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 need to calculate the gravitational forces exerted by each of the other two masses separately and then add them vectorially.
The formula to calculate the gravitational force between two point masses is given by Newton's Law of Universal Gravitation:
F = G * (m1 * m2) / r^2
where F is the gravitational force, G is the gravitational constant (6.674 × 10^-11 N m^2/kg^2), m1 and m2 are the masses of the two objects, and r is the distance between their centers of mass.
(a) Gravitational force exerted on the 4 kg mass by the 2 kg mass:
Distance (r1) between the 4 kg mass at (4, 0) m and the 2 kg mass at the origin (0, 0) m is 4 m.
F1 = (6.674 × 10^-11 N m^2/kg^2) * ((4 kg) * (2 kg)) / (4 m)^2
= 2.22 × 10^-11 N
(b) Gravitational force exerted on the 4 kg mass by the 3 kg mass:
Distance (r2) between the 4 kg mass at (4, 0) m and the 3 kg mass at (2, 0) m is 2 m.
F2 = (6.674 × 10^-11 N m^2/kg^2) * ((4 kg) * (3 kg)) / (2 m)^2
= 8.003 × 10^-11 N
To find the net (vectorial) gravitational force on the 4 kg mass, we need to add the forces F1 and F2 by taking into account their directions:
Net force on the 4 kg mass = F1 - F2
Net force on the 4 kg mass = 2.22 × 10^-11 N - 8.003 × 10^-11 N
= -5.783 × 10^-11 N
So, the gravitational force exerted on the 4 kg mass by the other two masses is approximately -5.783 × 10^-11 N.
To find the magnitude and direction of the gravitational force exerted on the 3 kg mass by the other two masses, we follow the same procedure as above.
Distance (r3) between the 3 kg mass at (2, 0) m and the 2 kg mass at the origin (0, 0) m is 2 m.
F3 = (6.674 × 10^-11 N m^2/kg^2) * ((3 kg) * (2 kg)) / (2 m)^2
= 5.56 × 10^-11 N
Distance (r4) between the 3 kg mass at (2, 0) m and the 4 kg mass at (4, 0) m is also 2 m.
F4 = (6.674 × 10^-11 N m^2/kg^2) * ((3 kg) * (4 kg)) / (2 m)^2
= 2.22 × 10^-11 N
To find the net (vectorial) gravitational force on the 3 kg mass, we need to add the forces F3 and F4 by taking into account their directions:
Net force on the 3 kg mass = F3 - F4
Net force on the 3 kg mass = 5.56 × 10^-11 N - 2.22 × 10^-11 N
= 3.34 × 10^-11 N
So, the magnitude of the gravitational force exerted on the 3 kg mass by the other two masses is approximately 3.34 × 10^-11 N.
To find the direction, we can consider the x-axis as positive to the right. The net force is positive, which means it is acting in the positive x-direction. Therefore, the direction of the gravitational force on the 3 kg mass is to the right.
Therefore, the answers are:
(a) Gravitational force exerted on the 4 kg mass by the other two masses is approximately 2.33 × 10^-11 N.
(b) Magnitude and direction of gravitational force exerted on the 3 kg mass by the other two masses is approximately 3.34 × 10^-11 N to the right.