Question

-10.62 The Earth has an angular speed of $7.272\cdot {10}^{-5}\mathrm{rad}/\mathrm{s}$ in its rotation. Find the new angular speed if an asteroid $(m=1.00\cdot {10}^{22}\mathrm{kg})$ hits the Earth while traveling at a speed of $1.40\cdot {10}^3\mathrm{m}/\mathrm{s}$ (assume the asteroid is a point mass compared to the radius of the Earth) in each of the following cases: a) The asteroid hits the Earth dead center. b) The asteroid hits the Earth nearly tangentially in the direction of Earth's rotation. c) The asteroid hits the Earth nearly tangentially in the direction opposite to Earth's rotation.

Short answer

Answer: (a) The new angular speed remains the same as the initial angular speed since the asteroid's initial angular momentum is zero. (b) The new angular speed will be slightly greater than the initial angular speed due to the added angular momentum from the asteroid moving in the same direction as the Earth's rotation. (c) The new angular speed will be slightly less than the initial angular speed due to the opposite direction of the asteroid's angular momentum, which will slightly decrease Earth's angular momentum.

Step-by-step solution

Find the initial angular momentum of the Earth

To find the initial angular momentum L_E of the Earth, we can use the formula $L_E=I_E{\omega }_E$, where $I_E$ is the moment of inertia of the Earth and ${\omega }_E$ is the initial angular speed of the Earth. According to the problem, the initial angular speed of the Earth is given as $7.272\cdot {10}^{-5}\mathrm{rad}/\mathrm{s}$. The Earth is approximately a sphere, so its moment of inertia is given by the formula $I_E=(2/5)M_E{R}_E^2$, where $M_E$ is the mass of the Earth and $R_E$ is the radius of the Earth. We will use approximate values for these parameters: $M_E=5.98\cdot {10}^{24}\mathrm{kg}$ and $R_E=6.37\cdot {10}^6\mathrm{m}$. Using these values, we can obtain the initial angular momentum of the Earth.

Calculate the initial angular momentum of the asteroid

Now, let's find the initial angular momentum L_A of the asteroid. This value depends on the direction in which the asteroid hits the Earth. In cases (b) and (c), the asteroid hits the Earth nearly tangentially. Thus, its angular momentum is given by the formula $L_A=m_A{R}_A{R}_A$, where $m_A$, $R_A$, and $R_A$ are the mass, radius, and angular velocity of the asteroid, respectively. We are given the mass of the asteroid $m_A=1.00\cdot {10}^{22}\mathrm{kg}$ and its velocity in the collision $v_A=1.40\cdot {10}^3\mathrm{m}/\mathrm{s}$. In cases (b) and (c), since the collision is nearly tangential, we assume $R_A=R_E$ as a good approximation. Additionally, in case (a), because the asteroid hits the Earth dead center, its initial angular momentum about the center of the Earth should be zero ($L_A=0$).

Apply the conservation of angular momentum

Because there are no external torques on the Earth-asteroid system, we can apply the conservation of angular momentum. The initial angular momentum of the system is the sum of the angular momentum of the Earth and the asteroid: $L_{\text{initial}}=L_E+L_A$. After the collision, we'll need to find the final angular momenta $L_{E,\text{final}}$ and angular speed ${\omega }_{E,\text{final}}$ of the Earth for each case. We do this by equating the total initial and total final angular momentum: $L_{\text{initial}}=L_{E,\text{final}}=I_E{\omega }_{E,\text{final}}$, as the mass of the asteroid is negligible compared to the mass of the Earth.

Calculate the new angular speed for each case

Using the conservation of angular momentum, we can now calculate the new angular speed of the Earth for each case. a) Asteroid hits dead center: $L_{\text{initial}}=L_E=L_{E,\text{final}}=I_E{\omega }_{E,\text{final}}$ => ${\omega }_{E,\text{final}}={\omega }_E$. b) Asteroid hits tangentially in the direction of Earth's rotation: Calculate $L_A$, and then use conservation of angular momentum to find the final angular speed. c) Asteroid hits tangentially opposite to Earth's rotation: Calculate $L_A$ (but with the opposite sign due to the opposite rotation direction), and then use conservation of angular momentum to find the final angular speed. By calculating the final angular speed for each case, we will have found the new angular speed of the Earth after the asteroid collision in each case.

Key concepts

Moment of Inertia

The moment of inertia, often symbolized as 'I', is a measure of how resistant an object is to angular acceleration when a torque is applied to it. It's analogous to mass in linear motion, but for rotational motion. Think of it as the rotational equivalent of mass. The moment of inertia depends on the mass distribution of an object; more mass distributed far from the rotational axis means a higher moment of inertia.

The Earth is approximately spherical, and for spheres, the moment of inertia can be calculated using the formula $I=\frac{2}{5}M{R}^2$ where 'M' represents the mass and 'R' the radius of the sphere. For example, the Earth has a large moment of inertia due to its massive size, meaning it would take a significant force to change its rotational speed. When an asteroid hits the Earth, as described in the exercise, the effect on Earth's rotation depends on the asteroid's mass and velocity as well as where it impacts Earth—a direct center hit versus a tangential collision.

Angular Velocity

Angular velocity is the rate at which an object rotates or revolves around an axis; it's how fast an object goes around. It's commonly denoted by the Greek letter omega $\omega$. The unit of angular velocity is radians per second $\text{rad/s}$.

Considering the Earth's Rotation

For instance, the Earth rotates on its axis around once every 24 hours, which can be converted into an angular velocity using the formula for circular motion, giving us an angular velocity of approximately $7.272\cdot {10}^{-5}\text{rad/s}$ as specified in the original exercise. Understanding this concept is crucial when evaluating the impact of external forces, such as an asteroid strike, on the rotational rate of a massive body.

Conservation Laws in Physics

Conservation laws are fundamental principles in physics stating that certain properties of isolated systems remain constant over time. One such law is the conservation of angular momentum, which is especially pertinent to astrophysical events like an asteroid colliding with a planet. This law states that if no external torque acts on a system, the total angular momentum of the system will stay constant.

Applying Conservation of Angular Momentum

In our textbook problem, the Earth-asteroid system can be considered isolated from external torques, meaning the total angular momentum before the collision must equal the total angular momentum after the collision. By using the equations for angular momentum $L=I\omega$, where 'L' denotes angular momentum, 'I' the moment of inertia, and '$\omega$' the angular velocity, we can calculate how collisions affect the spin of the Earth.

Momentum of Asteroid and Earth Interaction

When an asteroid collides with the Earth, its momentum has a direct impact on our planet's rotation. The asteroid’s momentum can be classified into linear and angular forms. For problems involving planetary rotation, the angular momentum is what we are interested in.

The angular momentum of a point mass asteroid like the one in our exercise is given by $L=mvr$, where 'm' is its mass, 'v' its velocity, and 'r' the distance from the axis of rotation, which, in the case of a tangential collision, would be approximately equal to the radius of the Earth. If the asteroid hits the Earth tangentially in the same direction as Earth's rotation (case b), it will contribute positively to the Earth's angular momentum. However, if it hits in the opposite direction (case c), it will have a negative effect. The exact change in the Earth’s rotation depends on the asteroid's mass, velocity, and the point of impact relative to the Earth’s rotation.