Mountain pull. A large mountain can slightly affect the direction of “down” as determined by a plumb line. Assume that we can model a mountain as a sphere of radius R = 3.00 km and density (mass per unit volume) 2.6 × 103 kg/m3. Assume also that we hang a 0.250 m plumb line at a distance of 3R from the sphere's center and such that the sphere pulls horizontally on the lower end. How far would the lower end move toward the sphere?
figure the force of attraction to the sphere, and the force of attration to the Earth.
TanTheta= forcetoSphere/ForceGravity
figure out theta
use acceleration of gravitation=gravitational constant*mass/(r^2)
use the radius and density to find the mass of the large mountain.
Find the distance from the center of the sphere (mountain) to the plumb.
Find the acceleration due to gravity in the x horizontal direction on the plumb using the equation.
Draw a right triangle with the vertical leg as 9.8 m/ss and the horizontal leg as the acceleration you found.
Use inverse tan to find the angle in the top corner of the triangle. Use the angle and the sin function to find the length of the horizontal leg. (the hypotenuse is 0.250m)
Remember to use proper units and convert km to m!!! check through your work for small mistakes.
There has been so much said, and on the whole so well said, that I shall not occupy the time.
In other words, refer to the answers above and perhaps to others on the internet.
You're welcome!
To determine how far the lower end of the plumb line moves toward the sphere, we need to calculate the gravitational force exerted by the mountain on the plumb line.
The force between two objects due to gravity is given by Newton's law of universal gravitation:
F = G * (m1 * m2) / r^2
Where:
F is the force of gravity,
G is the gravitational constant,
m1 and m2 are the masses of the two objects, and
r is the distance between the centers of the two objects.
In this case, we need to find the force exerted by the mountain on the plumb line. The mass of the plumb line's lower end is negligible compared to that of the mountain, so we can assume that the mass of the plumb line is negligible.
The gravitational force is responsible for the movement of the plumb line toward the mountain. We can calculate its magnitude using the formula above.
The distance between the lower end of the plumb line and the center of the mountain is given as 3R, where R is the radius of the mountain (3.00 km). In this case, the distance is 3 * 3.00 km = 9.00 km = 9000 m.
The gravitational constant G is approximately 6.674 × 10^-11 m^3 / (kg * s^2).
Now, we need to calculate the mass of the mountain. To do this, we can use the formula for the volume of a sphere:
V = (4/3) * π * r^3
Where V is the volume and r is the radius.
The volume V is given by:
V = (4/3) * π * R^3
Substituting the given radius R = 3.00 km = 3000 m, we can calculate the volume.
V = (4/3) * π * (3000 m)^3
V = (4/3) * π * (27 * 10^9 m^3)
V ≈ 113.097 × 10^9 m^3
The mass of the mountain can be calculated using the density (mass per unit volume) given as 2.6 × 10^3 kg/m^3. We can multiply the density by the volume to get the mass:
M = density * volume
M = (2.6 × 10^3 kg/m^3) * (113.097 × 10^9 m^3)
M ≈ 293.433 × 10^12 kg
Now we have all the values needed to calculate the gravitational force exerted by the mountain on the plumb line.
F = G * (m1 * m2) / r^2
F = (6.674 × 10^-11 m^3 / (kg * s^2)) * (0.25 kg) * (293.433 × 10^12 kg) / (9000 m)^2
Now we can calculate the magnitude of the gravitational force between the mountain and the plumb line.
F ≈ (6.674 × 10^-11 m^3 / (kg * s^2)) * (0.25 kg) * (293.433 × 10^12 kg) / (9000 m)^2
After evaluating the expression, we find that the magnitude of the gravitational force is approximately 4.646 × 10^-6 N.
Finally, we can use this force to calculate the displacement of the lower end of the plumb line toward the sphere. We can use Hooke's law, which states that the displacement is proportional to the force applied:
F = -k * x
Where F is the force, x is the displacement, and k is the spring constant.
In this case, the spring constant can be calculated using:
k = F / x
Substituting the force we calculated and the given displacement of 0.250 m, we can solve for the spring constant:
k = (4.646 × 10^-6 N) / (0.250 m)
After evaluating the expression, we find that the spring constant is approximately 1.8584 × 10^-5 N/m.
Now, we can calculate the displacement of the lower end of the plumb line using Hooke's law:
F = -k * x
Since we know F and k, we can solve for x:
x = -F / k
Substituting the values we've obtained, we can calculate the displacement:
x = -(4.646 × 10^-6 N) / (1.8584 × 10^-5 N/m)
After evaluating the expression, we find that the displacement of the lower end of the plumb line toward the sphere is approximately -0.250 m (negative sign indicates the direction toward the sphere).