With reference to moment of inertia, angular momentum, and simple harmonic motion under mechanics, consider the following statements:
- 1. The moment of inertia of a rigid body does not depend only on the mass of the body and the perpendicular distances of particles from the axis of rotation, but it also depends on where and in what direction the position of the axis of rotation is in space.
- 2. When the external torque on a rotating body is zero, its total angular momentum is conserved, which is called the 'law of conservation of angular momentum', and due to this principle, when a spinning ballet dancer tucks her arms close to her body, her angular velocity increases.
- 3. The time period of a simple pendulum depends on the mass and amplitude of its bob, and if the pendulum is taken to the center of the earth, its time period will become infinite due to zero effective acceleration due to gravity.
- 4. In the simple harmonic motion of a spring-mass system, when the body passes through its mean position, its kinetic energy is maximum and potential energy is minimum (zero), while the total mechanical energy is always conserved.
Which of the above statement(s) is/are completely true?
Detailed Explanation & Solution
Statement 1 is true because moment of inertia is not a constant quantity, but depends on the choice of the axis of rotation (perpendicular axis theorem and parallel axis theorem are examples of this). Statement 2 is true because according to $\tau = \frac{dL}{dt}$, when $\tau = 0$, $L$ remains constant; tucking arms decreases the moment of inertia, which increases the angular velocity. Statement 3 is false because the time period of a simple pendulum is $T = 2\pi\sqrt{L/g}$, which *does not* depend on the mass of the bob and small amplitudes; at the center of the earth $g = 0$ makes the time period infinite, but the part about mass/amplitude is incorrect. Statement 4 is true because velocity is maximum at the mean position, making kinetic energy maximum and potential energy minimum.