Physics 106 Lesson 6 Density Buoyancy and Archimedes

Physics 106 Lesson #6 Density, Buoyancy and Archimedes’ Principle Dr. Andrew Tomasch 2405 Randall Lab atomasch@umich. edu

Last Time: Work and Energy ≡ the ability to do work Work is a scalar • Definition: Work is the action of a force through a distance: Example: You exert a horizontal force F to push an object through a horizontal displacement d F d Work equals parallel force times distance.

Work by Conservative Forces • • The work a conservative force does on an object in moving it from a point A to a point B is path independent. It depends only on the path’s end points. For conservative it is possible Conservative Forces: forces to define a potential energy function 1. Gravity D 2. Elastic (spring) Force E 3. Electromagnetic Force The work done by gravity is the same for path A-B-C-D-E as it is for path A-E B A C Equivalently, the work done on an object traveling in a closed loop is zero.

Work Done by the Gravitational Force hinitial mg hfinal mg PE=0 The work done by gravity is just the difference in potential energy between where you start and where you end in space, and only the difference in height matters

Energy Accounting The Work-Energy Theorem is energy conservation and turns mechanics into accounting. The Work-Energy “Balance Sheet” The Work-Energy Balance Sheet automatically accounts for the work done by gravity as differences In gravitational potential energy. Wnc ≡ Work done by nonconservative forces other than gravity → Friction • KE ≡ Cash • PE ≡ Deposits • Wfriction ≡ Taxes!

A Bob-Sled Run The bob-sled exchanges gravitational potential energy for kinetic energy as it slides without friction down the hill. Total energy, the sum of kinetic and potential energy, remains constant at every point along the motion.

Density • Density (Greek letter rho): The mass of uniform object is its density times its volume: m = V Caution Quiz Ahead water = 1000 kg/m 3 (SI) water = 1 g/cm 3 (CGS) The specific gravity is density expressed in units of the density of water. The specific gravity of x ≡ x / water

Concept Test #1 A ton of feathers and a ton of bricks have the same mass. The feathers make _____ pile than the bricks. A. a bigger B. a smaller C. the same

Buoyancy T + • Before: The upward tension in the string equals the downward weight of the block →scale reads true weight W • After: The upward buoyant force helps to balance the FB T + downward weight, reducing the tension in the string, which is the apparent weight of the block → apparent weight is less W than the true weight Read T Demonstration

Archimedes’ Principle Magnitude of Buoyant Force Weight of Displaced Fluid • The buoyant force acting on a body is equal to the weight of the fluid it displaces

Eureka! Caution Quiz Ahead • Archimedes deduced the equality between the weight of his body and the weight of the water that had overflowed from his bath tub • He yelled triumphantly, “Eureka!” (“I have found it” in Greek) Don’t try this at home!

Concept Test #2 A lead ball of mass m floats in liquid mercury and sinks in water. Which statement is true about the buoyant force acting on the lead ball? A. It is greater when the ball is floating in the mercury. B. It is greater when the ball is totally submerged in the water. C. It is the same in both cases. In mercury the buoyant force is equal to the weight and the ball floats. In water the buoyant force is less than the weight and the it sinks.

Archimedes’ Principle Applied • • FB = ρfluid Vobject g W = ρobject Vobject g object sinks if ρobject > ρfluid object floats if ρobject < ρfluid FB ρobject Vobject ρfluid W Completely immerse an object of volume Vobject with density ρobject in a fluid of density ρfluid

Floating Objects • An object floats with some portion of its volume protruding from the liquid • When Floating in Equilibrium: FB Vin W