The mass and coordinates of three objects are given below: m1 = 6.0 kg at (0.0, 0.0) m, m2 = 1.5 kg at (0.0, 4.1) m, and m3 = 4.0 kg at (1.9, 0.0) m. Determine where we should place a fourth object with a mass m4 = 7.9 kg so that the center of gravity of the four-object arrangement will be at (0.0, 0.0) m

The location of the center of gravity of the fourth mass is .

Explanation:

Vectorially speaking, the center of gravity with respect to origin (), measured in meters, is defined by the following formula:

(1)

Where:

, , , - Masses of the objects, measured in kilograms.

, , , - Location of the center of mass of each object with respect to origin, measured in meters.

If we know that , , , , , , and , then the equation is reduced into this:

The location of the center of gravity of the fourth mass is .

answer: at 24 february, 2014 the population gets twice of its initial population.

explanation:

since we have given that

the population for alpha city, t years after january 1, 2004 is given as

first we find out the initial population, i.e. t=0,

so, our quadratic equation becomes,

according to question, we have said that if the population has twice its initial population, then it becomes

so, time taken to reach this above mentioned population is given by

now, we know that time can't be negative so, we take t=9.149 years.

hence, at 24 february, 2014 the population gets twice of its initial population.

scientific laws are the result of testing theories and proving them to be facts.