A centrifuge in a medical laboratory rotates at an angular speed of 3500 rev/ min. When switched off, it rotates through 41.0 revolutions before coming to rest. (a) Find the angular speed in rad/s. (b) Find the displacement in radians. (c) Find the constant angular acceleration of the centrifuge.

n₀=3500 rev/min =3500/60 rev/s =58.33 rev/s

N= 41 rev,
(a) ω=2π•n₀=2π•58.33=366.5 rad/s
(b) φ =2π•N =2π•41 =257.6 rad
(c) φ= ω²/2•ε,
ε= ω²/2•φ=58.33²/2•257.6=6.6 rad/s²

To solve this problem, we will use the following formulas:

(a) Angular speed in rad/s = Angular speed in rev/min * (2 * π / 1 rev)

(b) Displacement in radians = Number of revolutions * (2 * π)

(c) Constant angular acceleration = Final angular speed / Time taken

Now let's apply these formulas to find the answers:

(a) Angular speed in rad/s = 3500 rev/min * (2 * π / 1 rev)
Angular speed in rad/s = 3500 * 2 * π

(b) Displacement in radians = 41.0 revolutions * (2 * π)
Displacement in radians = 41.0 * 2 * π

(c) To find the constant angular acceleration, we need the time taken. Since the problem doesn't give us the time, we need to use one of the kinematic equations for rotational motion to derive it.

The equation we can use is:
Displacement (θ) = Initial angular speed (ω₀) * Time taken + (1/2) * Angular acceleration (α) * Time taken^2

Given that the initial angular speed is the angular speed in radians, which we found in part (a) as 3500 * 2 * π. We also know that the displacement is the displacement in radians, which we found in part (b) as 41.0 * 2 * π. By substituting these values into the equation, we can solve for the constant angular acceleration.

θ = ω₀ * t + (1/2) * α * t^2
41.0 * 2 * π = (3500 * 2 * π) * t + (1/2) * α * t^2

Simplifying the equation:
82.0 * π = 7000 * π * t + α * t^2

Since the centrifuge comes to rest, the final angular speed is 0 rad/s.

Using this information, we can rearrange the equation:

82.0 * π = 7000 * π * t + α * t^2
α * t^2 + 7000 * π * t - 82.0 * π = 0

Solving this quadratic equation will give us the values of α and t.

Once we find the values of α and t, we can substitute them into the formula for constant angular acceleration to find the answer in part (c).

I hope this explanation helps! Let me know if you have any further questions.