Question

A phonograph record accelerates from rest to 43.0 rpm in 4.63 s.

(a) What is its angular acceleration in rad/s2?
(b) How many revolutions does it go through in the process?

Answer 1

The angular acceleration of the phonograph record is approximately 0.97 rad/s², and during its acceleration, it goes through roughly 1.65 revolutions.

Step-by-step explanation:

To calculate the angular acceleration and the number of revolutions for a phonograph record that accelerates from rest to 43.0 rpm in 4.63 seconds, we first need to convert the final angular velocity to radians per second. The conversion factor is π radians per revolution and 60 seconds per minute, since 1 rpm = π/30 rad/s.

The angular velocity (ω) in rad/s is:

ω = 43.0 rpm × (π rad/rev) / (30 s/min) = 43 × π / 30 = 4.5 rad/s (rounded to 2 decimal places)

(a) Angular acceleration (α) is the change in angular velocity (ω) divided by the time (t):

α = ω / t = 4.5 rad/s / 4.63 s = 0.97 rad/s² (rounded to 2 decimal places)

(b) To find the number of revolutions, we use the formula for angular displacement (θ) under constant acceleration, θ = 0.5 × α × t²:

θ = 0.5 × 0.97 rad/s² × (4.63 s)² = 10.35 radians

Converting radians to revolutions, we have:

Number of revolutions = θ / (2π) = 10.35 rad / (2π) = 1.65 revolutions (rounded to 2 decimal places)

Answer 2

The angular acceleration of the phonograph record is 0.974 rad/s². It goes through approximately 0.718 revolutions.

Step-by-step explanation:

(a) What is its angular acceleration in rad/s²?

To find the angular acceleration, we can use the formula:

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

Given that the initial angular velocity is 0 and the final angular velocity is 43.0 rpm (which is equivalent to 4.506 rad/s), and the time is 4.63 s, we can substitute these values into the formula:

Angular acceleration = (4.506 rad/s - 0 rad/s) / 4.63 s = 0.974 rad/s²

Therefore, the angular acceleration is 0.974 rad/s².

(b) How many revolutions does it go through in the process?

To find the number of revolutions, we can use the formula:

Number of revolutions = final angular velocity / (2π)

Substituting the values, we get:

Number of revolutions = 4.506 rad/s / (2π) = 0.718 revolutions

Therefore, it goes through approximately 0.718 revolutions.