A heavier mass m1 and a lighter mass m2 are 17.5 cm apart and experience a gravitational force of attraction that is 9.80 10-9 N in magnitude. The two masses have a combined value of 5.00 kg. Determine the value of each individual mass
To determine the value of each individual mass (m1 and m2), we can use the formula for gravitational force:
F = (G * m1 * m2) / r^2
where F is the gravitational force, G is the gravitational constant (approximately 6.67 × 10^-11 N*m^2/kg^2), m1 and m2 are the masses, and r is the distance between the masses.
In this case, we are given the following information:
- The gravitational force (F) is 9.80 × 10^-9 N.
- The distance between the masses (r) is 17.5 cm, which is equal to 0.175 m.
- The combined mass of the two objects (m1 + m2) is 5.00 kg.
Now, we can rearrange the formula to solve for individual masses:
F * r^2 = G * m1 * m2
Let's substitute the values into the equation:
(9.80 × 10^-9 N) * (0.175 m)^2 = (6.67 × 10^-11 N*m^2/kg^2) * m1 * m2
(9.80 × 10^-9 N) * (0.030625 m^2) = (6.67 × 10^-11 N*m^2/kg^2) * m1 * m2
2.99625 × 10^-10 N*m^2 = (6.67 × 10^-11 N*m^2/kg^2) * m1 * m2
Now, we know that the sum of the individual masses (m1 + m2) is 5.00 kg. We can assign a variable, such as x, to one of the masses and express the other mass in terms of x:
m1 = x
m2 = 5.00 kg - x
Substituting these expressions into the equation:
2.99625 × 10^-10 N*m^2 = (6.67 × 10^-11 N*m^2/kg^2) * x * (5.00 kg - x)
Now, we can solve for x, which will give us the value of one of the masses. Let's perform the calculations:
2.99625 × 10^-10 N*m^2 = (6.67 × 10^-11 N*m^2/kg^2) * (5.00 kg * x - x^2)
Divide both sides of the equation by (6.67 × 10^-11 N*m^2/kg^2):
(2.99625 × 10^-10 N*m^2) / (6.67 × 10^-11 N*m^2/kg^2) = 5.00 kg * x - x^2
4.495252738 N/kg = 5.00 kg * x - x^2
Rearrange the equation:
x^2 - 5.00 kg * x + 4.495252738 N/kg = 0
Now we can use the quadratic formula to solve for x:
x = (-b ± √(b^2 - 4ac)) / 2a
where a = 1, b = -5.00 kg, and c = 4.495252738 N/kg:
x = (-(-5.00 kg) ± √((-5.00 kg)^2 - 4 * 1 * 4.495252738 N/kg)) / (2 * 1)
x = (5.00 kg ± √(25.00 kg^2 - 17.981011952 N/kg)) / 2
x = (5.00 kg ± √(384.019011952 N/kg)) / 2
x = (5.00 kg ± 19.595316338 N/kg) / 2
Now we have two possible values for x:
x1 = (5.00 kg + 19.595316338 N/kg) / 2
x2 = (5.00 kg - 19.595316338 N/kg) / 2
Simplifying further:
x1 ≈ 24.60 kg / 2 ≈ 12.30 kg
x2 ≈ -14.60 kg / 2 ≈ -7.30 kg
The negative value is not physically meaningful in this context, so we can discard x2. Therefore, the value of m1 is approximately 12.30 kg, and m2 is approximately 5.00 kg - 12.30 kg = -7.30 kg.