If a wheel revolving at 1800 r.p.m. slows down uniformly to 1200 r.p.m. in 25s, calculate the angular acceleration of the wheel and the number of revolutions it makes in this time.
To calculate the angular acceleration of the wheel, we can use the formula:
Angular acceleration (α) = (final angular velocity - initial angular velocity) / time
Given:
Initial angular velocity (ω1) = 1800 revolutions per minute (r.p.m.)
Final angular velocity (ω2) = 1200 revolutions per minute (r.p.m.)
Time (t) = 25 seconds
First, let's convert the initial and final angular velocities from r.p.m. to revolutions per second (r.p.s.), as we need the values in a consistent unit:
Initial angular velocity (ω1) = 1800 r.p.m. * 1/60 minutes per second = 30 r.p.s.
Final angular velocity (ω2) = 1200 r.p.m. * 1/60 minutes per second = 20 r.p.s.
Now, let's substitute these values into the formula to calculate the angular acceleration:
α = (ω2 - ω1) / t
= (20 r.p.s. - 30 r.p.s.) / 25 seconds
= -10 r.p.s. / 25 seconds
= -0.4 r.p.s² (rounded to one decimal place)
Therefore, the angular acceleration of the wheel is -0.4 r.p.s².
To calculate the number of revolutions the wheel makes in this time, we can use the formula:
Number of revolutions = (final angular velocity - initial angular velocity) * time
Substituting the values:
Number of revolutions = (ω2 - ω1) * t
= (20 r.p.s. - 30 r.p.s.) * 25 seconds
= -10 r.p.s. * 25 seconds
= -250 revolutions (rounded to the nearest whole number)
Since the value is negative, it indicates that the wheel is slowing down and completing 250 fewer revolutions in this time.
Therefore, the number of revolutions the wheel makes in this time is -250.
To calculate the angular acceleration of the wheel, we need to use the formula:
angular acceleration (α) = (final angular velocity - initial angular velocity) / time
Given that the initial angular velocity (ω1) is 1800 revolutions per minute (r.p.m.), the final angular velocity (ω2) is 1200 r.p.m., and the time (t) is 25 seconds, we can substitute these values into the formula:
α = (1200 - 1800) / 25
Simplifying the equation:
α = -600 / 25
α = -24 radians per second squared
Therefore, the angular acceleration of the wheel is -24 radians per second squared (note the negative sign indicates a decrease in angular velocity).
Next, let's calculate the number of revolutions the wheel makes in this time period.
The number of revolutions (N) can be calculated using the formula:
N = (final angular velocity - initial angular velocity) / (2π)
Given that the initial angular velocity (ω1) is 1800 r.p.m. and the final angular velocity (ω2) is 1200 r.p.m., we can substitute these values into the formula:
N = (1200 - 1800) / (2π)
Simplifying the equation:
N = -600 / (2π)
N ≈ -95.49
Since we cannot have a negative number of revolutions, we take the absolute value:
N ≈ 95.49
Therefore, the wheel makes approximately 95.49 revolutions in this time period.
1800 rev/min * 2 pi rad/rev * 1 min/60 s = Omegai
1200 similarly ---> Omegaf
d omega = (Omegaf-omegai)
dt = 25 s
so
d Omega/dt = alpha = d omega/dt
average angular velocity
= (Omagaf+Omegai)/2
total radians = average vel * 25 sec
total revs = total radians/2pi