A uniform, solid, spherical asteroid with mass 1.2 * 10^13 kg and radius 1.0 km is rotating with a period of 4.6 hours. A meteoroid moving in the asteroid's equatorial plane crashes into the equator at 9.0 km/s. It hits at a 58° angle to the vertical and embeds itself at the surface. After the impact the asteroid is rotating in the same direction with a period of 3.0 hours. Find the meteoroid's mass.
I know this has to do with angular momentum, but I'm really lost on how to go about this >.< please help.
Well, well, well, looks like we have an asteroid collision on our hands! Don't worry, my gravity-defying friend, I'm here to help you navigate through this cosmic conundrum with a dash of humor.
First, let's remind ourselves of the wise words of Sir Isaac Newton: "For every action, there is an equal and opposite reaction." This means that the initial angular momentum of the asteroid-meteoroid system should be conserved.
Now, the asteroid is like a cosmic top, spinning happily with a period of 4.6 hours. But then along comes our mischievous meteoroid, crashing into the asteroid's equator at a not-so-subtle speed of 9.0 km/s.
The meteoroid hits at a 58° angle to the vertical, so it imparts a tangential force to the asteroid, altering its angular momentum. As a result, the asteroid adjusts its rotation speed to a new period of 3.0 hours.
To find the meteoroid's mass, we'll have to use the principle of conservation of angular momentum. Let's call the initial angular momentum of the system "L1" and the final angular momentum "L2".
L1 = L2
The initial angular momentum (L1) of the asteroid-meteoroid system is given by the formula:
L1 = I1 * ω1
Where I1 is the moment of inertia of the asteroid (which for a solid sphere, you can find as 2/5 * m * r^2) and ω1 is the initial angular velocity, given by 2π / T1, where T1 is the initial period of rotation (4.6 hours).
Now, the final angular momentum (L2) can be calculated as:
L2 = I2 * ω2
Where I2 is the new moment of inertia of the asteroid (which will be the same as I1 because there are no external torques acting on the asteroid) and ω2 is the final angular velocity, given by 2π / T2, where T2 is the final period of rotation (3.0 hours).
Since L1 = L2, we can equate the two equations:
I1 * ω1 = I2 * ω2
Now plug in the values and solve for the unknown mass (m) of the meteoroid:
(2/5 * m * r^2) * (2π / T1) = (2/5 * m * r^2) * (2π / T2)
Cancel out the common factors and solve for m, and voila! You'll have the mass of the meteoroid.
Remember, my friend, in the vastness of the universe, even complex problems can be tackled with a sprinkle of humor and a dash of curiosity. Good luck with your calculations, and may your cosmic collisions lead to astronomical discoveries!
To solve this problem, we can use the conservation of angular momentum. The angular momentum of the asteroid-meteoroid system before the impact should be equal to the angular momentum of the asteroid after the impact.
Step 1: Find the initial angular momentum of the asteroid-meteoroid system before the impact.
The angular momentum is given by the formula: L = Iω, where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity.
The moment of inertia of a solid sphere is given by the formula: I = (2/5) * M * R^2, where M is the mass of the asteroid and R is its radius.
The angular velocity is given by the formula: ω = 2π / T, where T is the period of rotation.
The initial angular momentum (L_initial) is therefore: L_initial = I * ω_initial
Step 2: Find the final angular momentum of the asteroid after the impact.
The final angular velocity (ω_final) is given by the formula: ω_final = 2π / T_final, where T_final is the period of rotation after the impact.
The final angular momentum (L_final) is therefore: L_final = I * ω_final
Step 3: Set the initial and final angular momenta equal to each other and solve for the mass of the meteoroid.
L_initial = L_final
I * ω_initial = I * ω_final
Since the moment of inertia (I) cancels out, we can write the equation as:
M_asteroid * R^2 * ω_initial = M_asteroid * R^2 * ω_final
Simplifying further:
M_asteroid * ω_initial = M_asteroid * ω_final
Now we can solve for the mass of the meteoroid (M_meteoroid):
M_asteroid * ω_initial = M_meteoroid * ω_final
M_meteoroid = (M_asteroid * ω_initial) / ω_final
Substitute the given values into the equation and calculate:
M_meteoroid = (1.2 * 10^13 kg * (2π / (4.6 hours * 3600 s/hour))) / (2π / (3.0 hours * 3600 s/hour))
M_meteoroid ≈ 9.82 * 10^8 kg
So, the mass of the meteoroid is approximately 9.82 * 10^8 kg.
To tackle this problem, we need to consider the principle of conservation of angular momentum. The angular momentum of a rotating object is given by the product of its moment of inertia and its angular velocity.
Let's break down the problem into steps:
Step 1: Calculate the initial angular momentum of the asteroid before the impact.
To find the initial angular momentum, we'll use the formula:
Angular Momentum = Moment of Inertia * Angular Velocity
The moment of inertia of a uniform, solid sphere rotating around its axis is given by the formula:
Moment of Inertia = (2/5) * Mass * Radius^2
The angular velocity can be calculated using the formula:
Angular Velocity = 2 * π / Period
Given:
Mass (M) = 1.2 * 10^13 kg
Radius (R) = 1.0 km = 1000 m
Period (T) = 4.6 hours = 4.6 * 60 * 60 seconds
First, calculate the moment of inertia:
Moment of Inertia = (2/5) * M * R^2
Next, calculate the angular velocity:
Angular Velocity = 2 * π / T
Finally, calculate the initial angular momentum:
Initial Angular Momentum = Moment of Inertia * Angular Velocity
Step 2: Calculate the final angular momentum of the asteroid after the impact.
After the impact, the asteroid's angular velocity changes. We are given the new period, so we can calculate the new angular velocity using the same formula as above.
Given:
New Period (T') = 3.0 hours = 3.0 * 60 * 60 seconds
Calculate the final angular velocity:
Final Angular Velocity = 2 * π / T'
Step 3: Calculate the change in angular momentum.
The change in angular momentum is the difference between the final and initial angular momenta.
Change in Angular Momentum = Final Angular Momentum - Initial Angular Momentum
Step 4: Calculate the meteoroid's mass.
The change in angular momentum is caused by the embedded meteoroid. We'll assume the meteoroid is a point mass and changes the moment of inertia of the asteroid negligibly. Hence, the change in angular momentum is solely due to the meteoroid.
Given:
Impact velocity of the meteoroid (v) = 9.0 km/s = 9000 m/s
The angular momentum of the meteoroid is given by the formula:
Angular Momentum of Meteoroid = Mass of Meteoroid * Impact Distance * v
The impact distance can be calculated using the diagram and the given angle. The impact distance is the horizontal component of the distance from the center of the asteroid to the point of impact. It can be calculated as follows:
Impact Distance = Radius * sin(angle)
Finally, use the formula for the change in angular momentum:
Change in Angular Momentum = Angular Momentum of Meteoroid
Set the two expressions for the change in angular momentum equal to each other and solve for the mass of the meteoroid.