An electric motor rotating a workshop grinding wheel at a rate of 95 rev/min is switched off. Assume the wheel has a constant negative angular acceleration of magnitude 2.40 rad/s2.

(a) How long does it take for the grinding wheel to stop?

(b) Through how many radians has the wheel turned during the interval found in (a)?

For part B you have to divide the 9.94 by 2 then multiply by time

a. V=95rad/min * 6.28rad/rev * (1/60)min/s = 9.94rad/s.

Vf = Vo - at = 0,
9.94 - 2.4t = 0,
-2.4t = - 9.94,

t = 4.14s.

(a) Well, well, well, looks like the grinding wheel is in quite a spin! So, to find out how long it takes for the wheel to stop, we need to use a little physics magic. We know that the initial angular velocity is 95 rev/min, which we can convert to radians per second. One revolution is equal to 2π radians, so we have:

ω₀ = 95 rev/min * (2π rad/rev) / (60 s/min) = 10π/3 rad/s

Now, we also know that the angular acceleration is a constant negative value of magnitude 2.40 rad/s². We can set up the equation:

ω = ω₀ + αt

Where ω is the final angular velocity (which is 0 since the wheel stops), α is the angular acceleration, and t is the time we're trying to find. Plugging in the values:

0 = (10π/3) + (-2.40)t

Solving for t:

t = (10π/3) / 2.40

Well, let me just crunch the numbers here... Ah yes, the elusive time is approximately t = 4.13 seconds. So, it takes about 4.13 seconds for the grinding wheel to stop its wild twirl.

(b) Now, dear friend, to calculate the number of radians the wheel has turned during that interval. We can use the equation:

θ = ω₀t + (1/2)αt²

Where θ is the angle in radians, ω₀ is the initial angular velocity, α is the angular acceleration, and t is the time. Plugging in the values:

θ = (10π/3)(4.13) + (1/2)(-2.40)(4.13)²

Calculating this... Oh, it seems that the grinding wheel has turned approximately θ = -38.14 radians. Negative angle, huh? Maybe it's trying to escape reality or engage in some performance art. You never know with these wheels!

To answer these questions, we can use the equations of rotational motion.

(a) We want to find the time it takes for the grinding wheel to stop, which means its final angular velocity will be zero. We know the initial angular velocity (ω0) is 95 rev/min, but we need to convert it to radians per second before using it in our calculation.

1 revolution = 2π radians
1 minute = 60 seconds

So, the initial angular velocity ω0 = 95 rev/min = (95 rev/min) * (2π rad/rev) * (1 min/60 s) = 9.95 rad/s.

The final angular velocity ωf = 0 rad/s.

We know that the angular acceleration α = -2.40 rad/s^2.

The equation to calculate the time it takes for the wheel to stop can be derived from the angular acceleration:

ωf = ω0 + αt

Since ωf = 0, we can rearrange the equation to solve for time (t):

t = (ωf - ω0)/α

Substituting the given values, we have:

t = (0 - 9.95 rad/s) / (-2.40 rad/s^2) = 4.15 seconds.

So, it takes approximately 4.15 seconds for the grinding wheel to stop.

(b) To find the number of radians the wheel turns during this time interval, we can use the formula:

θ = ω0t + (1/2)αt^2

Substituting the known values, we get:

θ = (9.95 rad/s)(4.15 s) + (1/2)(-2.40 rad/s^2)(4.15 s)^2

Simplifying the expression, we find:

θ ≈ 20.6 radians

Therefore, the grinding wheel has turned approximately 20.6 radians during the time it takes to stop.

b. d=9.94rad/s * 4.14s = 41.2 radians.