Questions: Gyroscopic Motion, Precession, and Stability

Question 1

A gyroscope is spinning with its axis pointing east. A torque is applied pointing north. In which direction does the spin axis initially begin to move?

• A. Downward — the torque overcomes the spin and causes the axis to fall southward
• B. North — the spin axis rotates in the direction of the applied torque
• C. The spin axis rotates perpendicular to both the spin axis and the torque direction
• D. The spin axis does not move because the gyroscope rigidly resists all torques

Answer: (C)

The precession direction is given by dL = τ dt. L currently points east; τ points north; so dL points north, causing L (and the spin axis) to rotate northward — perpendicular to both the current L and τ directions. This is the core counterintuition: a torque does not cause the spin axis to tip toward the torque; it causes the axis to precess sideways. The gyroscope does not resist the torque — it responds to it in a direction 90° away from naive expectation.

Question 2

A bicycle wheel spins at 300 RPM and precesses at 0.5 rad/s under gravity. The spin rate is doubled to 600 RPM while the gravitational torque stays the same. What happens to the precession rate?

• A. It doubles to 1.0 rad/s, because faster spin produces more gyroscopic response
• B. It halves to 0.25 rad/s, because Ω = τ/L and L has doubled
• C. It stays at 0.5 rad/s, because the applied torque hasn't changed
• D. It initially increases, then decreases as nutation damping takes effect

Answer: (B)

Ω = τ/L = τ/(Iω). Doubling the spin rate ω doubles the angular momentum L (assuming constant I). With τ fixed and L doubled, Ω = τ/(2L) = half the original value = 0.25 rad/s. This is counterintuitive: faster spin makes the gyroscope precess more slowly, not faster, and appear more stable. A non-spinning top (L ≈ 0) would have Ω → ∞ — it falls immediately — the limiting case of zero stability.

Question 3

A spinning top subjected to a gravitational torque precesses rather than falling because the torque continuously rotates the angular momentum vector without changing its magnitude.

• T. True
• F. False

Answer: True

τ = dL/dt as a vector equation. If τ is perpendicular to L (as gravity's torque is on a horizontally displaced center of mass), then dL is perpendicular to L. A vector continuously receiving increments perpendicular to itself rotates — its direction changes but its magnitude stays constant. The top precesses because gravity continuously redirects the angular momentum vector sideways, causing the spin axis to sweep a cone, rather than producing the downward tipping expected from a non-spinning object.

Question 4

Increasing a gyroscope's spin rate increases its precession rate, making the gyroscope more active and less stable.

• T. True
• F. False

Answer: False

Precession rate Ω = τ/(Iω). A higher spin rate ω means larger angular momentum L = Iω, which reduces Ω. A faster-spinning gyroscope precesses more slowly, not faster, and holds its orientation more stubbornly against applied torques. This is the mechanism of gyroscopic stability: large angular momentum means any given torque produces a smaller angular change per unit time. The 'flywheel effect' of large L is precisely what makes spinning tops, bullets, and bicycle wheels stable.

Question 5

Explain, using the vector relationship τ = dL/dt, why a torque applied to a gyroscope causes precession rather than rotation about the torque axis.

Model answer: τ = dL/dt means the torque vector gives the rate of change of the angular momentum vector L. If L is large and points along the spin axis, and τ is perpendicular to L, then dL = τ dt is a small vector perpendicular to L. Adding a perpendicular increment to a vector rotates its direction while keeping its magnitude approximately constant. The spin axis therefore sweeps sideways in the direction of dL rather than tipping in the direction you'd expect from a non-spinning object. Only if L were zero (no spin) would the torque produce the naive expected rotation.

This is why spinning objects behave so surprisingly: the large angular momentum 'stores' a preferred direction, and any torque to change it is translated into a slow rotation of that stored direction rather than a rapid tumble. The faster the spin (larger |L|), the smaller the angular deflection per unit torque, and the more stable the gyroscope appears. The physics is entirely consistent with Newton's laws — it just requires treating angular momentum as a vector to see why the response is perpendicular to the forcing.