Problem structure
(a), 4, symbolic, Derivation of SHM from a non-linear force.
(b), 4, numerical, Application of SHM period to find physical constants.
(c), 6, numerical, Energy conservation in a non-linear potential field.
(d), 4, numerical, Logarithmic decrement and damping coefficients.
(e), 2, qualitative, Resonance in forced oscillations.
Plan your route before writing. Use equations, diagrams, units, and a clear final justification where needed.
A "magnetic spring" is constructed using two identical small neodymium magnets. One magnet is fixed to the base of a vertical glass tube. The second magnet, of mass $m = 12.0\text{ g}$, is placed inside the tube such that it is repelled by the fixed magnet and levitates at an equilibrium height $h_0$ above it.
The repulsive force $F_m$ between the magnets can be modelled by the power law $F_m = \frac{k}{h^n}$, where $h$ is the separation between the centers of the magnets, $k$ is a constant related to the magnetic strength, and $n$ is a dimensionless exponent.
(a)
4 marks
At equilibrium, the levitating magnet is at $h_0 = 4.50\text{ cm}$. If the magnet is displaced slightly by a distance $x$ from equilibrium ($x \ll h_0$), show that the resulting motion is approximately simple harmonic and derive an expression for the angular frequency $\omega$ in terms of $n, g,$ and $h_0$.
(b)
4 marks
An experimenter measures the period of small vertical oscillations to be $T = 0.350\text{ s}$. Using this data, determine the value of the exponent $n$.
(c)
6 marks
The magnet is now pushed down to a height $h_{min} = 3.00\text{ cm}$ and released from rest. Assuming the power law remains valid and ignoring air resistance, calculate the maximum height $h_{max}$ reached by the magnet.
(d)
4 marks
In a real setup, the magnet experiences a damping force $F_d = -bv$, where $v$ is the velocity. It is observed that after 10 full oscillations, the amplitude of the motion reduces by 40%. Calculate the damping constant $b$ in $\text{kg s}^{-1}$.
(e)
2 marks
If the base of the tube is now vibrated vertically with a very small amplitude $A_{drive}$ at a frequency of $2.86\text{ Hz}$, describe the resulting motion of the levitating magnet and explain why the amplitude of the magnet's oscillation might become unexpectedly large.
