A dentist causes the bit of a high-speed drill to accelerate from an angular speed of 1.50 x 104 rad/s to an angular speed of 3.76 x 104 rad/s. In the process, the bit turns through 3.13 x 104 rad. Assuming a constant angular acceleration, how long would it take the bit to reach its maximum speed of 9.49 x 104 rad/s, starting from rest?
To solve this problem, we can use the equation for rotational motion:
ωf = ωi + αt
where:
ωf = final angular speed
ωi = initial angular speed
α = angular acceleration
t = time
In this case, we know the initial and final angular speeds, as well as the angular acceleration. We want to find the time it takes for the bit to reach its maximum speed, starting from rest.
Let's break down the problem into two parts:
1. From rest to the maximum speed:
In this part, the initial angular speed (ωi) is 0 rad/s, and the final angular speed (ωf) is 9.49 x 10^4 rad/s.
ωf = ωi + αt
9.49 x 10^4 = 0 + αt
We can rearrange this equation to solve for α:
α = ωf / t
2. From the initial speed (1.50 x 10^4 rad/s) to the final speed (3.76 x 10^4 rad/s):
In this part, the initial angular speed (ωi) is 1.50 x 10^4 rad/s, and the final angular speed (ωf) is 3.76 x 10^4 rad/s.
ωf = ωi + αt
3.76 x 10^4 = 1.50 x 10^4 + αt
Again, we can rearrange this equation to solve for α:
α = (ωf - ωi) / t
Now, we have two equations with the same angular acceleration (α). We can equate these equations:
ωf / t = (ωf - ωi) / t
Simplifying the equation:
ωf = ωf - ωi
We can cancel out the t on both sides, leaving:
1 = (ωf - ωi) / α
Now, we can substitute the values we know into this equation:
1 = (9.49 x 10^4 - 0) / α
Simplifying, we find:
α = 9.49 x 10^4
Now, we can substitute this value of α into the first equation we derived earlier:
α = ωf / t
9.49 x 10^4 = 9.49 x 10^4 / t
We can solve for t:
t = 1 second
Therefore, it would take 1 second for the bit to reach its maximum speed of 9.49 x 10^4 rad/s, starting from rest.