Center of Mass and Density

Center of Mass and Density BY Jahi and Morgan

Density

Overview Density is shown as mass over volume, or Kg/cm^3 An objects density does not change with gravity, it is constant. Weight changes based on the forces exerted on the object

Pis the density of the fluid, gis the gravity, V is volume of the immersed part of the body in the fluid. h is the height of immersed part A is the area. Formula: F(b)=gpV=pghA Buoyancy The ability of an object to float (slightly submerged on the water.

Example 1 • If the Volume remains constant at 500 ml and the mass changes from 2.34 to 3.75 over the course of 30 seconds, Find the rate of change, and the Density at t=10,15, and 25

2.34/500=.00468 g/mL • 3.75/500= .0075 g/mL • @F(0) D=.00468 • @F(30) D=.0075 • F(30)-F(0)= .00282 • .00282/3= .00094= F(10) • .00282/2=.00141= F(15) • (.00282/6)*5=.00235= F(25)

2.34/500=.00468 g/mL3.75/500= .0075 g/mL@F(0) D=.00468@F(30) D=.0075F(30)-F(0)= .00282

Sample Problem • Question 1: A ice cube is having density of 0.5 g/cm3 is having a Buoyant force of 9 N is immersed in water. Calculate its Volume? • Formula: F(b)=gpV=pghA

Solution • Density of iceρ= 0.5 g/cm3, Buoyant force, Fb = 9 N, Buoyant Force is given by Fb =ρg V The Volume is given by V =Fbgρ=9N9.8m/s2×0.5×10−3Kg/cm3 = 1836 cm3.

Center of Mass

Equations

Example #1 • Determine the center of mass for the region bounded by y=and y=.

Example #1 • First we will need to find the area of the region. - =5/12

Example 1 - =

Example 1

Example 1 • Thus the center of mass is

Example 2 • Find the center of mass of the region bounded by the curves y = cosx, y = 0, x = 0, x = π/2

Example 2 • Find the area of the region.

Example 2

Example 2 =

Example 2 • So the center of mass is ((π/2) – 1, π/8)

Center of Mass and Density