Two objects attract each other gravitationally with a force of 2.3×10−10 when they are 0.21 apart. Their total mass is 4.50 . Find their individual masses.

To find the individual masses of the two objects, we can use Newton's Law of Universal Gravitation:

F = (G × m1 × m2) / r^2

Where:
F is the gravitational force between the objects,
G is the gravitational constant (approximately 6.67 × 10^-11 N m^2 / kg^2),
m1 and m2 are the masses of the two objects, and
r is the distance between the centers of the two objects.

Given that the force (F) is 2.3 × 10^-10 N and the distance (r) is 0.21 m, and we need to find the individual masses (m1 and m2).

Let's plug in the values into the formula and solve for the masses:

2.3 × 10^-10 = (6.67 × 10^-11) × (m1 × m2) / (0.21)^2

To make the calculations easier, we can rewrite this equation as:

2.3 × 10^-10 × (0.21)^2 = 6.67 × 10^-11 × (m1 × m2)

Now, let's solve for m1 × m2:

(2.3 × 10^-10 × 0.21^2) / (6.67 × 10^-11) = m1 × m2

m1 × m2 ≈ 1.9813 kg^2

To find the individual masses, we need to consider the total mass (4.50 kg). Let's solve for m1 and m2 by assuming that m1 and m2 add up to 4.50 kg:

m1 + m2 = 4.50

Now, we have two equations:
m1 × m2 = 1.9813
m1 + m2 = 4.50

To solve these equations simultaneously, we can use substitution or elimination method. Let's use the substitution method to find the individual masses.

From the second equation, m1 = 4.50 - m2. Substituting this into the first equation, we get:

(4.50 - m2) × m2 = 1.9813

Expanding and rearranging the equation to obtain a quadratic equation:

-(m2^2) + 4.50m2 - 1.9813 = 0

We can solve this equation for m2 using the quadratic formula. The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solutions for x are given by:

x = (-b ± √(b^2 - 4ac)) / (2a)

For our equation, a = -1, b = 4.50, and c = -1.9813. Let's substitute these values into the quadratic formula:

m2 = [-(4.50) ± √((4.50)^2 - 4(-1)(-1.9813))] / (2(-1))

Simplifying:

m2 = [-(4.50) ± √(20.25 - 7.9252)] / (-2)

m2 = [-(4.50) ± √(12.3248)] / (-2)

Now, we have two possible solutions for m2:

m2 = (4.50 + √(12.3248)) / -2 or m2 = (4.50 - √(12.3248)) / -2

Evaluating these expressions:

m2 ≈ -0.352 kg or m2 ≈ 4.852 kg

Since mass cannot be negative, we discard the negative value:

m2 ≈ 4.852 kg

Now, we can find m1 by substituting the value of m2 into the equation:

m1 = 4.50 - m2

m1 = 4.50 - 4.852

m1 ≈ -0.352 kg

Again, we discard the negative value:

m1 ≈ 0.352 kg

Therefore, the individual masses of the two objects are approximately 0.352 kg and 4.852 kg, respectively.