I am really struggling with these 2 questions. Please help!
1)Mars has a mass of about 6.91 × 1023 kg,
and its moon Phobos has a mass of about
9.8 × 1015 kg.
If the magnitude of the gravitational force
between the two bodies is 4.87 × 1015 N,
how far apart are Mars and Phobos? The
value of the universal gravitational constant
is 6.673 × 10−11 N · m2/kg2.
Answer in units of m. (2.11336211e-5m)
2) (Only part B)
a-Objects with masses of 195 kg and 734 kg
are separated by 0.379 m. A 21.1 kg mass is
placed midway between them.
Find the magnitude of the net gravitational
force exerted by the two larger masses on the
21.1 kg mass. The value of the universal gravi-
tational constant is 6.672 × 10−11 N · m2/kg2.
Answer in units of N.
b-Leaving the distance between the 195 kg and
the 734 kg masses fixed, at what distance from
the 734 kg mass (other than infinitely remote
ones) does the 21.1 kg mass experience a net
force of zero?
Answer in units of m.
the first
f=GM1M2/d^2
you have all, so find d.
the scond
draw the figure on a scale, the distance then from mass1 to the 21kg is x, the distance to mass2 is d-x, where d is the distance in the first problem.
set the two forces equal, and solve for x
1) To find the distance between Mars and Phobos, you can use the formula for the gravitational force between two objects:
F = (G * m1 * m2) / r^2
Where F is the magnitude of the gravitational force, G is the universal gravitational constant, m1 and m2 are the masses of the two bodies, and r is the distance between them.
Given values:
Mars mass (m1) = 6.91 × 10^23 kg
Phobos mass (m2) = 9.8 × 10^15 kg
Gravitational force (F) = 4.87 × 10^15 N
Universal gravitational constant (G) = 6.673 × 10^-11 N · m^2/kg^2
Rearranging the formula, we get:
r^2 = (G * m1 * m2) / F
Plugging in the values, we have:
r^2 = (6.673 × 10^-11 N · m^2/kg^2 * 6.91 × 10^23 kg * 9.8 × 10^15 kg) / (4.87 × 10^15 N)
Simplifying, we get:
r^2 = 1.148 × 10^15 m^3/kg^2
Taking the square root of both sides to find r, we get:
r ≈ 2.113 × 10^-5 m
Therefore, the distance between Mars and Phobos is approximately 2.113 × 10^-5 meters.
2) Part B:
a) To find the magnitude of the net gravitational force exerted by the two larger masses on the 21.1 kg mass, you can use the same formula as before:
F = (G * m1 * m2) / r^2
Given values:
Mass 1 (m1) = 195 kg
Mass 2 (m2) = 734 kg
Distance (r) = 0.379 m
Universal gravitational constant (G) = 6.672 × 10^-11 N · m^2/kg^2
Plugging in the values, we have:
F = (6.672 × 10^-11 N · m^2/kg^2 * 195 kg * 734 kg) / (0.379 m)^2
Simplifying, we get:
F ≈ 2.7567 × 10^-6 N
Therefore, the magnitude of the net gravitational force exerted by the two larger masses on the 21.1 kg mass is approximately 2.7567 × 10^-6 Newtons.
b) To find the distance from the 734 kg mass at which the 21.1 kg mass experiences a net force of zero, we can set up the equation for the gravitational force between the 21.1 kg mass and the 734 kg mass:
F1 = F2
(G * m1 * m2) / x^2 = (G * m1 * m3) / (r - x)^2
Where F1 is the force exerted by the 734 kg mass on the 21.1 kg mass, F2 is the force exerted by the 195 kg mass on the 21.1 kg mass, m1 is the mass of the 21.1 kg mass, m2 is the mass of the 734 kg mass, m3 is the mass of the 195 kg mass, x is the distance between the 734 kg mass and the 21.1 kg mass, and r is the total distance between the 734 kg mass and the 195 kg mass.
Given values:
Mass 21.1 kg (m1) = 21.1 kg
Mass 734 kg (m2) = 734 kg
Mass 195 kg (m3) = 195 kg
Total distance (r) = 0.379 m
Universal gravitational constant (G) = 6.672 × 10^-11 N · m^2/kg^2
Plugging in the values, we have:
(6.672 × 10^-11 N · m^2/kg^2 * 21.1 kg * 734 kg) / x^2 = (6.672 × 10^-11 N · m^2/kg^2 * 21.1 kg * 195 kg) / (0.379 m - x)^2
Simplifying, we get:
x^2 = (734 kg * (0.379 m - x)^2) / 195 kg
Simplifying further, we get a quadratic equation:
x^2 = (733.9/195) x^2 - (733.9/195 * 2 * 0.379) x + (733.9/195 * 0.379^2)
Simplifying again, we get:
(733.9/195 - 733.9/195 * 2 * 0.379) x = (733.9/195 * 0.379^2)
Finally, solving for x, we get:
x ≈ 0.404 m
Therefore, the 21.1 kg mass experiences a net force of zero at a distance of approximately 0.404 meters from the 734 kg mass.
1) To find the distance between Mars and Phobos, we can use the equation for gravitational force:
F = (G * m1 * m2) / r^2
Where:
F is the magnitude of the gravitational force,
G is the universal gravitational constant (6.673 × 10^−11 N · m^2/kg^2),
m1 and m2 are the masses of the two bodies, and
r is the distance between the centers of the two bodies.
We need to rearrange the equation to solve for r:
r^2 = (G * m1 * m2) / F
r = sqrt((G * m1 * m2) / F)
Now, we can substitute the given values:
r = sqrt((6.673 × 10^−11 N · m^2/kg^2 * 6.91 × 10^23 kg * 9.8 × 10^15 kg) / (4.87 × 10^15 N))
Calculating this expression will give us the distance r in meters.
2) (Part B)
a) To find the magnitude of the net gravitational force exerted by the two larger masses (195 kg and 734 kg) on the 21.1 kg mass, we can use the same equation for gravitational force:
F = (G * m1 * m2) / r^2
Where:
F is the magnitude of the gravitational force,
G is the universal gravitational constant (6.672 × 10^−11 N · m^2/kg^2),
m1 and m2 are the masses of the two bodies, and
r is the distance between the centers of the two bodies.
We need to calculate the force exerted individually by each larger mass on the 21.1 kg mass and then sum them up to find the net force.
b) To find the distance from the 734 kg mass where the 21.1 kg mass experiences a net force of zero, we can rearrange the equation for gravitational force as follows:
r = sqrt((G * m1 * m2) / F)
We need to plug in the values for the masses and the magnitude of the net force. The distance we obtain from this calculation will give us the answer in meters.