The blade in a food processor spins at 1700 rpm. After the machine is turned off, the blade comes to a stop in 3.7s. How many revolutions does it make during this time, assuming constant angular deceleration?
The average rpm during deceleration is 850 rpm. Multiply that by the time interval.
To find the number of revolutions the blade makes during the deceleration period, we need to first determine the initial angular velocity of the blade.
Given:
- Initial angular velocity (ω_0) = 1700 rpm
- Deceleration time (t) = 3.7 s
To get ω_0 in radians per second, we convert it from rpm to radians per second using the conversion factor:
1 rotation = 2π radians
Therefore, ω_0 = (1700 rpm) × (2π rad/1 min) × (1 min/60 s) = 178.48 rad/s.
Using the formula for angular deceleration:
ω = ω_0 - αt,
where ω is the final angular velocity (0 rad/s), α is the angular deceleration, and t is the deceleration time, we can solve for α:
0 rad/s = 178.48 rad/s - α × 3.7 s.
Simplifying the equation, we get:
α = 178.48 rad/s ÷ 3.7 s ≈ 48.23 rad/s^2.
We now have the angular deceleration (α) of the blade. To find the number of revolutions (N) made during the deceleration, we can use the equation:
N = (ω_0^2 - ω^2) / (2α),
where ω is the final angular velocity (0 rad/s).
Plugging in the values, we have:
N = (178.48 rad/s)^2 / (2 × 48.23 rad/s^2).
Calculating this expression:
N ≈ 1327 revolutions.
Therefore, during the deceleration period, the blade makes approximately 1327 revolutions.