Prior to the music CD, stereo systems had a phonographic turntable on which vinyl disk recordings were played. A particular phonographic turntable starts from rest and achieves a final constant angular speed of 33 1 3 rpm in a time of 4.5 s.

(a) How many rotations did the turntable undergo during that time?

(b) The classic Beatles album Abbey Road is 47 min and 7 s in duration. If the turntable requires 8 s to come to rest once the album is over, calculate the total number of rotations for the complete start-up, playing, and slow-down of the album.

Answer: a) 1.3 rev. b) 1,573 rev.

Explanation:

Let's split the process in 3 phases: start-up, playing, and slow-down.

Start-Up

As the turntable starts from rest, ω₀ = 0.

If we choose θ₀ = 0º, assuming a constant angular acceleration, we can write the following expression for θ (in rads):

θ = 1/2 γ t² (1)

The angular acceleration γ, by definition is the rate of change of the angular velocity, so:

γ = ωf - ω₀ / t. (2)

As ω₀ = 0, we can put the following:

γ = ωf / t

As we have ωf in rpm, it is advisable to convert it to rad/sec, as follows:

33 1/3 rpm (1 min/60 sec). (2π rad/1 rev) = 111/100π rad/sec

Replacing in (2), we have:

γ = 37/150 π rad/sec²

Replacing in (1):

θ₁ = 999/400 π rad = 1.3 rev

Playing

During this phase, the turntable rotates at constant angular speed, so we can apply the definition of angular velocity to get the angle rotated, as follows:

θ₂ = ωf. t = (111/100) π. 2,827 sec = 313797/100 π rad = 1,569 rev.

Slow-down

Assuming a constant deceleration, as ωf = 0, we can find the value of the angular acceleration during this phase, as follows:

γ = - 111/100 π / 8 rad/sec² = - 0.44 rad/sec²

Now, we can apply the kinematic equation for the angle rotated under constant angular acceleration condition, as follows:

θ₃ = ω₀ t - 1/2γ t²

Replacing by the values, we get:

θ₃ = 222/25 π - 111/25 π = 111/25 π rad = 2.2 rev

Adding the the three angles, we have:

θ = θ₁ + θ₂ + θ₃ = 1.3 + 1,569 + 2.2 = 1,573 rev