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We Are Designing a Simple Liquid-Mirror Telescope as Originally Proposed by Isaac Newton

Question

We are designing a simple liquid-mirror telescope, as originally proposed by Isaac Newton. The mirror is created by rotating a liquid at angular velocity ω\omega. Given a constant power source, derive the focal length as a function of time.

Deriving focal length from rotating fluid dynamics and constant power constraints.

Focal Length f(t)

gI/4Pt

f(t) = gI/(4Pt) — focal length decreases inversely with time

1Physics of a Rotating Liquid Mirror

When a liquid is rotated at angular velocity ω\omega, its surface forms a paraboloid due to the balance between centrifugal force and gravity: z(r)=ω2r22gz(r) = \frac{\omega^2 r^2}{2g}
  Cross-section of rotating liquid:

           |         
       ╱   |   ╲     z(r) = ω²r²/2g
      /    |    \
     /     |     \
  ───────────────────  liquid

  ←──── r ────→   ω = angular velocity

Natural Parabolic Mirror

A paraboloid naturally focuses parallel light to a single focal point. This is why rotating liquids (like mercury) can serve as telescope mirrors without any grinding or polishing.

2Derive the Focal Length

For a parabolic mirror z=r2/(4f)z = r^2/(4f), comparing with z=ω2r2/(2g)z = \omega^2 r^2/(2g): 14f=ω22g  ⟹  f=g2ω2\frac{1}{4f} = \frac{\omega^2}{2g} \implies f = \frac{g}{2\omega^2}

Inverse Square Relationship

Higher angular velocity ω\omega → shorter focal length (stronger focusing). Doubling the spin rate **quarters** the focal length.

3Apply Constant Power Constraint

With constant power PP driving the rotation: P=τω=Iω˙ωP = \tau \omega = I\dot{\omega}\omega where II is moment of inertia and τ\tau is torque. Separating variables: ω dω=PI dt\omega \, d\omega = \frac{P}{I} \, dt

4Solve for ω(t)

Integrating both sides: ω22=PIt  ⟹  ω(t)=2PtI\frac{\omega^2}{2} = \frac{P}{I}t \implies \omega(t) = \sqrt{\frac{2Pt}{I}}

5Final Result: f(t)

Substituting ω(t)\omega(t) into f=g/(2ω2)f = g/(2\omega^2): f(t)=g2⋅2Pt/I=gI4Ptf(t) = \frac{g}{2 \cdot 2Pt/I} = \frac{gI}{4Pt}
Paraboloid shapez = ω²r²/(2g)
Focal length (static)f = g/(2ω²)
ω under constant powerω(t) = √(2Pt/I)
FOCAL LENGTH f(t)gI/(4Pt)

Physical Interpretation

The focal length decreases as 1/t1/t: the longer the motor runs, the faster the mirror spins, and the shorter the focal length. Eventually, practical limits (turbulence, spillage) prevent further spin-up.

Quiz

Test your understanding with these questions.

1
What shape does the surface of a rotating liquid form?
2
How does the focal length f(t)f(t) change over time with constant power?