91Ó°ÊÓ

Linear acceleration in our context refers to the rate of change of velocity of the yo-yo as it descends along the string. This is crucial in understanding motion because it tells us how quickly the yo-yo speeds up from rest.
To find the linear acceleration, we use the formula: \[a = \frac{mg}{m + \frac{I}{r^2}}\] where:

• \( m \) is the mass of the yo-yo,
• \( g = 9.81 \, \text{m/s}^2 \) is the acceleration due to gravity,
• \( I \) is the rotational inertia,
• and \( r \) is the radius of the axle.

This equation arises because the yo-yo's motion must account for both its linear acceleration along the string and its rotation around its axle.
By substituting the known values, we calculate the linear acceleration as approximately \( 0.532 \, \text{m/s}^2 \). This result helps us predict how quickly the yo-yo gains speed along the string, and it's fundamental for calculating subsequent motions.

Translational kinetic energy is the energy that an object possesses because of its motion through space, specifically relating to the linear, or straight-line, movement. For the yo-yo, this involves its progress down the string.
The formula for translational kinetic energy is: \[KE_{\text{trans}} = \frac{1}{2}mv^2\] where:

• \( m \) is the mass of the object,
• \( v \) is the linear speed at which the object is moving.

Plugging in the values, we find that \( KE_{\text{trans}} \approx 0.0774 \, \text{J} \).
This tells us how much energy the yo-yo has due to its straight movement at the end of the string. Understanding this concept is important because it gives insight into the forces at work and helps gauge which part of the energy budget is "spent" moving the yo-yo in a straight line.

Rotational kinetic energy is the energy an object possesses due to its rotation around an axis. For the yo-yo, this means considering how it spins as it descends.
To calculate rotational kinetic energy, we use: \[KE_{\text{rot}} = \frac{1}{2}I\omega^2\] where:

• \( I \) is the rotational inertia,
• \( \omega \) is the angular speed.

The angular speed \( \omega \) is determined by the ratio of linear speed to the radius of the axle: \[\omega = \frac{v}{r}\] By substituting the given values, we calculate \( KE_{\text{rot}} \approx 6.00 \, \text{J} \).
This indicates how much energy the yo-yo uses up in spinning about its axle. Understanding rotational kinetic energy is key to getting a full picture of how energy is transferred and transformed in systems involving rotational motion, like our yo-yo here.

Angular speed describes how quickly an object rotates or spins, expressed in radians per second. For a yo-yo, angular speed is crucial to determine how fast the yo-yo is spinning as it reaches the end of the string.
The relationship between linear speed and angular speed is given by: \[\omega = \frac{v}{r}\] where:

• \( v \) is the linear speed,
• \( r \) is the radius of the axle.

Substituting these values provides \( \omega \approx 355.3 \, \text{rad/s} \).
Understanding angular speed is important as it shows how the linear motion of the string translates into the spinning motion of the yo-yo. Knowing a yo-yo's angular speed serves not only in practical applications but also supports the study of dynamics and rotational mechanics.