Answer:
τ =9.41 * 10⁻⁴ N*m
Explanation:
Kinematics of the CD
The CD rotates with constant angular acceleration and its angular acceleration is calculated as follows:
[tex]\alpha = \frac{\omega_{f}- \omega_{i}}{t}[/tex] Formula (1)
Where:
α : angular acceleration. (rad/s²)
ωf: final angular velocity (rad/s)
ωi : initial angular velocity (rad/s)
t = time interval (s)
Data
ωf= 20 rad/s
ωi =0
t = 0.65 s.
Calculating of the angular acceleration of the CD
We replace data in the formula (1)
[tex]\alpha = \frac{20 -0}{0.65}[/tex]
α = 30.77 rad/s²
Newton's second law in rotation:
F = ma has the equivalent for rotation:
τ = I * α Formula (2)
where:
τ : It is the net torque applied to the body. (N*m)
I : it is the moment of inertia of the body with respect to the axis of rotation (kg*m²)
α : It is angular acceleration. (rad/s²)
Calculating of the moment of inertia of the CD
The moment of inertia of a disk with respect to an axis perpendicular to the plane and passing through its center is calculated by the following formula:
I = (1/2) M*R² Formula (3)
Data
M= 17 g = 17/1000 kg = 0.017 kg : CD mass
R= 6.0 cm= 6/100 m = 0.06 m : CD radius
We replace data in the formula (3) :
I = (1/2) ( 0.017 kg)*(0.06 m)² = 3.06 * 10⁻⁵ kg*m²
Calculating of the net torque acting on the CD
Data
α = 30.77 rad/s²
I = 3.06 * 10⁻⁵ kg*m²
We replace data in the formula (2) :
τ = I * α
τ =( 3.06 * 10⁻⁵ kg*m²) * (30.77 rad/s²)
τ =9.41 * 10⁻⁴ N*m
