An electric motor rotating a workshop grinding wheel at a rate of 9.80 101 rev/min is switched off. Assume the wheel has a constant negative angular acceleration of magnitude 2.00 rad/s2.
(a) How long does it take for the grinding wheel to stop?
s
(b) Through how many radians has the wheel turned during the interval found in (a)?
rad
n₀= 9.80 101 rev/min =….rev/s
ε=2 rad/s²
2•π•n=2•π•n₀ - ε•t,
since n=0, 2•π•n₀ - ε•t=0,
t=2•π•n₀/ ε=…
φ=2•π•n₀•t- ε•t²/2=....
To solve this problem, we need to use the equations of rotational motion. The relevant equations are:
1. ω = ω0 + αt
2. θ = θ0 + ω0t + 0.5αt^2
3. ω^2 = ω0^2 + 2α(θ - θ0)
Given data:
ω0 = 9.80 * 10^1 rev/min
α = -2.00 rad/s^2
(a) To find the time it takes for the grinding wheel to stop, we need to find the time when ω = 0.
Using equation 1:
0 = 9.80 * 10^1 + (-2.00)t
Solving for t, we get:
t = -(9.80 * 10^1) / (-2.00)
t ≈ 49.0 s
Therefore, it takes approximately 49.0 seconds for the grinding wheel to stop.
(b) To find the angle turned by the wheel during this time interval, we need to use equation 2.
Using equation 2:
θ = θ0 + ω0t + 0.5αt^2
Since ω0 is given in rev/min, we need to convert it to rad/s:
ω0 = (9.80 * 10^1 rev/min) * (2π rad/1 rev) * (1 min/60 s)
ω0 ≈ 10.28 rad/s
Substituting the given values:
θ = 0 + (10.28 rad/s)(49.0 s) + 0.5*(-2.00 rad/s^2)(49.0 s)^2
Simplifying this equation will give us the angle turned by the wheel.