A pulley of radius 0.5m, on a motor is turning at 30Rev/s and slows down uniformly in 2.0s.calculate ;(a)the angular acceleration of the motor. (b) the number of revolution it makes in this time. (C) the length of the belt it winds in this time
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(a) Well, I don't mean to go in circles, but let's calculate the angular acceleration! We can use the formula:
Angular acceleration = (final angular velocity - initial angular velocity) / time
The initial angular velocity is 30 Rev/s, the final angular velocity is 0 Rev/s after slowing down uniformly, and the time is 2.0 seconds. Plugging these values into the formula, we have:
Angular acceleration = (0 Rev/s - 30 Rev/s) / 2.0 s
Simplifying this, we get:
Angular acceleration = -15 Rev/s^2
So, the angular acceleration of the motor is -15 Rev/s^2. Negative because it's slowing down. Don't worry, though, it's not getting dizzy!
(b) Now, the number of revolutions it makes in this time is simply the initial angular velocity multiplied by the time. So, we have:
Number of revolutions = initial angular velocity * time
Plugging in the values, we get:
Number of revolutions = 30 Rev/s * 2.0 s
Simplifying this, we find:
Number of revolutions = 60 Rev
So, the pulley makes 60 revolutions in this time. That's quite a spin!
(c) Lastly, to find the length of the belt it winds in this time, we need to consider the circumference of the pulley. The formula is:
Length of belt = number of revolutions * circumference of pulley
The number of revolutions is 60 Rev, and the radius of the pulley is 0.5 m. Using the formula for circumference, which is 2πr, we have:
Length of belt = 60 Rev * 2π(0.5 m)
Simplifying this, we get:
Length of belt = 60π m
So, the length of the belt it winds in this time is 60π meters. That's quite a long belt!
To solve this problem, we can use the following formulas:
(a) Angular acceleration (α) = Change in angular velocity (Δω) / Time (Δt)
(b) Number of revolutions (N) = Angular velocity (ω) * Time (t) / 2π
(c) Length of the belt (L) = Radius (r) * Number of revolutions (N) * 2π
Given:
Radius of the pulley (r) = 0.5m
Initial angular velocity (ω0) = 30 rev/s
Time (Δt) = 2.0s
(a) Angular acceleration (α) = ?
To find the angular acceleration, we need to calculate the change in angular velocity first.
Δω = Final angular velocity (ω) - Initial angular velocity (ω0)
The final angular velocity can be calculated using the formula:
ω = ω0 + α * Δt
Substituting the given values:
Final angular velocity (ω) = 0 (as the motor slows down to a stop)
Initial angular velocity (ω0) = 30 rev/s
Time (Δt) = 2.0s
So, Δω = 0 - 30 = -30 rev/s
Now, substituting the values into the formula for angular acceleration:
Angular acceleration (α) = Δω / Δt
= -30 rev/s / 2.0s
= -15 rev/s^2
Therefore, the angular acceleration of the motor is -15 rev/s^2.
(b) Number of revolutions (N) = ?
Using the formula for the number of revolutions:
N = ω * t / 2π
Substituting the given values:
ω = 0 (as the motor slows down to a stop)
Time (t) = 2.0s
So, N = 0 * 2.0s / 2π
= 0 revolutions
Therefore, the number of revolutions made by the pulley in 2.0 seconds is 0.
(c) Length of the belt (L) = ?
Using the formula for the length of the belt:
L = r * N * 2π
Substituting the given values:
Radius (r) = 0.5m
Number of revolutions (N) = 0
So, L = 0.5m * 0 * 2π
= 0 meters
Therefore, the length of the belt wound in 2.0 seconds is 0 meters.
To calculate the angular acceleration of the motor, we need to use the formula:
Angular acceleration (α) = (Change in angular velocity) / (Change in time)
Given:
Radius (r) = 0.5m
Initial angular velocity (ω1) = 30 rev/s
Final angular velocity (ω2) = 0 rev/s
Time (t) = 2.0s
Step 1: Convert rev/s to rad/s.
Since radian measure is used for angular acceleration, we need to convert revolutions to radians. There are 2π radians in one revolution, so:
ω1 (in rad/s) = 30 rev/s * 2π rad/rev = 60π rad/s
ω2 (in rad/s) = 0 rev/s * 2π rad/rev = 0 rad/s
Step 2: Calculate the change in angular velocity.
Change in angular velocity (Δω) = ω2 - ω1 = 0 rad/s - 60π rad/s = -60π rad/s
Step 3: Calculate the angular acceleration.
Angular acceleration (α) = Δω / t = (-60π rad/s) / 2.0s ≈ -30π rad/s²
Therefore, the angular acceleration of the motor is approximately -30π rad/s².
To calculate the number of revolutions the pulley makes in this time, we can use the formula:
Number of revolutions = (Final angular velocity - Initial angular velocity) * Time / (2π)
Given:
Initial angular velocity (ω1) = 30 rev/s
Final angular velocity (ω2) = 0 rev/s
Time (t) = 2.0s
Number of revolutions = (0 - 30) * 2.0s / (2π) ≈ -30/π ≈ -9.55 revolutions
Since the number of revolutions cannot be negative, the pulley makes approximately 9.55 revolutions in this time.
To calculate the length of the belt the pulley winds in this time, we need to find the distance traveled along the circumference of the pulley.
Given:
Radius (r) = 0.5m
Number of revolutions = 9.55 revolutions
Step 1: Convert revolutions to radians.
Number of radians = Number of revolutions * 2π = 9.55 * 2π ≈ 60.01 radians
Step 2: Calculate the length of the belt.
Length of the belt = Number of radians * radius = 60.01 radians * 0.5m = 30.005m
Therefore, the length of the belt the pulley winds in this time is approximately 30.005m.