F=m ⃗g with ⃗g=−9.81 N/kg ̂
k is an approximation of Newton's law of universal gravitation.
(a) Up to which height h above the ground is the deviation between the field strength g(h) and the value g(h=0)
less than 0.10%? [2]
(b) When you move on the ground, the direction of ⃗g changes due to the curvature of Earth's surface. How far
can you travel from your orignial point with the sideways component of the new, tilted ⃗g ' , being less than
0.10% of the original field strength? [2]
(c) Estimate the volume above the ground (in km3
) where ⃗g is a constant vector to within 0.10%. [2]
To answer these questions, we need to determine the appropriate equations and apply them using the given information.
(a) The equation F = m⃗g represents the force due to gravity. Here, m represents the mass of the object, ⃗g represents the gravitational field strength, and F is the force. We are interested in the deviation between the field strength g(h) and the value g(h=0) when this deviation is less than 0.10%.
To find the height h above the ground, where the deviation is less than 0.10%, we can use the following approximation:
g(h) ≈ g(h=0) * (1 - h/R)
In this equation, R is the radius of the Earth. We want to solve for h when the deviation is less than 0.10%.
Therefore, we need to determine the maximum value of h that satisfies:
| g(h) - g(h=0) | / g(h=0) ≤ 0.001
(b) When we move on the ground, the direction of the gravitational field ⃗g changes due to the curvature of the Earth's surface. In this case, we want to find the distance we can travel from the original point with the sideways component of the new, tilted ⃗g' being less than 0.10% of the original field strength. To determine this, we need to consider the relationship between the tilt angle and the displacement.
(c) To estimate the volume above the ground where ⃗g is a constant vector to within 0.10%, we can use the equation for the gravitational field strength:
g = G * M / R^2
In this equation, G represents the gravitational constant, M is the mass of the Earth, and R is the radius of the Earth. We need to determine the volume in which the gravitational field strength does not deviate by more than 0.10%. There are several approaches to estimating this volume, such as considering different shapes (spheres, cubes, etc.) or using integral calculus to determine the volume of a region.