A wheel has a constant angular acceleration of 2.5 rad/s2. Starting from rest, it turns through 200 rad.

(a)
What is its final angular velocity (in rad/s)? (Enter the magnitude.)
(b)
How much time elapses (in s) while it turns through the 200 radians?

To find the final angular velocity and the time elapsed, we can use the equations of rotational motion.

(a) To determine the final angular velocity, we can use the equation:

ωf = ωi + αt

where:
ωf is the final angular velocity,
ωi is the initial angular velocity (which is 0 since starting from rest),
α is the angular acceleration,
and t is the time.

Substituting the given values, we have:

ωf = 0 + (2.5 rad/s^2) × t

Since the initial angular velocity is 0, the equation simplifies to:

ωf = 2.5t rad/s

To solve for ωf, we need to find the value of t first.

(b) To determine the time elapsed, we can use the equation:

θ = ωit + 0.5αt^2

where:
θ is the angular displacement (200 radians in this case),
ωi is the initial angular velocity (0),
α is the angular acceleration (2.5 rad/s^2),
and t is the time.

Substituting the given values, we have:

200 = 0 × t + 0.5(2.5)(t^2)

Simplifying further:

200 = 1.25t^2

Dividing both sides by 1.25:

t^2 = 160

Taking the square root of both sides:

t ≈ ±12.65 (taking the positive value since time cannot be negative)

Now we can substitute the value of t into the equation for angular velocity to find ωf:

ωf = 2.5 × 12.65 ≈ 31.625 rad/s

Therefore, the answers are:
(a) The final angular velocity is approximately 31.625 rad/s.
(b) The time elapsed while turning through 200 radians is approximately 12.65 seconds.

To find the final angular velocity and the time it takes for the wheel to turn through 200 radians, we can use the equations of rotational motion.

(a) Final Angular Velocity:
The equation that relates angular acceleration (α), initial angular velocity (ω₀), final angular velocity (ω), and time (t) is:
ω = ω₀ + αt

Given that the initial angular velocity is zero (starting from rest) and the angular acceleration is 2.5 rad/s², we can substitute these values into the equation:
ω = 0 + 2.5t
ω = 2.5t

To find the final angular velocity, we need to know the time it takes to turn through 200 radians.
Now, let's solve for time (t) first.

(b) Time Elapsed:
The equation that relates angular displacement (θ), initial angular velocity (ω₀), final angular velocity (ω), angular acceleration (α), and time (t) is:
θ = ω₀t + 0.5αt²

Given that the initial angular velocity is zero and the angular displacement is 200 radians, we can substitute these values into the equation:
200 = 0.5(2.5)t²
200 = 1.25t²

Solving for t:
t² = 200 / 1.25
t² = 160
t = √160
t ≈ 12.65 s

Now that we know the time (t), we can substitute this value into the equation for angular velocity to find the final angular velocity:

ω = 2.5t
ω = 2.5(12.65)
ω ≈ 31.63 rad/s

So, the final angular velocity of the wheel is approximately 31.63 rad/s, and it takes approximately 12.65 seconds to turn through 200 radians.

I dont understand your question on where you are having difficulty. Is there something specific you don't understand on these questions?