As an astronaut, you observe a small planet to be spherical. After landing on the planet, you set off, walking always straight ahead, and find yourself returning to your spacecraft from the opposite side after completing a lap of 24.0 km (assume this to be the maximum circumference). You hold a hammer and a falcon feather at a height of 1.33 m, release them, and observe that they fall together to the surface in 28.8 s. Determine the mass of the planet.
To determine the mass of the planet, we can make use of Newton's Law of Universal Gravitation and the concept of Period of Revolution.
First, let's focus on the information about the planet's circumference. We are given that the astronaut completes a lap of 24.0 km, which we assume to be the maximum circumference of the planet. The circumference of a sphere is given by the formula C = 2πr, where r is the radius of the sphere. Therefore, we can solve for the radius as follows:
C = 2πr
24.0 km = 2πr (make sure to convert km to the appropriate unit)
24.0 km = 2π(1000 m/km)r (since the radius needs to be in meters)
24.0 km = 2000πr
r = (24.0 km) / (2000π)
r ≈ 0.006043 km (approximately)
Next, let's consider the observation of the falling objects. The time it takes for the hammer and feather to fall together is given as 28.8 seconds. This observation allows us to determine the acceleration due to gravity (g) on the planet. The acceleration due to gravity can be calculated using the formula:
g = (2d) / (t^2)
where d is the distance traveled (height) and t is the time taken.
In this case, the distance traveled (height) is given as 1.33 m, and the time taken is 28.8 seconds. Substituting these values into the formula, we can calculate the acceleration due to gravity:
g = (2 * 1.33 m) / (28.8 s)^2
g = 0.011 m/s^2 (approximately)
Now that we know the acceleration due to gravity, we can determine the mass of the planet using Newton's Law of Universal Gravitation. The formula for gravitational force is given by:
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 objects, and r is the distance between them.
In this case, we have the force of gravity acting on the falling objects, which is equal to their combined weight. Weight can be calculated using the formula:
Weight = mass * g
where g is the acceleration due to gravity.
Since both the hammer and feather experience the same acceleration due to gravity and fall together, we can set up the following equation:
m_hammer * g = m_feather * g
Since g is common to both sides, it cancels out, giving:
m_hammer = m_feather
Therefore, we can conclude that the masses of the hammer and feather are equal.
To find the mass of the planet, we can rearrange the formula for gravitational force:
F = (G * m1 * m2) / r^2
m1 = (F * r^2) / (G * m2)
In this case, m2 represents the mass of the planet, m1 represents the mass of the falling object (hammer or feather), and F is the weight of the object, which is given by:
Weight = mass * g
Let's assume the mass of the falling object (m1) is M (which is unknown) and the distance (r) between the astronaut and the planet is the radius we calculated earlier.
Substituting the values into the formula, we can solve for the mass of the planet (m2):
m2 = (F * r^2) / (G * M)
However, we need another piece of information to calculate the mass of the planet. We need to know the weight of the falling object (F). Without that information, we cannot determine the mass of the planet.
To determine the mass of the planet, we need to use the concept of the gravitational force acting on the hammer and the feather.
Step 1: Calculate the acceleration due to gravity on the planet.
The acceleration due to gravity on the planet can be calculated using the formula:
g = (4π²R) / T²
where:
- g is the acceleration due to gravity
- R is the radius of the planet
- T is the time taken to complete one lap around the planet
From the given information, the time taken to complete one lap (T) is 28.8 seconds. We know that the circumference of the planet is the maximum circumference of 24.0 km.
So, the radius of the planet (R) can be calculated as:
R = circumference / (2π)
R = 24,000 m / (2π)
R = 3,819.7 meters
Now, we can calculate the acceleration due to gravity (g):
g = (4π²(3,819.7)) / (28.8)²
g ≈ 1.63 m/s²
Step 2: Calculate the mass of the planet.
The gravitational force acting on the hammer and the feather is given by the formula:
F = mg
where:
- F is the gravitational force
- m is the mass of the object
- g is the acceleration due to gravity
In this case, both the hammer and the feather experience the same acceleration due to gravity, so we can equate their gravitational forces:
m₁g = m₂g
Dividing by g on both sides gives:
m₁ = m₂
This means that the mass of the hammer is equal to the mass of the feather.
Step 3: Determine the mass of the hammer and the feather.
As we don't have the mass of either the hammer or the feather, we need to determine the combined mass of the hammer and the feather.
We can use the formula to calculate the combined mass:
m₁ + m₂ = F / g
Since both the hammer and the feather fell together, the force (F) acting on them is the same.
Step 4: Calculate the time taken for the hammer and the feather to fall to the surface.
The time taken for the hammer and the feather to fall to the surface is given as 28.8 seconds.
Step 5: Calculate the mass of the planet.
Now we can calculate the combined mass of the hammer and the feather using the formula:
m₁ + m₂ = F / g
m₁ + m₂ = (Falls together) / g
m₁ + m₂ = 28.8 s / 1.63 m/s²
m₁ + m₂ ≈ 17.67 kg
Since the mass of the hammer is equal to the mass of the feather, we can divide the combined mass by 2 to get the mass of each object:
m₁ ≈ m₂ ≈ 17.67 kg / 2 ≈ 8.83 kg
Therefore, the mass of the planet is approximately 8.83 kg.