(1 pt) The acceleration due to gravity, g, is given by
g = (G*M)/(r^2)
where M is the mass of the Earth, r is the distance from the center of the Earth, and G is the uniform gravitational constant.
(a) Suppose that we change from our distance from the center of the Earth by a distance \Delta r = x. Use a linear approximation to find an approximation to the resulting change in g, as a fraction of the original acceleration:
(Your answer will be a function of x and r.)
(b) Is this change positive or negative?
(c) What is the percentage change in g when moving from sea level to the top of Mount Elbert (a mountain over 14,000 feet tall in Colorado; in km, its height is 4.29 km; assume the radius of the Earth is 6400 km)?
percent change =
a) -2(x/r)
b) Negative
c) 2(4.29 / 6400) * 100
(a) Alright, let's do some approximation math, shall we? We want to find an approximation to the resulting change in g when we change our distance from the center of the Earth by a distance x. To do this, we can use a linear approximation.
The linear approximation tells us that if we have a function, f(x), and we change x by a small amount, Δx, then the resulting change in f will be approximately equal to the derivative of f evaluated at x, times Δx. Here, f(x) represents the acceleration due to gravity, g(x).
So, the resulting change in g, Δg, is approximately equal to the derivative of g with respect to x, times Δx. Since g = (G * M) / (r^2), we need to find the derivative of g with respect to x. To do this, we use the chain rule:
d(g)/d(x) = d(g)/d(r) * d(r)/d(x)
Now, d(g)/d(r) is simply the derivative of g with respect to r, which we can find by differentiating g with respect to r. This gives us:
d(g)/d(r) = -2*(G * M)/(r^3)
Next, we need to find d(r)/d(x), which represents the change in r with respect to x. In this case, since we are simply changing our distance from the center of the Earth by a distance x, d(r)/d(x) is just equal to 1.
Finally, we can plug everything into the linear approximation formula:
Δg ≈ (-2*(G * M)/(r^3)) * x
So, the approximate change in g as a fraction of the original acceleration is approximately -2*(G * M)/(r^3) times x.
(b) Now, let's answer the question about the sign of this change. The negative sign in the formula for Δg indicates that the change in g is negative. In other words, when we move away from the center of the Earth, the acceleration due to gravity decreases.
(c) Moving on to the percentage change in g when going from sea level to the top of Mount Elbert. We have the height of Mount Elbert, h = 4.29 km, and the radius of the Earth, r = 6400 km.
To find the percentage change, we need to calculate the change in g and then express it as a percentage of the original acceleration. We know from part (a) that the change in g is approximately -2*(G * M)/(r^3) times x.
When moving from sea level to the top of Mount Elbert, the change in r is equal to the height of Mount Elbert, x = 4.29 km. Plugging these values into the formula, we get:
Δg ≈ -2*(G * M)/(r^3) * 4.29 km
To express this change as a percentage of the original acceleration, we can divide Δg by the original acceleration and multiply by 100:
Percent change ≈ (Δg / g) * 100
So, the percentage change in g when moving from sea level to the top of Mount Elbert is approximately (-2*(G * M)/(r^3) * 4.29 km) / g * 100.
(a) To find an approximation to the resulting change in g, we can use the linear approximation formula:
Δg ≈ g'(r) * Δr
where g'(r) is the derivative of g with respect to r (the rate of change of g with respect to r). Taking the derivative of g:
g'(r) = d/dx (G*M/r^2)
= -2*G*M/r^3
Using the linear approximation formula, our approximation for the change in g is:
Δg ≈ -2*G*M/r^3 * Δr
≈ -2*G*M/r^3 * x
Therefore, the approximation to the resulting change in g, as a fraction of the original acceleration, is -2*G*M/r^3 * x.
(b) The change in g depends on the sign of Δr (x). If the distance from the center of the Earth increases (Δr > 0), then the change in g will be negative. If the distance from the center of the Earth decreases (Δr < 0), then the change in g will be positive.
(c) To find the percentage change in g when moving from sea level to the top of Mount Elbert, we need to calculate the change in g and express it as a percentage of the original acceleration.
The radius of the Earth, r = 6400 km
The height of Mount Elbert, Δr = 4.29 km
G is the uniform gravitational constant.
Using the formula from part (a), the change in g is approximately:
Δg = -2*G*M/r^3 * Δr
To find the percentage change, we can calculate:
Percent change = (|Δg| / g) * 100
Substituting the respective values:
Percent change = (|-2*G*M/r^3 * Δr| / g) * 100
Keep in mind that the value of G, M, and g need to be provided in the problem to calculate the exact percentage change.
To find the approximation to the resulting change in g, we can use a linear approximation.
(a) The linear approximation is given by the formula:
\(\Delta g \approx \frac{{dg}}{{dr}} \cdot \Delta r\)
To find \(\frac{{dg}}{{dr}}\), we take the derivative of the equation for g with respect to r:
\(\frac{{dg}}{{dr}} = \frac{{d}}{{dr}} \left( \frac{{G \cdot M}}{{r^2}} \right) \)
Using the power rule for differentiation, we get:
\(\frac{{dg}}{{dr}} = -2 \cdot \frac{{G \cdot M}}{{r^3}}\)
Substituting this back into the linear approximation formula, we have:
\(\Delta g \approx -2 \cdot \frac{{G \cdot M}}{{r^3}} \cdot \Delta r\)
(b) The change in g will be negative because as we move farther away from the center of the Earth, the force of gravity decreases.
(c) To find the percentage change in g when moving from sea level to the top of Mount Elbert, we need to calculate the values of g at those two locations and then find the percentage difference.
At sea level, the distance from the center of the Earth is equal to the radius of the Earth, r = 6400 km. The acceleration due to gravity is given by:
\(g_{\text{{sea level}}} = \frac{{G \cdot M}}{{r^2}}\)
At the top of Mount Elbert, the distance from the center of the Earth is equal to the radius of the Earth plus the height of the mountain, r = 6400 km + 4.29 km. The acceleration due to gravity at the top of Mount Elbert is given by:
\(g_{\text{{Mount Elbert}}} = \frac{{G \cdot M}}{{(r + \Delta r)^2}}\)
The percentage change in g is then calculated using the formula:
\(\text{{percent change}} = \frac{{g_{\text{{sea level}}} - g_{\text{{Mount Elbert}}}}}{{g_{\text{{sea level}}}}} \cdot 100\)
You can substitute the values of G, M, and r into the equations and calculate the percentage change.