Simple Harmonic Motion-should be easy

[mattmannmf]

A particle's position in centimeters is given by the expression x = 40 cos(45 t + 8), where t is given in seconds.

angular frequency= 45 rad/sec
period= .139 s
amplitude= 40 cm

Heres the part i can't get but seems so simple...

where is the particle at t= .05 sec?

what is its velocity?

what is its accleration?

what is the maximum speed attained by the particle?

So what i figured on doing is just plugging .05 into my original equation x = 40 cos(45 t + 8) to solve for distance...where t= .05, distance= 39.36...which came out to be wrong for some reason...(all is in cm)
then for the rest i just add a "-w" infront of the equation for velocity
and a "-w^2" for acceleration and then solve when a=0, velocity has a max or min..

any ideas?

 

[denverdoc]

First issue, the argument of the cosine is in radians--make sure to set your calculator for radians. I got something like -27.

See if that helps with the remaining questions which are modified by -w and w^2 as you note.

 

[kerimek]

I would like to clarify a few things about Simple Harmonic Motion. First, the equation provided is not a complete representation of simple harmonic motion. It is only valid for a particle moving in a straight line with simple harmonic motion, and it assumes the particle starts at its equilibrium position (x=0) at t=0. Additionally, the given expression does not have units, which are important in understanding the physical meaning of the values.

To answer the questions, we can use the given equation and the principles of simple harmonic motion. At t=0.05 s, the particle's position can be found by plugging in t=0.05 into the equation:
x = 40 cos(45*0.05 + 8) = 39.36 cm. This is the correct answer, so it seems like there may have been an error in the calculation.

To find the velocity at t=0.05 s, we can use the derivative of the position equation:
v = -40*45*sin(45*0.05 + 8) = -179.93 cm/s. The negative sign indicates that the particle is moving in the opposite direction of the positive x-axis.

Similarly, the acceleration at t=0.05 s can be found by taking the derivative of the velocity equation:
a = -40*45^2*cos(45*0.05 + 8) = -8096.85 cm/s^2. Again, the negative sign indicates that the acceleration is in the opposite direction of the positive x-axis.

The maximum speed attained by the particle can be found by setting the velocity equation equal to zero and solving for t:
0 = -40*45*sin(45t + 8)
sin(45t + 8) = 0
45t + 8 = 0 or 45t + 8 = π
t = -8/45 or t = (π-8)/45
Since t cannot be negative, the maximum speed occurs at t = (π-8)/45 s. Plugging this value into the velocity equation gives us the maximum speed:
v = -40*45*sin(45*(π-8)/45 + 8) = 40 cm/s. This is the same as the amplitude, which makes sense since the particle reaches its maximum speed at the equilibrium position.

In summary, simple harmonic motion is a type of periodic motion where