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**The problem statement, with all known data/variables.**

A 500 g object is moving a horizontal frictionless surface. Its displacement from the origin is given by the equation: [tex]x(t) = (3.50m)\sin{[(\frac{\pi}{2})t + \frac{5\pi}{4}]}[/tex].

a)what kind of motion is this?

b)what is the amplitude of this motion?

c)what is the period of this motion?

d)what is the frequency of this motion?

e)what is the linear velocity of this motion?

f)what is the linear acceleration of this motion?

g)what is the maximun kinetic energy of this system?

h)What is the maximum potential energy of this system?

i)What is the total mechanical energy of this system?

**Given :**

postion function x(t)

Amplitude: A=3.50m

mass = 500g= .500kg

no friction is present

## Homework Equations

[tex]x'(t)=v(t)[/tex]

[tex]v'(t)=a(t)[/tex]

[tex]KE(t)= \frac{1}{2} m{[v(t)]}^2[/tex]

[tex]PE(t)= \frac{1}{2} k{[x(t)]}^2[/tex]

[tex]E_{mech} = \frac{1}{2} k{A}^2[/tex]

[tex]T= \frac{2\pi}{\omega}[/tex]

[tex]x(t)= A\sin{[ t\omega + \phi]}[/tex]

[tex]f=\frac{1}{T}[/tex]

[tex]\omega = \sqrt{\frac{k}{m}[/tex]

## The Attempt at a Solution

a) since the postion function is sinusoidal, this reflects simple harmonic motion.

b) Given: A=3.50m (sinusoidal function)

c)[tex]T= \frac{2\pi}{\omega}[/tex]

[tex]T= \frac{2\pi}{\frac{\pi}{2}}[/tex]

[tex]T=4[/tex]

d) linear frequency [tex]f=\frac{1}{T}[/tex]

[tex]f=\frac{1}{4}[/tex]

e)differentiating x(t) gives [tex]v(t)= 3.50(\frac{\pi}{2})\cos{(\frac{\pi}{2} t + \frac{5\pi}{4})}[/tex]

f)differentiating v(t) gives [tex]a(t)= -3.50(\frac{{\pi}^2}{4})sin{(\frac{\pi}{2} t + \frac{5\pi}{4})}[/tex]

g) using the kinetic energy formula gives [tex]KE(t)= \frac{1}{2} (.500kg){[3.50(\frac{\pi}{2}) \cos(\frac{\pi}{2} t + \frac{5\pi}{4})]}^2[/tex]

To find max KE, we to find the time that velocity is greatest. Using velocity graph, we see max values at t=0,4: Therefore v is greatest at t=4. Plugging this in KE(t), we get KE(4)=3.778 Joules

h)First solve for k, [tex]T= \frac{2\pi}{\sqrt{\frac{k}{m}}}[/tex]

I get k= .20264, and I get stuck here.

i) Using total mech energy eqution I obtain total energy to be 1.24117 J

I sense that something is not right with parts (g),(h), and (i). And I am unsure if parts (a-f) are correct. Thanks in advance!