At a distance H above the surface of a planet, the true weight of a remote probe is six percent less than its true weight on the surface. The radius of the planet is R. Find the ratio H/R.

just plug and chug. The weight is GMm/r^2 and GMm is constant, so

(1/(R+H)^2) / 0.94 (1/R^2)

It's a bit messy to solve that for H, but since H=kR for some k, we just want to solve for k. H/R = k.

(1/((1+k)R)^2) / 0.94 (1/R^2)
R^2/(R(1+k))^2) = 0.94
1/(1+k)^2 = 0.94
(1+k)^2 = 1/0.94
1+k = 1/√0.94
k = 1/√0.94 - 1 = 0.0314

Why did the remote probe go on a diet? Because it wanted to weigh less than its true weight on the surface of the planet! But don't worry, I'm here to help with your question.

Let's call the true weight of the remote probe on the surface W. According to the information given, its weight at a distance H above the surface is 6% less than W. So, the weight of the remote probe at the distance H is 0.94W.

The weight of an object at a distance H above the surface of a planet can be found using the equation:

Weight = (Mass * Gravitational Acceleration) / (Radius + H)^2

Since we know that the weight at the distance H is 0.94W, we can set up an equation:

0.94W = (Mass * Gravitational Acceleration) / (R + H)^2

Now, let's simplify this equation. We can cancel out the Mass and Gravitational Acceleration on both sides:

0.94W = 1 / (R + H)^2

We can rearrange this equation to solve for (R + H)^2:

(R + H)^2 = 1 / 0.94W

Taking the square root of both sides gives us:

R + H = sqrt(1 / 0.94W)

Now, we can subtract R from both sides to solve for H:

H = sqrt(1 / 0.94W) - R

Finally, to find the ratio H/R, we can divide both sides by R:

H/R = (sqrt(1 / 0.94W) - R) / R

And there you have it! The ratio H/R for the remote probe above the surface of the planet. Just plug in the values of W, Mass, Gravitational Acceleration, and the radius of the planet to get the specific ratio.

To find the ratio H/R, we can set up the following equation based on the given information:

(Weight at height H) = (Weight on the surface) - 6% of (Weight on the surface)

Let's assume the weight on the surface is W.

(Weight at height H) = W - 0.06W

Simplifying the equation, we get:

(Weight at height H) = 0.94W

Now, let's consider the gravitational force equation:

(Weight at height H) = (Gravitational constant) * (Mass of the probe) / (Distance from the center of the planet)^2

Substituting the weight at height H with 0.94W and the distance from the center of the planet with R + H, we have:

0.94W = (Gravitational constant) * (Mass of the probe) / (R + H)^2

Now, let's solve for H/R:

H/R = (Distance from the center of the planet) / (Radius of the planet)
= (R + H) / R

Rearranging the equation from earlier, we have:

0.94W = (Gravitational constant) * (Mass of the probe) / (R + H)^2

Simplifying the equation, we get:

(R + H) = sqrt[(Gravitational constant) * (Mass of the probe) / 0.94W]

Hence, the ratio H/R is:

H/R = [(sqrt[(Gravitational constant) * (Mass of the probe) / 0.94W]) - R] / R

Please note that we cannot determine the exact value of the ratio H/R without knowing the specific values of the gravitational constant, the mass of the probe, and the weight on the surface (W).

To solve this problem, we can use the concept of gravitational force. The weight of an object is given by the equation W = mg, where W is the weight, m is the mass, and g is the acceleration due to gravity.

Let's denote the true weight of the remote probe on the surface as W_surface. Thus, its weight at a distance H above the surface would be W_above = (1-0.06)W_surface = 0.94W_surface.

The acceleration due to gravity is inversely proportional to the square of the distance from the center of the planet. So, at the surface of the planet, the acceleration due to gravity is g_surface = G(M/R^2), where G is the gravitational constant, M is the mass of the planet, and R is the radius.

At a distance H above the surface, the acceleration due to gravity is g_above = G(M/(R+H)^2).

Since weight is directly proportional to mass and acceleration due to gravity, we can write the following equation:

W_surface / W_above = g_surface / g_above

Substituting the values we have:

W_surface / (0.94W_surface) = g_surface / g_above

Canceling out W_surface:

1 / 0.94 = g_surface / g_above

Using the expressions for g_surface and g_above:

1 / 0.94 = (G(M/R^2)) / (G(M/(R+H)^2))

Canceling out G and M:

1 / 0.94 = R^2 / (R+H)^2

Cross-multiplying:

(R+H)^2 = 0.94R^2

Taking the square root of both sides:

R+H = sqrt(0.94) * R

Rearranging the equation:

H = sqrt(0.94) * R - R

Now, we can find the ratio H/R by dividing H by R:

H/R = (sqrt(0.94) - 1)

Calculating the value:

H/R ≈ (0.9697 - 1)

H/R ≈ -0.0303

Therefore, the ratio H/R is approximately -0.0303.