On this page
A block bobbing in water and mercury sloshing in a U-tube look different, but both oscillate because gravity acts on a displaced column of liquid. Both appear as exercises in NCERT, and both reduce to the same three steps: find the restoring force, find the moving mass, take the ratio. This post works through each, with numbers.
Is this in the syllabus?
NCERT's chapter Oscillations asks you to show the motion is simple harmonic in two exercises: 13.17, a floating cylinder of cork, and 13.18, mercury in a U-tube. The Syllabus for JEE (Main) 2026 lists "simple harmonic motion (S.H.M.) and its equation" and "restoring force and force constant", and the JEE (Advanced) 2026 Information Brochure lists "Linear and angular simple harmonic motions" and "Buoyancy". Check the current year's documents before your exam.
What is the recipe?
For any linear SHM, displace the body by a small y, write the net restoring force as F = −Ky, and identify the moving mass m. Then ω = √(K/m) and T = 2π√(m/K). The skill is finding K.
Why does a floating block oscillate?
Take a block of uniform cross-section A, height h and density ρ, floating upright in a liquid of density σ. At equilibrium its weight equals the buoyant force, so the submerged depth d satisfies ρAh = σAd, giving d = ρh/σ.
Push it down a further y. The extra buoyant force is σAgy, upward, so K = σAg. The mass is m = ρAh. Then
T = 2π√(ρAh/(σAg)) = 2π√(ρh/(σg)) = 2π√(d/g)
The middle form is NCERT's answer to Exercise 13.17. The last form is the useful one: the period depends only on how deep the block floats.
Worked example 1: a wooden block
A block 20 cm tall with density 600 kg/m³ floats upright in water of density 1000 kg/m³. Take g = 9.8 m/s².
Submerged depth: d = (600/1000) × 20 cm = 12 cm = 0.12 m.
Period: T = 2π√(0.12/9.8) ≈ 2π × 0.111 ≈ 0.70 s.
In denser salt water, d is smaller, so the block bobs faster. The motion stays simple harmonic only while the block neither leaves the water nor goes fully under; beyond that, the restoring force stops growing with y.
Why does liquid in a U-tube oscillate?
Let a U-tube of uniform cross-section A hold a liquid column of total length L, measured along the tube, with density ρ. Push the level down by y on one side; it rises by y on the other. The levels now differ by 2y, so an unbalanced pressure ρg(2y) acts on area A:
restoring force: F = −2ρgAy, so K = 2ρgA
moving mass: the whole column, m = ρAL
period: T = 2π√(L/(2g))
Density and cross-section cancel. Mercury and water columns of the same length oscillate with the same period.
If the arms are tilted at angles θ₁ and θ₂ to the horizontal, a displacement y along the tube raises and lowers the levels by y sin θ₁ and y sin θ₂, so T = 2π√(L/(g(sin θ₁ + sin θ₂))). Vertical arms bring back the factor 2g.
Worked example 2: a mercury column
A U-tube holds a mercury column 40 cm long in total.
T = 2π√(0.40/(2 × 9.8)) = 2π√0.0204 ≈ 2π × 0.143 ≈ 0.90 s.
Replace the mercury with a water column of the same total length: the period is still about 0.90 s.
Which mistakes cost marks?
Using the block's full height instead of its submerged depth in T = 2π√(d/g).
Dropping the factor 2 in the U-tube. The level difference is twice the displacement.
Using the length of one arm instead of the whole column.
Counting only the liquid that moves up. The whole column moves, including the bend.
Treating large displacements as SHM after the block has left the liquid or gone under.
Why do buoyancy questions feel slippery?
Loverude, Kautz and Heron (2003) found that standard instruction on hydrostatics left many university science and engineering students unable to predict and explain the sinking and floating of simple objects, and that some difficulties came from ideas usually assumed to be secure before hydrostatics begins. If the equilibrium step, weight equals buoyant force, is shaky, the oscillation step will be too. Our post on pressure, buoyancy and Bernoulli mistakes is a quick check.
A practice routine
Day 1: derive T = 2π√(d/g) and T = 2π√(L/(2g)) from a blank page.
Day 2: three floating-body problems in different liquids, finding d first each time.
Day 3: two U-tube problems, one with tilted arms.
Day 4: a mixed set with a spring problem from our post on two blocks joined by a spring, so you must find K for each.
If you prefer to watch the levels move, the Fermi Simulation Lab has interactive simulations for ideas that are easier to see than to read.
Today, measure how deep a floating object sits in a glass of water and predict its bobbing period from T = 2π√(d/g).
