5. I measure time using a pendulum. I find that a pendulum swinging from a fixed
support oscillates with a different period than one whose support is accelerated
horizontally with respect to the laboratory frame. This is mainly because of:
a. The effect of Earth’s rotation
b. Lorentz contraction
c. Doppler effect
d. None of the above

Pendulum Period Horizontally Accelerated Support

Pendulum period horizontally accelerated support differs from fixed setups due to non-inertial frames creating pseudo-forces. In accelerated frames like trolleys or cars, the bob experiences horizontal pseudo-acceleration -a, combining vectorially with gravity g downward. Resultant g_eff = √(g² + a²) tilts equilibrium by θ = tan⁻¹(a/g).

Small oscillations about this tilted mean position yield period T = 2π √(L/g_eff), shorter than fixed-support T₀ = 2π √(L/g) since g_eff > g. Example: Car acceleration drops period from 4s to 3.99s implies small a ≈ g(T₀² – T²)/(2T₀²).

Key Applications

• Train/trolley problems in exams (JEE/CSIR).
• Inclinometers measuring acceleration.
• Seismographs detecting horizontal motion.

Derivation Insight

In accelerating frame, equations of motion show restoring torque proportional to g_eff sin ϕ, where ϕ is angular displacement from tilted equilibrium, confirming SHM with modified frequency.

Pendulum Period Changes with Horizontally Accelerated Support

A pendulum swinging from a fixed support has a different period than one from a horizontally accelerated support due to the effective gravity increase in the non-inertial frame.

Option Analysis

a. Earth’s rotation: Earth’s rotation causes minor Foucault pendulum precession but negligible period change for typical lab pendulums (period ≈ 2 seconds), as rotational effects scale with T² cos ϕ / R, where ϕ is latitude and R is Earth radius. This does not explain the observed difference.

b. Lorentz contraction: Relativistic length contraction occurs at speeds near light speed (v ≈ c), irrelevant for lab accelerations (<< c). Pendulum periods follow classical mechanics here.

c. Doppler effect: Doppler shift affects wave frequencies from relative motion, not mechanical oscillation periods of pendulums. No frequency mixing applies.

d. None of the above: Correct. The period differs because horizontal acceleration a tilts equilibrium, making effective gravity g_eff = √(g² + a²) > g. Period T = 2π √(L/g_eff) decreases since g_eff > g. Fixed support uses g.