Two spherical balls are placed so their centers are 2.6 m apart. The force between the two balls is 2.75 x 10-12 N. What is the mass of each ball if one ball is twice the mass of the other ball? Please show how to solve it with the numbers.
Well, let's just say these two balls are getting a little too close for comfort! But, fear not, I'm here to help with your physics problem.
To solve this, let's call the mass of the smaller ball 'm' and the mass of the larger ball '2m'. We'll also need to know the gravitational constant, which is approximately 6.67 x 10^-11 N(m/kg)^2.
Now, the force between these balls is given by the equation:
F = (G * m * (2m)) / (2.6)^2,
where F is the force, G is the gravitational constant, and (2.6)^2 is the squared distance between their centers.
Plugging in the values, we have:
2.75 x 10^-12 = (6.67 x 10^-11 * m * (2m)) / (2.6)^2.
Now, let's simplify this equation a bit. We can cancel out the (2.6)^2 on both sides and rearrange the equation to solve for 'm':
2.75 x 10^-12 * (2.6)^2 = 6.67 x 10^-11 * m * (2m).
After simplifying, we get:
2.75 x 10^-12 * (2.6)^2 = 6.67 x 10^-11 * 2m^2.
Now, let's solve for 'm'. Dividing both sides by 2m^2 and rearranging, we find:
m^2 = (2.75 x 10^-12 * (2.6)^2) / (6.67 x 10^-11).
Using a calculator, the value inside the brackets is approximately 6.698 x 10^-23.
Now, taking the square root of both sides, we find:
m ≈ √(6.698 x 10^-23).
Calculating this, we get m ≈ 2.59 x 10^-12 kg.
So, the mass of the smaller ball, m, is approximately 2.59 x 10^-12 kg, and the mass of the larger ball, which is twice the mass, is approximately 5.18 x 10^-12 kg.
Voila! We've solved the puzzle. Just be careful not to let those balls get too close together, or we might have a forceful situation on our hands!
To solve this problem, we need to use Newton's Law of Universal Gravitation, which states that the force between two objects is proportional to the product of their masses and inversely proportional to the square of the distance between their centers.
The formula for the force (F) is given by:
F = G * (m1 * m2) / r^2
Where:
F is the force between the two objects
G is the gravitational constant (approximated as 6.674 x 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 that:
F = 2.75 x 10^-12 N (force between the two balls)
r = 2.6 m (distance between their centers)
m1 = x (mass of one ball)
m2 = 2x (mass of the other ball, which is twice the mass of the first ball)
Now, we can substitute the given values into the formula:
2.75 x 10^-12 N = (6.674 x 10^-11 N · m^2 / kg^2) * (x * 2x) / (2.6 m)^2
Simplifying the equation:
2.75 x 10^-12 N = (6.674 x 10^-11 N · m^2 / kg^2) * 2x^2 / (2.6 m)^2
Multiply both sides of the equation by (2.6 m)^2:
(2.75 x 10^-12 N) * (2.6 m)^2 = (6.674 x 10^-11 N · m^2 / kg^2) * 2x^2
Rearranging the equation:
(2.75 x 10^-12 N) * (2.6 m)^2 / (6.674 x 10^-11 N · m^2 / kg^2) = 2x^2
Now we can solve for x by dividing both sides of the equation by the value in front of x^2:
x^2 = (2.75 x 10^-12 N) * (2.6 m)^2 / (6.674 x 10^-11 N · m^2 / kg^2)
x^2 = 0.000274 N · m^2 / (kg^2 / m^2)
To solve for x, we can take the square root of both sides:
x = sqrt(0.000274 N · m^2 / (kg^2 / m^2))
Using a calculator, we find that:
x ≈ 0.016 kg
Since m2 is twice the mass of m1, m2 = 2 * 0.016 kg = 0.032 kg
Therefore, the mass of one ball is approximately 0.016 kg, and the mass of the other ball is approximately 0.032 kg.
To solve this problem, we can use Newton's law of universal gravitation, which states that the force of gravity between two objects is given by the equation:
F = G * (m1 * m2) / r^2
where F is the force between the objects, G is the gravitational constant (approximately 6.67430 x 10^-11 N * (m/kg)^2), m1 and m2 are the masses of the two objects, and r is the distance between their centers.
Given that the force F is 2.75 x 10^-12 N, the distance between the centers of the two balls (r) is 2.6 m, and one ball is twice the mass of the other ball, we can set up the following equations:
F = G * (m1 * m2) / r^2
m1 = 2 * m2
Substituting the values into the equation, we have:
2.75 x 10^-12 N = (6.67430 x 10^-11 N * (m/kg)^2) * (2 * m2 * m2) / (2.6 m)^2
Simplifying the equation, we get:
2.75 x 10^-12 N = 6.67430 x 10^-11 N * 2 * m2^2 / 6.76 m^2
Let's solve for m2:
m2^2 = (2.75 x 10^-12 N * 6.76 m^2) / (6.67430 x 10^-11 N * 2)
m2^2 = 0.1375 kg^2
Taking the square root of both sides, we find:
m2 = 0.37 kg
Since m1 is twice the mass of m2:
m1 = 2 * 0.37 kg
m1 = 0.74 kg
Therefore, one ball has a mass of 0.74 kg, and the other ball has a mass of 0.37 kg.
F = GMm/r^2
so plug in your numbers
6.67*10^-11 * m * 2m / 2.6^2 = 2.75*10^-12
m^2 = 2.75*10^-12 * 2.6^2 / 13.34*10^-11 = 0.139
m = 0.373 kg