A plumb bob (a mass m hanging on a string) is deflected from the vertical by an angle θ due to a massive mountain.
A) Make an estimate of the mass of Mt. Everest, assuming it has the shape of a cone 3910 m high and base of diameter 3910 m. Assume its mass per unit volume is 3000 kg per m^3.
B)Estimate the angle of the plumb bob if it is 5.20 km from the center of Mt. Everest.
mass estimate= volume*density=1/3 base area*height*density
= 1/3 (PI *(3910/2)^2)*3910*3000kg/m^3
=4.69483305 × 1013
So the angle must then be
sidways force: GMbob*4.69E13/5200^2
downward force: GMbob*Me/radiusearth^2
the angle sideways is
arctan*forcesideways/forcegravity
= (4.69E13/5200^2)/(Me/Re^2)
= arc tan(4.69E13/Me * (Re/5200)^2)
Thanks a lot. For B the answer was 6.83x10^-4 deg.
A) To estimate the mass of Mt. Everest, we need to calculate the volume of the mountain first and then multiply it by the given mass per unit volume.
The formula to calculate the volume of a cone is V = (1/3) * π * r^2 * h, where V is the volume, r is the radius of the base, and h is the height.
Given that the diameter of the base of Mt. Everest is 3910 m, the radius (r) is half of that, which is 3910 / 2 = 1955 m.
The height (h) of the cone is given as 3910 m.
Now, let's calculate the volume of Mt. Everest using the formula:
V = (1/3) * π * (1955^2) * 3910
Calculating the above formula, we get:
V ≈ 1.611 x 10^10 m^3
Now, we can estimate the mass of Mt. Everest by multiplying the volume by the given mass per unit volume:
Mass = Volume * Mass per unit volume
Mass = (1.611 x 10^10) * (3000)
Calculating the above formula, we get:
Mass ≈ 4.833 x 10^13 kg
Therefore, the estimated mass of Mt. Everest is approximately 4.833 x 10^13 kg.
B) To estimate the angle of the plumb bob, we can use the concept of gravitational potential energy. The potential energy due to gravity at a distance r from the center of Mt. Everest is given by the formula:
PE = - (G * M * m) / r
where PE is the potential energy, G is the gravitational constant, M is the mass of Mt. Everest, m is the mass of the plumb bob, and r is the distance from the center of Mt. Everest.
To find the angle θ, we can equate the potential energy to the potential energy at the vertical position, which is zero. So:
0 = - (G * M * m) / r0
where r0 is the distance from the center of Mt. Everest to the plumb bob when it is at the vertical position.
If the plumb bob is 5.20 km from the center of Mt. Everest, we can convert it to meters by multiplying by 1000:
r0 = 5.20 km * 1000 = 5200 m
Now, let's substitute the values into the formula and solve for the angle θ:
0 = - (G * M * m) / 5200
Solving for θ, we get:
θ = atan(-(G * M) / (5200 * g))
where atan represents the inverse tangent function and g is the acceleration due to gravity.
Substituting the known values into the equation and using the appropriate units:
θ ≈ atan(-(6.67 x 10^-11 N*m^2/kg^2 * 4.833 x 10^13 kg) / (5200 m * 9.8 m/s^2))
Calculating the above equation, we get:
θ ≈ atan(-1.0717)
Now, calculating the inverse tangent of -1.0717, we find:
θ ≈ -47.4 degrees
Therefore, the estimated angle of the plumb bob is approximately -47.4 degrees. Note that the negative sign indicates that the plumb bob is deflected in the opposite direction of the mountain.
To estimate the mass of Mt. Everest, we can use the following steps:
A) Estimate the volume of the mountain:
1. First, we need to calculate the radius of the base of the cone. Given that the diameter is 3910 m, the radius would be half of that, which is 1955 m.
2. Next, we can calculate the volume of a cone using the formula V = (1/3)πr^2h, where V is the volume, π is the constant pi, r is the radius of the base, and h is the height of the cone.
Plugging in the values, V = (1/3)π(1955)^2(3910).
Therefore, the estimated volume of Mt. Everest is approximately (1/3)π(1955)^2(3910) cubic meters.
3. Multiply the volume by the mass per unit volume to get an estimate of the mass of the mountain. Given that the mass per unit volume is 3000 kg per m^3, we can multiply the volume by this value.
Mass = Volume * Mass per unit volume.
Therefore, the estimated mass of Mt. Everest would be approximately ((1/3)π(1955)^2(3910)) * 3000 kg.
Now, let's move on to estimate the angle of the plumb bob from the mountain:
B) To estimate the angle of the plumb bob, we can consider it as a right triangle, with the distance from the center of Mt. Everest to the plumb bob as the hypotenuse, and the vertical deflection caused by Mt. Everest as the opposite side of the angle.
1. Given the distance from the center of Mt. Everest to the plumb bob is 5.20 km (or 5200 m), and the height of the cone-shaped mountain is 3910 m, we can directly use trigonometry to find the angle.
Tangent(theta) = Opposite side / Hypotenuse
Therefore, the estimated angle of the plumb bob would be arctan(3910 / 5200).
By following these steps, you can estimate the mass of Mt. Everest and the angle of the plumb bob caused by the mountain.