When is the angular momentum of a system constant? A) When the moment of inertia is constant B) When the linear momentum and the energy are constant C) When the total kinetic energy is constant D) When no net external force acts on the system E) When no torque acts on the system
The correct answer and explanation is:
The correct answer is E) When no torque acts on the system.
Angular momentum is defined as the product of the moment of inertia and the angular velocity of a rotating object, and it can be expressed as $L = I \cdot \omega$, where $L$ is angular momentum, $I$ is the moment of inertia, and $\omega$ is angular velocity. The principle of conservation of angular momentum states that if no external torque is applied to a system, the angular momentum of the system remains constant over time.
In the absence of external torque, there is no force to change the system’s rotation, and thus the angular momentum remains unchanged. This is because torque is the rate of change of angular momentum, given by $\tau = \frac{dL}{dt}$. If no external torque is present, the time derivative of angular momentum is zero, implying that $L$ is conserved.
Now, let’s consider the other options:
- A) When the moment of inertia is constant: While the moment of inertia is related to the distribution of mass within the system, the angular momentum can still change if there is an external torque. Thus, just having a constant moment of inertia does not guarantee that angular momentum is conserved.
- B) When the linear momentum and the energy are constant: Linear momentum and energy conservation do not directly imply conservation of angular momentum. They are separate concepts, and each has different conditions under which they are conserved.
- C) When the total kinetic energy is constant: Kinetic energy conservation is not a requirement for angular momentum conservation. For example, an object can undergo inelastic collisions (which conserve angular momentum but not kinetic energy) or experience changes in shape and velocity while maintaining angular momentum.
- D) When no net external force acts on the system: This is related to linear momentum, not angular momentum. Linear momentum is conserved when no external forces act on a system, but angular momentum is conserved when no external torque acts on the system, even if external forces are present, as long as the net torque is zero.
In conclusion, angular momentum remains constant when no external torque is applied to the system, as torque is the key factor that can change angular momentum.