Three masses are aligned along the x axis of a rectangular coordinate system so that a 2 kg mass is at the origin, a 3kg is at (2,0) m, and a 4 kg mass is at (4,0) m. Find (a) the gravitational force exerted on the 4 kg mass by the other two masses and (b) the magnitude and direction of the gravitational exerted on the 3 kg mass by the other two. (answer: 2.33x10^-3N, 1x10^-10N)
To find the gravitational force exerted on the 4 kg mass by the other two masses, we can use Newton's Law of Universal Gravitation:
F = G * (m1 * m2) / r^2
where:
F is the gravitational force
G is the universal gravitational constant (approximately 6.67430 x 10^-11 N·m^2/kg^2)
m1 and m2 are the masses
r is the distance between the centers of the masses
(a) Let's calculate the gravitational force exerted on the 4 kg mass by the 2 kg mass at the origin:
m1 = 2 kg (mass at the origin)
m2 = 4 kg (mass under consideration)
r = distance between the origin and the point (4,0) m, which is 4 m
Using the formula, we have:
F1 = G * (m1 * m2) / r^2
= (6.67430 x 10^-11 N·m^2/kg^2) * (2 kg * 4 kg) / (4 m)^2
= 5.336 x 10^-11 N
Now let's calculate the gravitational force exerted on the 4 kg mass by the 3 kg mass at (2,0) m:
m1 = 3 kg (mass at (2,0) m)
m2 = 4 kg (mass under consideration)
r = distance between (2,0) m and (4,0) m, which is 2 m
Using the formula:
F2 = G * (m1 * m2) / r^2
= (6.67430 x 10^-11 N·m^2/kg^2) * (3 kg * 4 kg) / (2 m)^2
= 2.669 x 10^-11 N
To find the total force exerted on the 4 kg mass by the other two masses, we can sum up F1 and F2:
Total force = F1 + F2
= 5.336 x 10^-11 N + 2.669 x 10^-11 N
= 8.005 x 10^-11 N
≈ 8.01 x 10^-11 N
So, the gravitational force exerted on the 4 kg mass by the other two masses is approximately 8.01 x 10^-11 N.
(b) To find the magnitude and direction of the gravitational force exerted on the 3 kg mass by the other two masses, we can follow the same steps as above but switching the masses and distances:
m1 = 2 kg (mass at the origin)
m2 = 3 kg (mass under consideration)
r = distance between the origin and the point (2,0) m, which is 2 m
Using the formula:
F3 = G * (m1 * m2) / r^2
= (6.67430 x 10^-11 N·m^2/kg^2) * (2 kg * 3 kg) / (2 m)^2
= 6.674 x 10^-11 N
The magnitude of the gravitational force exerted on the 3 kg mass by the other two masses is approximately 6.674 x 10^-11 N (6.67 x 10^-11 N, rounded to two significant figures).
Since this force is attractive, its direction will be towards the origin (negative x-axis), since the 2 kg mass is positioned at the origin.