A disk at rest experiences a constant angular acceleration for t = 85 s, at the end of which it is spinning with a frequency of f = 45 rpm. Please answer the following questions.

A)Write an expression for the magnitude of the angular velocity of the disk after the acceleration in terms of the frequency.
B)Calculate the magnitude of the angular velocity of the disk in rad/s at time t = 85 s.

B)

45 revs/ min * 2 pi radians/rev * 1 min/60 s) = omega final
= 4.71 rad/s

A)
omega = alpha t
4.71 = alpha (85)
alpha = ang acceleration = .0554 radians/s^2
so
omega = .0554 t

A) The magnitude of the angular velocity of the disk after the acceleration can be expressed as ω = 2πf, where ω represents the angular velocity in rad/s and f represents the frequency in rpm.

B) To calculate the magnitude of the angular velocity of the disk at time t = 85 s, we can use the formula ω = ω0 + αt, where ω0 is the initial angular velocity, α is the angular acceleration, and t is the time.

Since the disk is at rest initially (ω0 = 0), we can rewrite the formula as ω = αt.

Given that the duration of constant angular acceleration is t = 85 s and the final frequency is f = 45 rpm, we need to find the angular acceleration first.

To convert the frequency from rpm to rad/s, we use the conversion factor:
1 rpm = 2π/60 rad/s

So, the final angular velocity in rad/s is:
ω = 2πf = 2π(45/60) = π/2 rad/s

Now, we can plug in the values into the formula ω = αt and solve for α:
π/2 = α(85)
α = (π/2) / 85

Finally, we substitute the value of α into the formula ω = αt to find the magnitude of the angular velocity at t = 85 s:
ω = (π/2) / 85 * 85 = π/2 rad/s

Therefore, the magnitude of the angular velocity of the disk at time t = 85 s is π/2 rad/s.

A) The magnitude of the angular velocity (ω) of the disk after the acceleration can be written in terms of frequency (f) using the relationship:

ω = 2πf

where ω is the angular velocity in rad/s and f is the frequency in rpm.

B) To calculate the magnitude of the angular velocity of the disk in rad/s at time t = 85 s, we need to convert the frequency from rpm to rad/s.

Given:
f = 45 rpm

Using the relationship above, we have:

ω = 2πf

ω = 2π(45 rpm) [converting from rpm to rad/s]

To convert rpm to rad/s, we need to multiply by a conversion factor of (2π/60) since there are 2π radians in a complete revolution and 60 seconds in a minute.

Therefore, the magnitude of the angular velocity of the disk at t = 85 s is:

ω = (2π)(45 rpm)(2π/60) = 9π rad/s

A) To find the magnitude of the angular velocity of the disk after the acceleration in terms of the frequency, we can use the formula:

Angular velocity (ω) = 2π × frequency (f)

Since the frequency is given as 45 rpm (revolutions per minute), we need to convert it to radians per second (rad/s). There are 2π radians in one revolution and 60 seconds in one minute, so the conversion factor is:

1 rpm = (2π rad) / (60 s)

Now we can substitute the given frequency into the formula to find the magnitude of the angular velocity:

Angular velocity = 2π × 45 rpm × (2π rad) / (60 s)
= 4π² rad/s

Therefore, the expression for the magnitude of the angular velocity of the disk after the acceleration in terms of the frequency is 4π² rad/s.

B) To calculate the magnitude of the angular velocity of the disk in rad/s at time t = 85 s, we can use the formula for angular acceleration:

Angular acceleration (α) = change in angular velocity (Δω) / time (Δt)

Since the disk experiences a constant angular acceleration, the change in angular velocity is the final angular velocity minus the initial angular velocity. However, the initial angular velocity is not given. Instead, we are given the time (t) when the acceleration ends and the final angular velocity (ω). We need to find the initial angular velocity first using the formula:

Final angular velocity (ωf) = Initial angular velocity (ωi) + angular acceleration (α) × time (t)

In this case, the angular acceleration is constant, so we can rearrange the formula to solve for the initial angular velocity:

Initial angular velocity (ωi) = Final angular velocity (ωf) - angular acceleration (α) × time (t)

Substituting the known values into the formula, we get:

Initial angular velocity = (45 rpm × 2π rad / (60 s)) - 4π² rad/s² × 85 s
= 3π rad/s - 340π rad/s
= -337π rad/s

Now that we have the initial angular velocity, we can find the magnitude of the angular velocity at time t = 85 s:

Angular velocity = |Initial angular velocity| + angular acceleration × time
= |(-337π rad/s)| + 4π² rad/s² × 85 s
= 337π rad/s + 340π rad/s
= 677π rad/s

Therefore, the magnitude of the angular velocity of the disk at time t = 85 s is 677π rad/s.