A heavier mass m1 and a lighter mass m2 are 20.5 cm apart and experience a gravitational force of attraction that is 8.60 10-9 N in magnitude. The two masses have a combined value of 5.80 kg. Determine the value of each individual mass.
To determine the value of each individual mass, we can use the formulas related to the force of gravity.
The force of gravity between two objects can be calculated using the equation:
F = G * (m1 * m2) / r^2
Where:
- F is the magnitude of the gravitational force
- G is the gravitational constant (approximately 6.67430 × 10^-11 N m^2/kg^2)
- m1 and m2 are the masses of the two objects
- r is the distance between the centers of the two objects
In this case, we are given:
F = 8.60 × 10^-9 N
r = 20.5 cm = 0.205 m (converting centimeters to meters)
The total mass of the two objects is given as 5.80 kg, so we can express m2 in terms of m1:
m1 + m2 = 5.80 kg --> m2 = 5.80 kg - m1
Now, we can substitute these values into the gravitational force equation:
8.60 × 10^-9 N = (6.67430 × 10^-11 N m^2/kg^2) * (m1 * (5.80 kg - m1)) / (0.205 m)^2
Simplifying the equation, we solve for m1:
8.60 × 10^-9 N = (6.67430 × 10^-11 N m^2/kg^2) * (m1 * (5.80 kg - m1)) / 0.042025 m^2
Multiply both sides of the equation by 0.042025 m^2 to simplify further:
(8.60 × 10^-9 N) * (0.042025 m^2) = (6.67430 × 10^-11 N m^2/kg^2) * (m1 * (5.80 kg - m1))
Now, rearrange the equation to solve for m1:
4.31075 × 10^-10 N m = (6.67430 × 10^-11 N m^2/kg^2) * (5.80 m1 kg - m1^2 kg)
Simplify further and rearrange into a quadratic equation form:
4.31075 × 10^-10 N m = (6.67430 × 10^-11 N m^2/kg^2) * (5.80 m1 kg) - (6.67430 × 10^-11 N m^2/kg^2) * (m1^2 kg)
Now, set it equal to zero:
(6.67430 × 10^-11 N m^2/kg^2) * (m1^2 kg) - (6.67430 × 10^-11 N m^2/kg^2) * (5.80 m1 kg) + 4.31075 × 10^-10 N m = 0
By solving this quadratic equation, we can find the values for m1.