A 69-cm-diameter wheel accelerates uniformly about its center from 150rpm to 320rpm in 4.4s . Determine angular acceleration?
change in rpm = 170
170 rev/min *1 min/60 s * 2 pi radians/rev = 17.8 radians/second change
which is delta omega
alpha = change in omega/ change in time
-17.8 radians/second / 4.4 seconds
= 4.05 radians/second^2
To determine the angular acceleration of the wheel, we can use the formula:
angular acceleration = (final angular velocity - initial angular velocity) / time
Given:
Initial angular velocity (ω₁) = 150 rpm
Final angular velocity (ω₂) = 320 rpm
Time (t) = 4.4 s
First, let's convert the angular velocities to radians per second (rad/s). Since 1 revolution = 2π radians, we can use the conversion factor:
1 rpm = (2π rad) / 60 s
Initial angular velocity (ω₁) = 150 rpm = (150 * 2π) / 60 rad/s
Final angular velocity (ω₂) = 320 rpm = (320 * 2π) / 60 rad/s
Now we can substitute the values into the formula to find the angular acceleration:
angular acceleration = (ω₂ - ω₁) / t
angular acceleration = ((320 * 2π) / 60 - (150 * 2π) / 60) / 4.4
Simplifying the expression further:
angular acceleration = (320π - 150π) / (60 * 4.4)
angular acceleration = 170π / (60 * 4.4)
Finally, we can calculate the angular acceleration:
angular acceleration ≈ 1.899 rad/s²
To determine the angular acceleration of the wheel, we can use the formula:
Angular acceleration (α) = (Change in angular velocity (Δω)) / (Time taken (Δt))
First, let's convert the initial and final rotations per minute (rpm) to radians per second (rad/s):
Initial angular velocity (ω₁) = (Initial rpm) × (2π rad/1 min) × (1 min/60 s)
Final angular velocity (ω₂) = (Final rpm) × (2π rad/1 min) × (1 min/60 s)
Initial angular velocity (ω₁) = 150 rpm × (2π rad/1 min) × (1 min/60 s)
Final angular velocity (ω₂) = 320 rpm × (2π rad/1 min) × (1 min/60 s)
Next, we can calculate the change in angular velocity (Δω):
Change in angular velocity (Δω) = Final angular velocity (ω₂) - Initial angular velocity (ω₁)
Substituting the calculated values:
Change in angular velocity (Δω) = (320 rpm × (2π rad/1 min) × (1 min/60 s)) - (150 rpm × (2π rad/1 min) × (1 min/60 s))
Finally, we can use the formula to find the angular acceleration (α):
Angular acceleration (α) = Change in angular velocity (Δω) / Time taken (Δt)
Using the given time of 4.4 seconds and substituting the previously calculated value for Δω:
Angular acceleration (α) = (Δω) / 4.4s
By calculating these values, you can determine the angular acceleration of the wheel.