A CD has a mass of 17.4 g and a radius of 6.01 cm. When inserted into a player, the CD starts from rest and accelerates to an angular velocity of 20.0 rad/s in 0.901 s. Assuming the CD is a uniform solid disk, determine the net torque acting on it.
To determine the net torque acting on the CD, we can use Newton's second law for rotational motion:
Net Torque = Moment of Inertia * Angular Acceleration
First, let's calculate the moment of inertia of the CD using the formula for a solid disk:
Moment of Inertia (I) = (1/2) * m * r^2
where m is the mass of the CD and r is the radius.
Plugging in the values, we have:
I = (1/2) * (17.4 g) * (6.01 cm)^2
Convert the mass from grams to kilograms:
1 g = 0.001 kg
So, the mass of the CD is:
m = 17.4 g * 0.001 kg/g = 0.0174 kg
Also, convert the radius from centimeters to meters:
1 cm = 0.01 m
Therefore, the radius of the CD is:
r = 6.01 cm * 0.01 m/cm = 0.0601 m
Substituting these values, we have:
I = (1/2) * (0.0174 kg) * (0.0601 m)^2
Now, let's calculate the angular acceleration using the formula:
Angular Acceleration (α) = (Final Angular Velocity - Initial Angular Velocity) / Time
Plugging in the given values:
α = (20.0 rad/s - 0 rad/s) / 0.901 s
Finally, we can find the net torque by substituting the values of moment of inertia and angular acceleration into the equation:
Net Torque = (0.5) * I * α
Calculating the net torque gives us the answer.