this is the question :
A CD has a diameter of 12.0 cm and a mass of 15.8 g. The motor in a CD player causes the disk, initially at rest, to spin. When in operation, the CD spins in such a way that the laser scans the tracks at a constant tangential velocity of 1.20 m/s. The first track of the CD is at a radius of 20.0 mm from the centre of the disk. Calculate the work done by the motor in the CD player in bringing the disk from rest to the proper speed to read the first track. You may ignore the small circular hole at the CD’s centre. [Hint: Recall the work-kinetic energy theorem.]
I got an angular speed of 60 rad/s, and now I'm trying to figure out the moment of inertia (I) so I can use the formula K rotational = 1/2 I w^2, then I should be able to equate work with kinetic energy using the formula K final = K initial + work, where K initial is zero.
but I don't know how to find moment of inertia. Do I just use the total mass of the cd given?
To find the moment of inertia (I) of the CD, you need to take into account the mass distribution of the object. Since the CD is a flat disk, you can use the formula for moment of inertia of a thin, uniform disk.
The moment of inertia of a thin, uniform disk is given by the formula:
I = (1/2) * m * r^2
where I is the moment of inertia, m is the mass of the disk, and r is the radius of the disk.
In this case, you are given the mass of the CD (15.8 g) and the radius of the first track (20.0 mm). However, you need to convert the mass to kilograms and the radius to meters to ensure consistent units in the calculation:
Mass of CD (m) = 15.8 g = 0.0158 kg
Radius of first track (r) = 20.0 mm = 0.02 m
Now, you can plug these values into the formula:
I = (1/2) * m * r^2
I = (1/2) * 0.0158 kg * (0.02 m)^2
Simplifying the calculation:
I = 0.0000158 kg * 0.0004 m^2
I = 0.00000632 kg * m^2
Therefore, the moment of inertia (I) of the CD is 0.00000632 kg * m^2.
To calculate the work done by the motor in bringing the CD from rest to the proper speed, you can use the work-kinetic energy theorem. The formula for rotational kinetic energy is:
K rotational = (1/2) * I * w^2
where K rotational is the rotational kinetic energy, I is the moment of inertia, and w is the angular speed.
You have already calculated the angular speed (w) to be 60 rad/s. Now, you can use the formula:
K rotational = (1/2) * 0.00000632 kg * m^2 * (60 rad/s)^2
Simplifying the calculation:
K rotational = 0.00000632 kg * m^2 * 3600 rad^2/s^2
Therefore, the rotational kinetic energy (K rotational) of the CD is 0.135936 kg * m^2 * rad^2/s^2.
Since the CD starts from rest (K initial = 0) and the work done is equal to the change in kinetic energy, you can set up an equation:
K final = K initial + work
Substituting in the values:
0.135936 kg * m^2 * rad^2/s^2 = 0 + work
Therefore, the work done by the motor in the CD player to bring the CD from rest to the proper speed is 0.135936 kg * m^2 * rad^2/s^2.
To find the moment of inertia (I) of the CD, you need to consider the distribution of mass around the axis of rotation. The moment of inertia depends on both the mass and the shape of the object.
For a thin disk like a CD, you can use the moment of inertia formula for a solid disk, which is given by:
I = (1/2) * m * r^2
where:
- I is the moment of inertia
- m is the mass of the disk
- r is the radius of the disk
In this case, you have the mass of the CD (15.8 g) and the radius of the first track (20.0 mm = 0.020 m). So, you can calculate the moment of inertia using the formula:
I = (1/2) * m * r^2
I = (1/2) * 0.0158 kg * (0.020 m)^2
After calculating the moment of inertia, you can proceed with the rest of your calculations using the formulas you mentioned.