Question

For two identical satellites in circular motion around the Earth, which statement is true? a) The one in the lower orbit has less total energy. b) The one in the higher orbit has more kinetic energy. c) The one in the lower orbit has more total energy. d) Both have the same total energy.

Short answer

Answer: (c) The one in the lower orbit has more total energy.

Step-by-step solution

Formula for total energy

E = -G * M * m / (2 * r) Now, let's analyze each answer choice: #a) The one in the lower orbit has less total energy.# Let's compare the total energy of the satellite in a lower orbit (r1) with the total energy of the satellite in the higher orbit (r2). Since r1 < r2, using the formula above, we have E1 > E2. This means that the one in the lower orbit has MORE total energy. So this option is incorrect. #b) The one in the higher orbit has more kinetic energy.# Now, let's analyze the relationship between the kinetic energy and the radius of the orbit. Since the satellites are of equal mass and are in a circular motion around the Earth, their angular momentum is conserved: L = mvr. This implies: v = L / (mr). Let's now find the expression for kinetic energy:

Formula for kinetic energy

K = 0.5 * m * (L / (mr))^2 = L^2 / (2 * m^2 * r^2) Since m and L are constants, the relationship between kinetic energy and the radius of the orbit is:

Relation between K and r

K ∝ 1 / r^2 Since r1 < r2, we have K1 > K2. This means the one in the lower orbit has MORE kinetic energy. So this option is also incorrect. #c) The one in the lower orbit has more total energy.# As we have analyzed in option 'a)', it's true that the one in the lower orbit has more total energy. So this option is correct. #d) Both have the same total energy.# As analyzed earlier, the total energy depends on the radius of the orbit. Since the satellites are in different orbits, they cannot have the same total energy. So this option is incorrect. The correct answer is (c) The one in the lower orbit has more total energy.

Key concepts

Circular Motion

Imagine a satellite orbiting the Earth in a circular path. This is a classic example of circular motion, where the satellite is constantly changing direction, which means it is always accelerating towards the center of the circle, which in this case, is the center of the Earth. This type of acceleration is caused by the centripetal force, provided by Earth's gravitational pull.

For satellites in circular motion, the speed is constant, but the velocity is not, because velocity is a vector that includes both speed and direction. In the context of our exercise, understanding circular motion is crucial because it sets the stage for how other principles, like total energy, kinetic energy, and angular momentum, come into play in the environment of an orbiting satellite.

Total Energy in Orbit

The total energy in orbit for a satellite encompasses both its kinetic energy—energy due to its motion— and potential energy—energy due to its position in the Earth's gravitational field. The formula to calculate the total energy (\( E \) is \[ E = -\frac{G \cdot M \cdot m}{2 \cdot r} \) where \( G \) is the gravitational constant, \( M \) is the mass of Earth, \( m \) is the mass of the satellite, and \( r \) is the radius of the orbit.

In our exercise, two identical satellites are compared: one in a lower orbit and one in a higher orbit. Since their mass is the same and the gravitational constant and Earth's mass are constants, the total energy is inversely related to the orbit's radius. Therefore, a satellite in a lower orbit has more total energy because it is closer to Earth, and the negative sign in the formula indicates that it is gravitationally bound to the Earth.

Kinetic Energy Conservation

Kinetic energy conservation is a part of the broader physical principle of energy conservation, which states that in a closed system, energy cannot be created or destroyed, only transformed. In orbital mechanics, when considering an isolated two-body system like the Earth and a satellite, the total mechanical energy is conserved if there is no external force or energy added or removed from the system.

The kinetic energy (\( K \) for a satellite in orbit is given by the equation \[ K = \frac{1}{2} m v^2 \) where \( m \) is the mass of the satellite, and \( v \) is the orbital speed. For circular orbits, where angular momentum is conserved, any changes in the orbit's radius will inversely affect the satellite's speed, maintaining the balance of kinetic and potential energy—a beautiful demonstration of conservation principles in the cosmos.

Angular Momentum Conservation

The law of conservation of angular momentum states that if no external torque acts on a system, the total angular momentum of the system remains constant. In the context of a satellite orbiting Earth, the angular momentum (\( L \) is given by \[ L = m v r \) where \( m \) is the mass of the satellite, \( v \) is its orbital speed, and \( r \) is the radius of the orbit.

Our satellites in circular orbits have a fixed angular momentum because there are no external torques acting on them. This leads to an inverse relationship between the satellite's speed and the radius of its orbit. When a satellite moves to a higher orbit, it slows down. Conversely, as a satellite gets closer to the Earth, its speed increases. This principle is key to understanding why the satellite in a lower orbit has a higher kinetic energy and, as a result, a higher total energy than the one in a higher orbit.