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QUESTION 1 (20 MARKS) In a 2D anisotropic harmonic oscillator, a small object of mass \( m=100 \mathrm{~g} \) is attached on both sides to a pair of light springs with different spring constants along each of the Cartesian coordinate axis; \( x \) - and \( y \)-axis. The spring constant along the \( x \) - and \( y \)-axis is \( 0.4 \pi^{2} \) and \( 0.1 \pi^{2} \) \( \mathrm{N} / \mathrm{m} \), respectively. In all calculations, neglect the effects of weight and air resistance on the object, and set the acceleration due to gravity \( g=10 \mathrm{~ms}^{-2} \). a) At some instant of time, the object is located at position \( \vec{r}=x \hat{i}+y \hat{j} \) from its equilibrium position. i. Write the expression for the net force \( \vec{F}(\vec{r})=\vec{F}(x, y) \) acting on the object. ii. Write the components of the equation of motion (EoM) for the object. iii. Write the general solutions to the equations in part ii. above. b) The followings are the initial values for the oscillator: (5 marks) \[ x_{0}=0 \mathrm{~m} ; v_{x_{0}}=0.1 \mathrm{~ms}^{-1} \text { and } y_{0}=0.05 \mathrm{~m} ; v_{y_{0}}=0 \mathrm{~ms}^{-1} \text {. } \] i. Calculate the angular frequency, \( \omega \) and the period, \( T \) of the oscillation. ii. Determine the solution \( \vec{r}(t)=x(t) \hat{i}+y(t) \hat{j} \) for the oscillator. iii. Calculate the position \( \vec{r} \) and velocity \( \vec{v} \) of the object at \( t=1 \mathrm{~s} \).

          QUESTION 1 (20 MARKS)
In a 2D anisotropic harmonic oscillator, a small object of mass \( m=100 \mathrm{~g} \) is attached on both sides to a pair of light springs with different spring constants along each of the Cartesian coordinate axis; \( x \) - and \( y \)-axis. The spring constant along the \( x \) - and \( y \)-axis is \( 0.4 \pi^{2} \) and \( 0.1 \pi^{2} \) \( \mathrm{N} / \mathrm{m} \), respectively. In all calculations, neglect the effects of weight and air resistance on the object, and set the acceleration due to gravity \( g=10 \mathrm{~ms}^{-2} \).
a) At some instant of time, the object is located at position \( \vec{r}=x \hat{i}+y \hat{j} \) from its equilibrium position.
i. Write the expression for the net force \( \vec{F}(\vec{r})=\vec{F}(x, y) \) acting on the object.
ii. Write the components of the equation of motion (EoM) for the object.
iii. Write the general solutions to the equations in part ii. above.
b) The followings are the initial values for the oscillator:
(5 marks)
\[
x_{0}=0 \mathrm{~m} ; v_{x_{0}}=0.1 \mathrm{~ms}^{-1} \text { and } y_{0}=0.05 \mathrm{~m} ; v_{y_{0}}=0 \mathrm{~ms}^{-1} \text {. }
\]
i. Calculate the angular frequency, \( \omega \) and the period, \( T \) of the oscillation.
ii. Determine the solution \( \vec{r}(t)=x(t) \hat{i}+y(t) \hat{j} \) for the oscillator.
iii. Calculate the position \( \vec{r} \) and velocity \( \vec{v} \) of the object at \( t=1 \mathrm{~s} \).

QUESTION 1 (20 MARKS)
In a 2D anisotropic harmonic oscillator, a small object of mass m=100 g is attached on both sides to a pair of light springs with different spring constants along each of the Cartesian coordinate axis; x - and y-axis. The spring constant along the x - and y-axis is 0.4 π^2 and 0.1 π^2 N / m, respectively. In all calculations, neglect the effects of weight and air resistance on the object, and set the acceleration due to gravity g=10 ms^-2.
a) At some instant of time, the object is located at position r⃗=x î+y ĵ from its equilibrium position.
i. Write the expression for the net force F⃗(r⃗)=F⃗(x, y) acting on the object.
ii. Write the components of the equation of motion (EoM) for the object.
iii. Write the general solutions to the equations in part ii. above.
b) The followings are the initial values for the oscillator:
(5 marks)

x0=0 m ; vx0=0.1 ms^-1 and y0=0.05 m ; vy0=0 ms^-1.

i. Calculate the angular frequency, ω and the period, T of the oscillation.
ii. Determine the solution r⃗(t)=x(t) î+y(t) ĵ for the oscillator.
iii. Calculate the position r⃗ and velocity v⃗ of the object at t=1 s.

Added by Jennifer A.

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