91Ó°ÊÓ

Potential energy (PE) is the stored energy of an object due to its position or height. For our gymnast at her greatest height of 0.5 meters, her potential energy is calculated using the formula: \[PE = mgh\]where:

• \( m = 40 \) kg is her mass,
• \( g = 9.81 \) m/s² is the acceleration due to gravity, and
• \( h = 0.5 \) m is her height at release.

Plugging these values, we get:\[PE = 40 \times 9.81 \times 0.5 = 196.2 \text{ J}\]This energy is ready to be converted into motion as she falls, illustrating a critical concept of energy exchange in physics.

The principle of conservation of energy tells us that in a closed system, energy cannot be created or destroyed, only transformed. When our gymnast lets go, her potential energy transforms as she swings. At her maximum angle, her energy divides between a new smaller potential energy and rotational kinetic energy. The conservation of energy equation is expressed as:\[PE_{initial} = RE_{final} + PE_{final}\]Initially, she has all potential energy, but when she swings to an angle \( \theta = 135^\circ \), part of this is converted into rotational kinetic energy while she still has some potential energy depending on her height above the lowest swing point.This principle ensures that even as her position changes, the total energy within the system remains constant, allowing us to solve for quantities like her angular velocity.

Rotational kinetic energy (RE) comes into play when the gymnast swings around the bar. This form of energy depends on the object's moment of inertia (\( I \)) and its angular velocity (\( \omega \)). The equation to find rotational kinetic energy is:\[RE = \frac{1}{2} I \omega^2\]The moment of inertia quantifies how the mass is distributed relative to the axis of rotation. For our gymnast, it involves her body's mass and how it spreads from the pivot point (her hands on the bar). The energy previously stored as potential energy converts, giving speed and momentum to her swing. Different from linear kinetic energy due to movement along a path, rotational kinetic energy measures how quickly something spins, which is essential in gymnastics and machinery alike.

Angular velocity (\( \omega \)) measures how fast an object rotates around a fixed point, such as a gymnast swinging on a bar. It's akin to speed but in rotational terms—instead of kilometers per hour, we deal with radians per second.To find the gymnast's angular velocity, use the rotational kinetic energy and her moment of inertia:The equation \( RE = \frac{1}{2} I \omega^2 \) rearranges to solve for angular velocity:\[\omega = \sqrt{\frac{2 \times RE}{I}}\]Plugging in our values:- \( RE = 109.8 \) J- \( I = 30 \text{ kg} \cdot \text{m}^2 \)\[\omega \approx \sqrt{\frac{2 \times 109.8}{30}} \approx 2.71 \text{ rad/s}\]This figure represents her speed of rotation and highlights how energy changes within a swinging motion to produce physical dynamics encountered in everyday rotations.