A dentist causes the bit of a high-speed drill to accelerate from an angular speed of 1.44 x 104 rad/s to an angular speed of 5.05 x 104 rad/s. In the process, the bit turns through 1.77 x 104 rad. Assuming a constant angular acceleration, how long would it take the bit to reach its maximum speed of 8.95 x 104 rad/s, starting from rest?
Do I start by subtracting the two angular speeds?
ω=ωₒ+ε•t,
t= (ω-ωₒ)/ε……..(1).
Plug (1) in following
φ =ωₒ•t +ε•t²= (ω²-ω²ₒ)/2•ε .
ε= (ω²-ω²ₒ)/2• φ =(25.5-2.07) •10^8/2•1.77•10^4 =6,32•10^4 rad/s²
Now,
ω1=ωₒ+ε•t1,
t1 = (ω1-ωₒ)/ ε =(8.95-0) •10^4/ 6,32•10^4 =1.42 s.
Yes, to find out the change in angular speed, you would subtract the initial angular speed from the final angular speed. In this case, the change in angular speed would be:
Change in angular speed = Final angular speed - Initial angular speed
Change in angular speed = (5.05 x 104 rad/s) - (1.44 x 104 rad/s)
Next, you need to find the average angular acceleration. The average angular acceleration can be found using the formula:
Average angular acceleration = Change in angular speed / Change in time
However, in this problem, we don't know the change in time directly. We are given the change in angular displacement instead. So, we need to use an alternate formula.
The average angular acceleration can also be expressed as:
Average angular acceleration = (Final angular speed)^2 - (Initial angular speed)^2 / 2 x Change in angular displacement
Using this formula, we can calculate the average angular acceleration:
Average angular acceleration = ((5.05 x 104 rad/s)^2 - (1.44 x 104 rad/s)^2) / (2 x 1.77 x 104 rad)
Now that we have the average angular acceleration, we can determine how long it would take to reach the maximum speed. We can use the formula:
Final angular speed = Initial angular speed + (Average angular acceleration x Time)
Since the initial angular speed is zero (starting from rest), the equation becomes:
Final angular speed = Average angular acceleration x Time
Solving for time, we have:
Time = Final angular speed / Average angular acceleration
Plugging in the values, we get:
Time = (8.95 x 104 rad/s) / Average angular acceleration