A wheel has a rotational velocity of 18·rad/s counterclockwise. You place you hand against the wheel and the friction between your hand and the rim stops it.

(a) If it takes 4·s to stop the wheel, what is its rotational acceleration?
rad/s2

(b) How far does the wheel rotate while slowing to a stop?
rad

w=ar*t 18=4ar ar=4.5rad/s^2 (b)18^2=2ar*theta theta=36rad

To solve this problem, we need to use the equations of rotational motion. There are three key equations that we can use:

1. Angular velocity (ω) = Initial angular velocity (ω₀) + Angular acceleration (α) * Time (t)
2. Angular acceleration (α) = Change in angular velocity (Δω) / Time (t)
3. Angle (θ) = Initial angular velocity (ω₀) * Time (t) + 1/2 * Angular acceleration (α) * Time (t)²

Let's solve each part of the problem:

(a) To find the rotational acceleration, we can use equation (2):

α = Δω / t

Given that the initial angular velocity (ω₀) is 18 rad/s and it takes 4 s to stop the wheel, we need to find the change in angular velocity (Δω). Since the wheel stops, its final angular velocity is 0 rad/s. Therefore,

Δω = 0 - 18 = -18 rad/s

Substituting this value in equation (2), we have:

α = -18 rad/s / 4 s = -4.5 rad/s²

So, the rotational acceleration of the wheel is -4.5 rad/s².

(b) To find the angle the wheel rotates while slowing to a stop, we can use equation (3):

θ = ω₀ * t + 1/2 * α * t²

Substituting the values, we have:

θ = 18 rad/s * 4 s + 1/2 * (-4.5 rad/s²) * (4 s)²
= 72 rad + 1/2 * (-4.5 rad/s²) * 16 s²
= 72 rad + (-36 rad)
= 36 rad

So, the wheel rotates 36 radians while slowing to a stop.

Therefore, the answers are:
(a) The rotational acceleration is -4.5 rad/s².
(b) The wheel rotates 36 radians while slowing to a stop.