A wheel has a speed of 1,200 revolutions per minute and is made to slow down at a rate of 4 rad/s2. The number of revolutions it makes before coming to rest is:a.143b.722c.314d.272
Question
A wheel has a speed of 1,200 revolutions per minute and is made to slow down at a rate of 4 rad/s2. The number of revolutions it makes before coming to rest is:a.143b.722c.314d.272
Solution
First, we need to convert the speed from revolutions per minute to radians per second.
1 revolution = 2π radians 1 minute = 60 seconds
So, 1200 revolutions per minute = 1200 * 2π / 60 = 40π rad/s
The wheel is slowing down at a rate of 4 rad/s^2. We can use the formula for the final velocity in uniformly accelerated linear motion:
v_f = v_i - at
where: v_f is the final velocity (0 rad/s in this case, as the wheel comes to rest), v_i is the initial velocity (40π rad/s), a is the acceleration (-4 rad/s^2, negative because it's slowing down), t is the time.
Rearranging for t gives:
t = (v_f - v_i) / a = (0 - 40π) / -4 = 10π s
The total angle θ through which the wheel turns before coming to rest can be found using the formula for the displacement in uniformly accelerated linear motion:
θ = v_it - 0.5at^2 = 40π10π - 0.54(10π)^2 = 400π^2 - 200π^2 = 200π^2 rad
Converting this to revolutions gives:
200π^2 rad * 1 rev / 2π rad = 100π rev
So, the wheel makes approximately 314 revolutions before coming to rest. Therefore, the correct answer is c. 314.
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