A disk rotates about its central axis starting from rest and accelerates with constant angular acceleration. At one time it is rotating at 7.8 rad/s; 60 radians later, its angular speed is 19 rad/s. Calculate (a) the angular acceleration (rad/s2), (b) the time required to complete the 60 radians, (c) the time required to reach the 7.8 rad/s angular speed, and (d) the number of radians from rest until the time the disk reaches the 7.8 rad/s angular speed.

Let's solve each part step-by-step:

(a) To calculate the angular acceleration, we can use the formula:

ω^2 = ω0^2 + 2αθ

where ω is the final angular speed, ω0 is the initial angular speed, α is the angular acceleration, and θ is the angle covered.

Given that ω = 19 rad/s, ω0 = 7.8 rad/s, and θ = 60 radians, we can rearrange the formula to solve for α:

α = (ω^2 - ω0^2) / (2θ)

Plugging in the values:

α = (19^2 - 7.8^2) / (2 * 60)

Simplifying the calculation:

α = (361 - 60.84) / 120
α = 300.16 / 120
α = 2.50 rad/s^2

Therefore, the angular acceleration is 2.50 rad/s^2.

(b) To find the time required to complete the 60 radians, we can use the formula:

ω = ω0 + αt

where t is the time taken.

Given that ω = 19 rad/s, ω0 = 7.8 rad/s, and α = 2.50 rad/s^2, we can rearrange the formula to solve for t:

t = (ω - ω0) / α

Plugging in the values:

t = (19 - 7.8) / 2.50

Simplifying the calculation:

t = 11.2 / 2.50
t = 4.48 seconds

Therefore, the time required to complete the 60 radians is 4.48 seconds.

(c) To find the time required to reach the 7.8 rad/s angular speed, we can rearrange the same formula:

t = (ω - ω0) / α

Given that ω = 7.8 rad/s, ω0 = 0 rad/s (starting from rest), and α = 2.50 rad/s^2, we can solve for t:

t = (7.8 - 0) / 2.50
t = 7.8 / 2.50
t = 3.12 seconds

Therefore, the time required to reach the 7.8 rad/s angular speed is 3.12 seconds.

(d) To find the number of radians from rest until the time the disk reaches the 7.8 rad/s angular speed, we can use the formula:

θ = ω0t + 0.5αt^2

Given that ω0 = 0 rad/s (starting from rest), α = 2.50 rad/s^2, and t = 3.12 seconds (from part c), we can solve for θ:

θ = (0)(3.12) + 0.5(2.50)(3.12)^2

Simplifying the calculation:

θ = 0 + 0.5(2.50)(9.7344)
θ = 0 + 0.5(24.336)
θ = 0 + 12.168
θ ≈ 12.17 radians

Therefore, the number of radians from rest until the time the disk reaches the 7.8 rad/s angular speed is approximately 12.17 radians.

To solve this problem, we can use the equations of angular motion. Let's denote the initial angular velocity as ω₁ (starting from rest), the final angular velocity as ω₂, the angular acceleration as α, the time required to complete the 60 radians as t, and the number of radians from rest until the time the disk reaches the 7.8 rad/s angular speed as θ.

(a) To find the angular acceleration (α), we can use the equation:

ω₂ = ω₁ + αt

Given that ω₁ = 0 rad/s (starting from rest) and ω₂ = 19 rad/s, we can rearrange the equation to solve for α:

α = (ω₂ - ω₁) / t

Substituting the values, we have:

α = (19 rad/s - 0 rad/s) / t

Note: We do not have the value of t yet, but we will find it in the next part.

(b) To find the time required to complete the 60 radians, we can use the equation:

θ = ω₁t + (1/2)αt²

Given that θ = 60 radians and ω₁ = 0 rad/s (starting from rest), the equation becomes:

60 radians = 0 rad/s * t + (1/2)αt²

Simplifying the equation, we have:

30t² = 60

Dividing by 30, we get:

t² = 2

Taking the square root of both sides, we find:

t = √2

(c) To find the time required to reach the 7.8 rad/s angular speed, we can use the equation:

ω = ω₁ + αt

Given that ω = 7.8 rad/s, ω₁ = 0 rad/s (starting from rest), and α is the same as the value we found in part (a), we have:

7.8 rad/s = 0 rad/s + αt

Simplifying the equation, we find:

αt = 7.8

Substituting the value of α we found in part (a), we can solve for t:

[(19 rad/s - 0 rad/s) / t] * t = 7.8

19 rad/s = 7.8

This equation is not possible, so we need to re-evaluate our calculation in part (a). Let's solve for α again:

α = (ω₂ - ω₁) / t

α = (19 rad/s - 0 rad/s) / (√2)

Simplifying the equation, we get:

α = 19 rad/s / √2

(d) Now, we can find the number of radians from rest until the time the disk reaches the 7.8 rad/s angular speed (θ). We can use the equation:

θ = ω₁t + (1/2)αt²

Given that ω₁ = 0 rad/s (starting from rest), α is the same as the value we found in part (c), and t is the value we found in part (b), we can calculate θ:

θ = 0 rad/s * √2 + (1/2) * (19 rad/s / √2) * (√2)²

θ = 0 rad + (1/2) * 19 rad/s * 2

θ = 19 radians

Therefore, the answers are:
(a) The angular acceleration (α) is 19 rad/s / √2 rad/s².
(b) The time required to complete the 60 radians is √2 seconds.
(c) The time required to reach the 7.8 rad/s angular speed is not possible to determine based on the given information.
(d) The number of radians from rest until the time the disk reaches the 7.8 rad/s angular speed is 19 radians.