watch your units!

is x 639.6 mm

is x 639.6 mm


No.

Do you no what I'm doing wrong

is x 163.9

is x 163.9


Bingo! But the answer should properly be rounded to 164mm, since that's the level of accuracy of the data you were given.

I think I need the equation for angular velocity to find the frequency do you no what this is equation is

Go back to post #4. There were two equations in two unknowns: \omega and x_o. You have now solved for x_o. So you can use that knowledge to find \omega, by plugging the value for x_o into either one of those equations and solving for \omega.

do I have to rearrange the equation

do i have to rearrange the equation

Sure. For example if you use this one:


16 \frac m s = \omega \sqrt{x_0^2-(0.08m)^2}


Rearange to get \omega by itself, then plug in the value for x_o, and you've got it.

dose w = 106rad/s

Close: I get 112 rad/s

is it definitely 112 I get 115.4

square root of 0.16 squared - 0.08 squared then dived all that by 16 = 115.4

is it definetly 112 i get 115.4

square root of 0.16 squared - 0.08 squared then dived all that by 16 = 115.4

I don't understand what formula you are trying to write out. Do you mean this:


\frac {\sqrt { (0.16m)^2 - (0.08m)^2}} {16 m/s} \ ?


Note that the units of this formula would work out to be seconds, but you need an answer in radians/second. So once again - watch your units!

Using the formula I gave you in my previous post:


16 \frac m s = \omega \sqrt{x_o^2 - (0.08m)^2}


You know that x_o = 0.164 m. So rearrange to get:


\omega = \frac {16 \frac m s}{\sqrt{(0.164m)^2 - (0.08m)^2}} = 111.8 s^{-1}

so the frequency = 17.79Hz I tried to figer out the acceleration to I get 1499.9 meters per second squared

so the frequency = 17.79Hz

Good!


i tried to figer out the acceleration to i get 1499.9 meters per second squared

Right - except the exact answer is actually -1500 m/s^2. Note the minus sign, because for positive displacement the piston has a negative acceleration.