A flywheel with a diameter of 1 m is initially at rest. Its angular acceleration versus time is graphed in the figure.

(a) What is the angular separation between the initial position of a fixed point on the rim of the flywheel and the point's position 8 s after the wheel starts rotating? (Be sure to give angular separation rather than total angular displacement.)

(b) The point starts its motion at θ = 0. Calculate the linear position 8 s after the wheel starts rotating. (Let the initial position of the point be 0.5 .)
m

Calculate the linear velocity vector 8 s after the wheel starts rotating. (Enter the magnitude.)

Calculate the tangential acceleration vector after the wheel starts rotating. (Enter the magnitude.)

To answer these questions, we need to analyze the given graph of angular acceleration versus time for the flywheel.

(a) The angular separation between the initial position and the position after 8 seconds can be found by integrating the angular acceleration graph with respect to time.

First, let's determine the equation for angular acceleration as a function of time. We can see from the graph that the angular acceleration is constant for the first 4 seconds and zero afterwards.

Therefore, the equation for angular acceleration (α) can be written as:
α = 2 rad/s^2 (0 ≤ t ≤ 4s), and α = 0 rad/s^2 (4 ≤ t ≤ 8s)

Now, to find the angular separation, we need to integrate the angular acceleration equation with respect to time.

For the first 4 seconds:
∫ α dt = ∫ (2 rad/s^2) dt = 2t + C
Plugging in the limits of integration, for 0 ≤ t ≤ 4s, we get:
∫ α dt = 2(4) + C = 8 + C

For the next 4 seconds (4 ≤ t ≤ 8s), where α = 0, the angular acceleration is constant and does not contribute to angular separation. Hence, we can ignore this part in our calculation.

So, after integrating the angular acceleration equation, we have:
θ = 8t + C (0 ≤ t ≤ 4s)

Now, to find the value of the constant C, we can use the given initial condition, which states that the angular separation at t = 0 is 0 radians (θ = 0). Plugging in these values, we get:
0 = 8(0) + C
C = 0

Therefore, the equation for angular separation (θ) as a function of time is:
θ = 8t (0 ≤ t ≤ 4s)

To find the angular separation 8 seconds after the wheel starts rotating, we substitute t = 8s into the equation:
θ = 8(8) = 64 radians

Therefore, the angular separation between the initial position and the position after 8 seconds is 64 radians.

(b) To calculate the linear position at 8 seconds, we need to know the radius (r) of the flywheel. The diameter of the flywheel is given as 1m, so the radius can be calculated as:
r = 1m / 2 = 0.5m

Now, we can calculate the linear position (s) using the formula:
s = rθ

Plugging in the values, we get:
s = (0.5m)(64 radians) = 32m

Therefore, the linear position 8 seconds after the wheel starts rotating is 32 meters.

Next, to calculate the linear velocity, we need to know the relationship between linear velocity (v) and angular velocity (ω). Since the angular acceleration is constant, we can use the formula:
α = rα

Plugging in the values, we get:
α = (0.5m)(2 rad/s^2) = 1 m/s^2

Now, we can find the angular velocity (ω) using the formula:
ω = ω₀ + αt

At t = 0, the wheel starts from rest, so ω₀ = 0. Plugging in the values, we get:
ω = (1 m/s^2)(8s) = 8 rad/s

Finally, we can calculate the linear velocity (v):
v = rω
= (0.5m)(8 rad/s)
= 4 m/s

Therefore, the linear velocity 8 seconds after the wheel starts rotating is 4 m/s.

To calculate the tangential acceleration, we can use the formula:
at = rα

Plugging in the values, we get:
at = (0.5m)(2 rad/s^2)
= 1 m/s^2

Therefore, the magnitude of the tangential acceleration after the wheel starts rotating is 1 m/s^2.

To solve this problem, we need to analyze the given graph of angular acceleration versus time.

(a) To find the angular separation between the initial position and the position after 8 seconds, we need to calculate the area under the graph. However, since the graph is not provided, we cannot determine the exact angular acceleration versus time relationship.

(b) Assuming we know the angular acceleration, we can proceed to calculate the linear position, linear velocity, and tangential acceleration after 8 seconds.

To calculate the linear position 8 seconds after the wheel starts rotating, we can use the formula:

θ = ω_0 * t + (1/2) * α * t^2

where:
θ = angular displacement
ω_0 = initial angular velocity (0 since the wheel is initially at rest)
α = angular acceleration
t = time (8 seconds)

Since we do not have the actual values, we cannot calculate the exact linear position.

To calculate the linear velocity after 8 seconds, we can use the formula:

v = ω_0 * r + α * r * t

where:
v = linear velocity
r = radius of the flywheel (0.5 m)
ω_0 = initial angular velocity (0 since the wheel is initially at rest)
α = angular acceleration
t = time (8 seconds)

Since we do not have the exact values of α, we cannot calculate the exact linear velocity.

To calculate the tangential acceleration after 8 seconds, we can use the formula:

at = α * r

where:
at = tangential acceleration
α = angular acceleration
r = radius of the flywheel (0.5 m)

Since we do not have the exact value of α, we cannot calculate the exact tangential acceleration.