AP Physics 1 · Lesson 2

Buoyancy and forces in fluids

Know Know it

A fluid pushes on every surface of an object in it, and it pushes harder on the bottom because the bottom is deeper. The difference is an upward buoyant force.

Archimedes’ principle: the buoyant force equals the weight of the fluid the object pushes aside, Fb=ρfluidVdispgF_b = \rho_{fluid}V_{disp}g. For a fully submerged object, VdispV_{disp} is the object’s whole volume; for a floating object, it is only the part below the surface.

A floating object at rest has Fb=mgF_b = mg. Writing ρfluidVsubg=ρobjVg\rho_{fluid}V_{sub}g = \rho_{obj}Vg gives the fraction submerged, Vsub/V=ρobj/ρfluidV_{sub}/V = \rho_{obj}/\rho_{fluid}. An object denser than the fluid sinks. An object’s apparent weight in a fluid is its weight minus the buoyant force.

What happens when an object is placed in a fluid
Object’s average densityWhat it doesBuoyant force once at rest
less than the fluid’sfloats with part above the surfaceequal to its weight
equal to the fluid’sstays put at any depth when fully underequal to its weight
greater than the fluid’ssinks to the bottomless than its weight; the floor pushes up with the rest

Worked example Easy

A plastic fishing float with a volume of 30 cm³ (3.0 × 10⁻⁵ m³) is held completely under fresh water. Use g = 9.8 m/s² and water density 1000 kg/m³. What is the buoyant force on it?

  1. 0.030 N
  2. 2.9 N
  3. 290 N
  4. 0.29 NAnswer

How to solve it

  1. The float is fully under, so the displaced volume is its whole volume, 3.0 × 10⁻⁵ m³.
  2. Buoyant force = weight of that water = (1000)(3.0 × 10⁻⁵)(9.8) = 0.29 N.

Why each choice is right or wrong

  • A. Incorrect. 0.030 is the mass of displaced water in kilograms; the buoyant force is that water’s weight, so multiply by g.
  • B. Incorrect. 2.9 N uses a volume of 3.0 × 10⁻⁴ m³, ten times the float’s real volume.
  • C. Incorrect. 290 N treats 30 cm³ as 0.030 m³; a cubic centimeter is 10⁻⁶ m³, so 30 cm³ is 3.0 × 10⁻⁵ m³.
  • D. Correct. Fb=ρVg=(1000)(3.0×10−5)(9.8)=0.29F_b = \rho V g = (1000)(3.0 \times 10^{-5})(9.8) = 0.29 N.

Trap The buoyant force is not the object’s weight

For a fully submerged object, the buoyant force depends only on its volume and on the fluid’s density, not on what the object is made of. A lead ball and a plastic ball of the same size feel the same buoyant force; the lead ball sinks because its weight is larger.

Explain Explain it

Task verbs: predict, explain

For floating problems, write the force balance first, Fb=mgF_b = mg, and then decide what the displaced volume is. If the object floats freely, only the submerged part counts. If it is held under, the whole volume counts, and another force (a hand, a string) makes up the difference between the buoyant force and the weight.

A scale under a container counts everything the container holds up. When a floating object is added, the water pushes up on the object with the buoyant force, the object pushes down on the water just as hard, and that push reaches the scale.

Worked example Medium

An ice cube (density 917 kg/m³) floats at rest in a glass of fruit juice (density 1080 kg/m³). What percent of the ice cube’s volume is above the surface of the juice?

  1. 85 percent
  2. 15 percentAnswer
  3. 8.3 percent
  4. 18 percent

How to solve it

  1. Floating: ρjuiceVsubg=ρiceVg\rho_{juice}V_{sub}g = \rho_{ice}Vg, so Vsub/V=917/1080=0.849V_{sub}/V = 917/1080 = 0.849.
  2. The fraction above is 1 − 0.849 = 0.151, about 15 percent.

Why each choice is right or wrong

  • A. Incorrect. 85 percent is the fraction below the surface, not above it.
  • B. Correct. The submerged fraction is 917/1080 = 0.85, so 1 − 0.85 = 0.15, or 15 percent, is above the surface.
  • C. Incorrect. 8.3 percent uses the density of water, 1000 kg/m³, instead of the juice’s 1080 kg/m³.
  • D. Incorrect. 18 percent divides the juice density by the ice density, 1080/917 − 1, which inverts the ratio.

Predict Your turn

A rigid, sealed plastic bottle is held completely under water just below the surface, then pushed slowly down until it is 2 m deep. Treat the water as having the same density at every depth. Predict whether the buoyant force on the bottle at 2 m is greater than, less than or equal to the buoyant force just below the surface, and explain why.

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Compare with a model answer

A full answer does this

  • Says the buoyant force is the same at both depths
  • Gives a correct reason: the bottle displaces the same volume of water, or the pressure difference between its bottom and top stays the same

Model answer

The buoyant force is the same at both depths. The bottle is rigid, so it pushes aside the same volume of water wherever it is, and the buoyant force equals the weight of that displaced water. Deeper down, the pressure on the top and the pressure on the bottom both rise by the same amount, so the difference between them, which is what pushes up, does not change.

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