a pulley of 5cm radius on a motor is turning at 30 rev/s and slows down uniformly to 20 rev/s in 2s. calculate the angular acceleration of the motor number of revolution it makes in this time and lenght of belt it winds in this time
ω= ω₀- εt
2πn=2πn₀ - εt
ε=2π(n₀-n)/t =2π(30-20)/2=31.4 rad/s²
N= εt²/4π = 31.4•2²/4π =10 rev
L=2πR•N =2π•0.05•10 =3.14 m
Well, it seems like we have a pulley that's having a bit of a slowdown party! Let's crunch some numbers and find the answers you're looking for.
First, let's find the angular acceleration (α) of the motor. To do that, we can use the formula:
α = (ω₂ - ω₁) / t
Where ω₁ is the initial angular velocity (30 rev/s), ω₂ is the final angular velocity (20 rev/s), and t is the time it takes to slow down (2s).
Plugging in the values, we get:
α = (20 rev/s - 30 rev/s) / 2s
Calculating that, we have:
α = -5 rad/s²
Negative angular acceleration, huh? That's the motor's way of saying, "I'm slowing down, but not having the time of my life."
Now, let's find the number of revolutions the motor makes during this time. We can use the formula:
θ = ω₁ * t + 0.5 * α * t²
Where θ is the number of revolutions, ω₁ is the initial angular velocity (30 rev/s), α is the angular acceleration (-5 rad/s²), and t is the time (2s).
Plugging in the values, we have:
θ = 30 rev/s * 2s + 0.5 * (-5 rad/s²) * (2s)²
Calculating that, we get:
θ ≈ 60 rev
The motor managed to make 60 revolutions during this time. It's like it's saying, "Look at me go round and round!"
Finally, let's calculate the length of the belt wound during this time. We can use the formula:
l = r * θ
Where l is the length of the belt, r is the radius of the pulley (5cm or 0.05m), and θ is the number of revolutions (60 rev).
Plugging in the values, we have:
l = 0.05m * 60 rev
Calculating that, we get:
l = 3m
So, during this time, the motor wound about 3 meters of belt. That's like a mini fashion show for belts, wouldn't you say?
To calculate the angular acceleration of the motor, we can use the formula:
angular acceleration (α) = (final angular velocity - initial angular velocity) / time
Given:
Initial angular velocity (ω₁) = 30 rev/s
Final angular velocity (ω₂) = 20 rev/s
Time (t) = 2s
Plugging in the values into the formula, we get:
α = (20 rev/s - 30 rev/s) / 2s
= -10 rev/s / 2s
= -5 rev/s²
Therefore, the angular acceleration of the motor is -5 rev/s².
To find the number of revolutions the motor makes in this time, we can use the formula:
number of revolutions (N) = (final angular velocity - initial angular velocity) * time
Plugging in the given values, we have:
N = (20 rev/s - 30 rev/s) * 2s
= -10 rev/s * 2s
= -20 rev
Since revolutions cannot be negative, we take the absolute value:
N = |-20 rev|
= 20 rev
Therefore, the motor makes 20 revolutions in this time.
Now, to find the length of the belt it winds in this time, we can use the formula:
length of belt (L) = 2π * radius * number of revolutions
Given:
Radius of the pulley (r) = 5 cm = 0.05 m
Number of revolutions (N) = 20 rev
Plugging in the values into the formula, we get:
L = 2π * 0.05 m * 20 rev
= 2π * 0.05 m * 20
= π m
Therefore, the length of the belt it winds in this time is π meters.
To calculate the angular acceleration of the motor, we can use the formula:
Angular acceleration (α) = (final angular velocity - initial angular velocity) / time
The initial angular velocity is given as 30 rev/s and the final angular velocity is given as 20 rev/s. The time is given as 2s.
Substituting these values into the formula:
α = (20 rev/s - 30 rev/s) / 2s
= (-10 rev/s) / 2s
= -5 rev/s²
The negative sign indicates that the motor is slowing down.
Next, let's calculate the number of revolutions the motor makes in this time:
Number of revolutions = (final angular velocity + initial angular velocity) / 2 * time
Substituting the values:
Number of revolutions = (20 rev/s + 30 rev/s) / 2 * 2s
= 50 rev/s / 4s
= 12.5 rev
Finally, to calculate the length of the belt wound during this time, we need to use the formula:
Length of belt = circumference of pulley * number of revolutions
The circumference of the pulley can be calculated using the formula:
Circumference = 2π * radius
Given that the radius of the pulley is 5 cm, which is equivalent to 0.05 m, and the number of revolutions is 12.5 rev:
Circumference = 2π * 0.05 m
= 0.314 m
Length of belt = 0.314 m * 12.5 rev
= 3.925 m
Therefore, the angular acceleration of the motor is -5 rev/s², it makes 12.5 revolutions in 2 seconds, and it winds a length of 3.925 meters of belt in this time.