2. (4 pts) An electric motor rotating a grinding wheel at a rate of 126rev/min is switched off. Assume constant angular acceleration of magnitude 2.34rad/s^2 (a) How much time does it take for the grinding to stop? AND (b) Through what angle (in degrees) has the wheel turned before it finally comes to rest?

(a) To find the time it takes for the grinding wheel to stop, we can use the kinematic equation:

v = u + at

where:
v = final velocity (0 m/s, since the wheel comes to rest)
u = initial velocity
a = angular acceleration
t = time

First, we need to convert the initial velocity from rev/min to rad/s:

u = 126 rev/min * (2π rad/1 rev) * (1 min/60 s) = 13.2 rad/s

Now, we can plug in the values into the equation:

0 = 13.2 + (-2.34)t

Solving for t, we get:

t = 13.2 / 2.34 ≈ 5.65 s

So, it takes approximately 5.65 seconds for the grinding wheel to stop.

(b) To find the angle through which the wheel has turned before it comes to rest, we can use the kinematic equation:

θ = ut + 0.5at^2

where:
θ = angle in radians

Since the final velocity is 0, we can simplify the equation to:

θ = 0.5at^2

Plugging in the values, we get:

θ = 0.5 * (-2.34 rad/s^2) * (5.65 s)^2 ≈ -41.5 rad

The negative sign indicates that the angle is measured in the clockwise direction. To convert this to degrees, we can use the conversion factor 180°/π rad:

θ ≈ -41.5 rad * (180°/π rad) ≈ -2370°

So, the grinding wheel has turned approximately -2370 degrees before it finally comes to rest