An electric motor rotating a workshop grinding wheel at a rate of 114 rev/min is switched off. Assume the wheel has a constant negative angular acceleration of magnitude 2.10 rad/s2.
(a) How long does it take for the grinding wheel to stop?
wf=0=wi+2at
0=114*2PI/60 + 2(-2.10)t
solve for time, t
6.09
To find the time it takes for the grinding wheel to stop, we can use the equation for angular acceleration:
α = Δω / Δt
where:
α is the angular acceleration (given as -2.10 rad/s^2),
Δω is the change in angular velocity (equal to the final angular velocity minus the initial angular velocity), and
Δt is the change in time (the time taken for the wheel to stop).
In this case, the initial angular velocity is 114 rev/min, which can be converted to radians per second by multiplying by 2π/60:
ω_initial = (114 rev/min) x (2π rad/rev) / (60 s/min)
Calculating this:
ω_initial ≈ (114 x 2π) / 60 ≈ 3.77 rad/s
Since the wheel eventually comes to a stop, the final angular velocity is 0 rad/s.
Therefore, we have:
α = (Δω) / (Δt)
-2.10 rad/s^2 = (0 - 3.77 rad/s) / Δt
Simplifying:
-2.10 rad/s^2 = -3.77 rad/s / Δt
To find Δt, we can rearrange the equation:
Δt = -3.77 rad/s / -2.10 rad/s^2
Calculating this:
Δt ≈ 1.79 s
So, it takes approximately 1.79 seconds for the grinding wheel to stop.