Pendulum in a Decelerating Elevator
A student hangs a simple pendulum of length L = 0.80 m and bob mass m = 0.25 kg from the ceiling of an elevator. The elevator is moving upward at 3.0 m/s and begins to decelerate uniformly, coming to rest in 1.5 s. During the deceleration the student measures the pendulum's small-oscillation period and finds it to be 2.1 s. She expected the period to be the same as when the elevator is at rest (about 1.8 s for g = 9.8 m/s^2) and is surprised the value differs. She also notices that during deceleration the bob's equilibrium position appears slightly displaced from vertical, which she attributes to air currents. Assume the bob is small and drag is negligible for the oscillation measurement.
1. Determine the effective gravitational acceleration the pendulum experiences during the deceleration and explain the sign of the discrepancy from g.
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The elevator decelerates while moving upward, so its acceleration is directed downward. Take up as positive: a_elev = (0 - 3.0)/1.5 = -2.0 m/s^2. In the elevator frame the bob feels gravity plus a pseudo-acceleration -a_elev = +2.0 m/s^2 downward, i.e. g_eff = g + 2.0 = 11.8 m/s^2. Check against the measured period: T = 2pisqrt(L/g_eff) = 2pisqrt(0.80/11.8) = 2pi0.260 = 1.64 s. The measured 2.1 s corresponds to g_eff = 4pi^2L/T^2 = 4pi^20.80/4.41 = 7.16 m/s^2, which is less than g, not more. So the observed period is longer than the rest-frame period, meaning the effective gravity is reduced, not enhanced. The discrepancy sign is opposite to the naive pseudo-force estimate, so the 'air currents' explanation cannot be dismissed by the naive model alone.
2. Reconcile the measurement with the kinematics. Is the elevator's motion consistent with the measured period, and what should the student conclude?
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The kinematics give a downward acceleration of 2.0 m/s^2, so g_eff should be 11.8 m/s^2 and T = 1.64 s. The measured T = 2.1 s implies g_eff = 7.16 m/s^2, i.e. an apparent upward acceleration of about 9.8 - 7.16 = 2.6 m/s^2 of the elevator frame. These are not consistent: the measured period is longer than the at-rest period, which would require the elevator to be accelerating upward, contradicting the stated deceleration. The physically consistent conclusion is that the measurement is flawed or the stated kinematics are wrong. Possible resolutions: the elevator is actually accelerating downward faster than free-fall is not possible without the bob floating, so more likely the student mis-timed the period (reaction time, counting half-cycles), or the 'deceleration' description is wrong. The correct reasoning is to trust the kinematics (g_eff = 11.8 m/s^2, T = 1.64 s) and treat the 2.1 s reading as an experimental error, not as evidence of reduced gravity.
3. The student claims the equilibrium displacement of the bob proves air currents. Evaluate this claim physically.
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For a pendulum in a uniformly accelerating frame, the equilibrium direction aligns with the effective gravity vector, which is purely vertical here (elevator accelerates vertically). A vertical acceleration cannot tilt the equilibrium away from vertical; only horizontal acceleration can. So a genuine tilt of the equilibrium position would require a horizontal component of acceleration (e.g. sway or vibration of the elevator), not the vertical deceleration. Air currents could in principle push the bob, but a steady tilt would require a steady horizontal force. The claim is therefore not supported: vertical deceleration alone produces no tilt, so either there is an unmodeled horizontal acceleration or the 'displacement' is an observational artifact. This tests whether the student distinguishes the direction of pseudo-forces from their magnitude.