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**1. The problem statement, all variables and given/known data**

A particle of mass

*m*moves in a fixed plane along the trajectory [tex] \vec{r} = \hat{i} A cos(3 \omega t) + \hat{j} A cos(\omega t) [/tex].

(a) Sketch the trajectory of the particle.

(b) Find the force acting on the particle

(c) find its potential energy

(d) find its total energy as functions of time

(e) is the motion periodic? if so, find the period

**2. Relevant equations**

**3. The attempt at a solution**

Part (a) I won't worry about for now

Part (b): I think this is correct:

[tex] \vec{r} = \hat{i} A cos(\omega t) + \hat{j} A cos(3 \omega t) [/tex].

[tex] \vec{v} = -\hat{i} A \omega sin(\omega t) - \hat{j} 3 A \omega sin(3 \omega t) [/tex]

[tex] \vec{a} = -\hat{i} A \omega^2 cos(\omega t) - \hat{j} 9 A \omega^2 cos(3 \omega t}) [/tex]

[tex] \vec{F} = m \vec{a} [/tex]

[tex] \vec{F} = m (-\hat{i} A \omega^2 cos(\omega t) - \hat{j} 9 A \omega^2 cos(3 \omega t})) [/tex]

So, for part (b), this is where I'm wondering if I am correct:

[tex] F = -\frac{d U}{d t} [/tex]

[tex] U = m( A \omega sin(\omega t) + 3 A \omega sin(3 \omega t)) [/tex]

And, if this is correct, would this be the kinetic energy as a function of time?

[tex] K(t) = \frac{1}{2} m (-\hat{i} A \omega sin(\omega t) - \hat{j} 3 A \omega sin(3 \omega t))^2 [/tex]

Which would make part (c):

[tex] E = \frac{1}{2} m (-\hat{i} A \omega sin(\omega t) - \hat{j} 3 A \omega sin(3 \omega t))^2 + m( A \omega sin(\omega t) + 3 A \omega sin(3 \omega t)) [/tex]

Thanks

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