A yo-yo has a rotational inertia of 860 g·cm2 and a mass of 150 g. Its axle radius is 3.4 mm, and its string is 90 cm long. The yo-yo rolls from rest down to the end of the string.

(a) What is the magnitude of its linear acceleration?
1 cm/s2

(b) How long does it take to reach the end of the string?
2 s

(c) As it reaches the end of the string, what is its linear speed?
3 cm/s

(d) As it reaches the end of the string, what is its translational kinetic energy?
4 J

(e) As it reaches the end of the string, what is its rotational kinetic energy?
5 J

(f) As it reaches the end of the string, what is its angular speed?
6 rad/s

To find the answers to these questions, we need to use the concepts of rotational inertia, linear acceleration, linear speed, and kinetic energy. Here's how we can calculate each of these values:

(a) To find the magnitude of the linear acceleration, we can use the formula:

Linear acceleration = (rotational inertia * angular acceleration) / (mass * radius)

Given that the rotational inertia is 860 g·cm^2 (or 0.086 kg·m^2), the mass is 150 g (or 0.15 kg), and the radius is 3.4 mm (or 0.034 m), we can plug in these values into the above formula:

Linear acceleration = (0.086 * angular acceleration) / (0.15 * 0.034)

Since the yo-yo is rolling from rest, its initial angular speed is zero, and it undergoes a constant angular acceleration. We can use the equation:

Angular acceleration = (final angular speed - initial angular speed) / time

Since the yo-yo rolls from rest down to the end of the string, its final angular speed will be non-zero, and its initial angular speed is zero. We are not given the time it takes to reach the end of the string. Therefore, we cannot calculate the exact magnitude of the linear acceleration and time it takes for the yo-yo to reach the end of the string. Without this information, the correct answer cannot be determined.

(b) Without the time, we cannot calculate the time it takes for the yo-yo to reach the end of the string. The correct answer cannot be determined.

(c) Without the linear acceleration and time, we cannot calculate the linear speed of the yo-yo as it reaches the end of the string. The correct answer cannot be determined.

(d) The translational kinetic energy is given by the formula:

Translational kinetic energy = (1/2) * mass * (linear speed)^2

Since we don't have the linear speed of the yo-yo, we can't calculate its translational kinetic energy. The correct answer cannot be determined.

(e) The rotational kinetic energy is given by the formula:

Rotational kinetic energy = (1/2) * rotational inertia * (angular speed)^2

Again, we don't have the angular speed of the yo-yo, so we cannot calculate its rotational kinetic energy. The correct answer cannot be determined.

(f) Without the angular acceleration and time, we cannot calculate the angular speed of the yo-yo as it reaches the end of the string. The correct answer cannot be determined.

In conclusion, we do not have enough information to determine the values for (a), (b), (c), (d), (e), and (f).