91Ó°ÊÓ

The principle of conservation of energy is key in solving problems involving motion, including rotational motion. It states that the total energy in a closed system remains constant.
In the case of the falling chimney, its gravitational potential energy when upright converts into rotational kinetic energy as it falls. We start by identifying the potential energy when the chimney is upright, which is given by the formula:
\[ U = Mg\left(\frac{H}{2}\right) \]
Here, M is the mass of the chimney, H is the height, and g is the acceleration due to gravity. As the chimney tips over, this energy transforms into rotational kinetic energy, expressed as:
\[ K = \frac{1}{2}I \omega^2 \]
where I is the moment of inertia and Ó¬ is the angular velocity. For energy conservation, we set the potential energy equal to the kinetic energy and solve for Ó¬.

Rotational kinetic energy is the energy an object possesses due to its rotation. For our chimney, it's crucial to express this energy in terms of its rotational inertia and speed. The formula for rotational kinetic energy is:
\[ K = \frac{1}{2}I \omega^2 \]
In this expression, I represents the moment of inertia, and Ó¬ is the angular velocity. For a thin rod rotating about one end, the moment of inertia I is given by:
\[ I = \frac{1}{3}MH^2 \]
Combining this with our energy conservation equation, we substitute and solve for angular velocity Ó¬ by setting the potential energy equal to the kinetic energy, leading to:
\[ Mg\left(\frac{H}{2}\right) = \frac{1}{2}\left(\frac{1}{3}MH^2\right) \omega^2 \]
Solving for Ó¬, we get:
\[ \omega = \sqrt{\frac{3g}{H} \cos\theta} \]
This indicates how the rotational speed of the chimney depends on its angle with the vertical.

The moment of inertia is a fundamental concept in rotational motion, representing how mass is distributed relative to the axis of rotation. It's often called the rotational analog of mass in linear motion.
For a thin rod rotating about one end, the moment of inertia I is calculated using:
\[ I = \frac{1}{3}MH^2 \]
This means that the resistance to rotational acceleration depends on both the mass and the length of the rod. In our problem, the moment of inertia helps us relate the chimney's mass and height to its rotational kinetic energy.
The larger the moment of inertia, the slower the object will rotate for a given amount of energy, making it a crucial factor in analyzing the motion of our falling chimney.

Tangential acceleration is the rate of change of the tangential velocity of a point on a rotating object. For our chimney, the tangential acceleration at the top is particularly important. It can be found using the relation:
\[ a_t = \alpha H \]
where α is the angular acceleration and H is the height of the chimney. Angular acceleration α is related to the change in angular velocity, given by:
\[ \alpha = \frac{d\omega}{dt} \]
By differentiating the rotational speed Ó¬ with respect to time and angle, we get:
\[ \alpha = \frac{d}{d\theta} \left( \sqrt{\frac{3g}{H} \cos\theta} \right) \omega \] Simplifying and solving leads to the tangential acceleration formula:
\[ a_t = g \sin\theta \]
When the tangential acceleration equals the acceleration due to gravity g, it occurs at an angle θ where:

\[ \sin\theta = 1 \]
This results in θ being 90 degrees.