91Ó°ÊÓ

The moment of inertia is a measure of an object's resistance to changes in its rotation. It's similar to how mass measures resistance to changes in linear motion. For a rotating object, the distribution of mass matters. The further the mass is from the axis of rotation, the greater the moment of inertia.

The moment of inertia for different shapes is calculated using specific formulas. For a solid cylinder, like a gymnast spinner, the formula is:

• \( I = \frac{1}{2} m r^2 \)

where \( I \) represents the moment of inertia, \( m \) is the mass, and \( r \) is the radius.
This tells us that increasing the mass or radius will increase the moment of inertia. In our example, the mass is 75 kg and the radius is 0.5 m, leading to a moment of inertia of 9.375 kg·m².

A solid cylinder is a three-dimensional object with uniform mass distribution. Its characteristics are crucial in understanding its rotational motion.

Here's why the solid cylinder is important:

• The uniform mass makes calculations for inertia simpler.
• The fixed shape allows for a standard formula for rotational inertia.

A tumbler modeled as a solid cylinder has dimensions that influence how energy is divided between rotational and translational movements. In the given exercise, the cylinder is used to approximate the gymnast's body, where its uniform structure allows us to apply the formula for calculating moment of inertia and further kinetic energies.

Angular velocity describes how fast an object rotates around an axis. It's a vector quantity, typically measured in radians per second (rad/s).

• To convert from revolutions per second to rad/s, multiply by \( 2\pi \) (since there are \( 2\pi \) radians in a full circle).
• In our exercise, 0.5 revolutions per second equates to \( \pi \) rad/s.

Understanding angular velocity is crucial for computing rotational kinetic energy, as it's a part of the formula: \( KE_{rot} = \frac{1}{2} I \omega^2 \).
This tells us that the faster something spins, given a constant moment of inertia, the more rotational kinetic energy it possesses.

Translational kinetic energy is associated with the linear motion of an object, as opposed to rotational kinetic energy which relates to spinning or circular motion.

For translational kinetic energy, the formula used is:

• \( KE_{trans} = \frac{1}{2} m v^2 \)

Where \( m \) is the mass and \( v \) is the linear velocity of the object. In our example:

• The linear velocity \( v \) is found by multiplying the angular velocity by the radius (\( v = \omega \times r \)).
• With \( \omega = \pi \) rad/s and \( r = 0.5 \) m, \( v \) equals 1.57 m/s.

Using these values, the gymnast's translational kinetic energy is calculated to be 92.44 J. This energy represents the motion as the body moves forward, in contrast to the energy of spinning.