On this page
A pendulum clock that keeps perfect time on the ground runs fast in a lift that is speeding up on its way up. In a car that accelerates, the bob hangs at an angle. Both follow from one change to T = 2π√(L/g): replace g with the effective gravity in the frame of the support. This post shows how, with numbers for five situations.
Is this in the syllabus?
NCERT's chapter Oscillations derives T = 2π√(L/g) for small swings in section 13.8, and its Exercise 13.16 asks for the period of a pendulum in a car moving on a circular track. The Syllabus for JEE (Main) 2026 lists "simple pendulum: derivation of expression for its time period", and the JEE (Advanced) 2026 Information Brochure lists "Inertial and uniformly accelerated frames of reference" and "Linear and angular simple harmonic motions". Check the current year's documents before your exam.
What is effective g?
In the frame of a support accelerating at a, the bob feels its weight mg and a pseudo force −ma. Together they act like a single gravity:
g_eff = g − a (vector subtraction)
The bob's equilibrium position lines up with g_eff, and the period becomes T = 2π√(L/g_eff). NCERT's derivation is unchanged; only the restoring force uses g_eff instead of g.
What happens in a lift?
Accelerating upward (speeding up going up, or slowing down going down): g_eff = g + a, so the period is shorter.
Accelerating downward (speeding up going down, or slowing down going up): g_eff = g − a, so the period is longer.
Free fall (a = g): g_eff = 0. The bob does not oscillate; displaced, it stays where you leave it relative to the lift.
The same g_eff sets the reading of a weighing machine in a lift, which NCERT's Laws of Motion chapter asks about in Exercise 4.13.
What happens in an accelerating car?
If a car accelerates horizontally at a, the vectors g and −a are at right angles:
g_eff = √(g² + a²), always larger than g, so the period is shorter
the bob hangs back at an angle θ from the vertical, with tan θ = a/g
On a circular track at speed v and radius R, the acceleration is v²/R toward the centre, so g_eff = √(g² + v⁴/R²). That answers NCERT's Exercise 13.16.
Worked example: one pendulum, five rides
A simple pendulum is 1.0 m long. Take g = 9.8 m/s².
On the ground: T = 2π√(1.0/9.8) ≈ 2.01 s.
Lift accelerating upward at 2.2 m/s²: g_eff = 12.0 m/s², T = 2π√(1.0/12.0) ≈ 1.81 s.
Lift accelerating downward at 2.2 m/s²: g_eff = 7.6 m/s², T = 2π√(1.0/7.6) ≈ 2.28 s.
Car accelerating at 3.0 m/s²: g_eff = √(9.8² + 3.0²) = √105.04 ≈ 10.25 m/s², so T ≈ 1.96 s, and the string hangs back at tan⁻¹(3.0/9.8) ≈ 17°.
Trolley sliding freely down a smooth 30° incline: its acceleration g sin 30° is along the slope, leaving g_eff = g cos 30° ≈ 8.49 m/s², perpendicular to the incline. T ≈ 2.16 s, and the string hangs perpendicular to the incline.
Does the clock gain or lose?
A pendulum clock counts swings. If each swing is shorter, it counts more swings than it should and gains time. In case 2, each swing takes 1.81 s instead of 2.01 s, so the clock runs fast while the lift accelerates upward. In case 3 it runs slow.
Which mistakes cost marks?
Changing the period for a lift moving at steady speed.
Reading the direction from the motion. "Moving down" does not tell you the sign; "accelerating down" does. A lift slowing down on its way down is accelerating upward.
Adding g and a as numbers in a car. They are perpendicular, so use Pythagoras.
Tilting the bob forward in a car. It hangs back, opposite to the acceleration.
Using T = 2π√(L/g) for large swings. NCERT's derivation assumes small angles, where sin θ ≈ θ.
How should you practise this?
Practice testing, recalling answers without looking, was one of two techniques rated high utility in a review of ten study techniques by Dunlosky and colleagues (2013). Use it here:
Day 1: write, from memory, g_eff for six situations: lift up, lift down, free fall, car, circular track and incline. Then check against this post.
Day 2: four numerical problems, each starting with a vector sketch.
Day 3: one clock problem: does it gain or lose, and by how much over a given time?
Day 4: the same effective g for a liquid, from our post on liquid in an accelerating container.
For pendulums that are not point masses, see physical pendulum and angular SHM. If the vector picture is hard to hold in your head, the Fermi Simulation Lab has interactive simulations for ideas that are easier to see than to read.
Tonight, write g_eff for a lift slowing down on its way down, and say whether a pendulum inside swings faster or slower.
