A dentist causes the bit of a high-speed drill to accelerate from an angular speed of 1.20 104 rad/s to an angular speed of 3.14 104 rad/s. In the process, the bit turns through 1.92 104 rad. Assuming a constant angular acceleration, how long would it take the bit to reach its maximum speed of 7.85 104 rad/s, starting from rest?

3.14*10^4 - 1.2*10^4 = 1.94*10^4 rad/s. = Change in velocity.

t=1.92*10^4rad * 1s/1.94*10^4rad=0.99s.

T=7.85*10^4rad/s/1.94*10^4rad/s * 0.99s= 4.05 Seconds.

We can solve this problem using the equations of rotational kinematics. The equation relating angular velocity (ω), angular acceleration (α), and time (t) is:

ω = ω0 + αt

where ω0 is the initial angular velocity.

Given:
Initial angular velocity, ω0 = 1.20 × 10^4 rad/s
Final angular velocity, ω = 7.85 × 10^4 rad/s
Total angular displacement, θ = 1.92 × 10^4 rad

We need to find the time taken to reach maximum speed, so we substitute ω0 = 0 and ω = 7.85 × 10^4 rad/s into the equation:

7.85 × 10^4 rad/s = 0 + αt

Now, let's find the angular acceleration (α). We can use the equation:

ω² = ω0² + 2αθ

Substituting the given values:

(7.85 × 10^4 rad/s)² = (1.20 × 10^4 rad/s)² + 2α(1.92 × 10^4 rad)

(7.85 × 10^4)² = (1.20 × 10^4)² + 2α(1.92 × 10^4)

615022500 = 144000000 + 38400α

38400α = 615022500 - 144000000

38400α = 471022500

α = (471022500) / (38400)

α ≈ 12263.17 rad/s²

Now, we can substitute the value of α into the equation for ω = 7.85 × 10^4 rad/s:

7.85 × 10^4 rad/s = 12263.17 rad/s² × t

Simplifying for t:

t = (7.85 × 10^4 rad/s) / (12263.17 rad/s²)

t ≈ 6.40 s

Therefore, it would take approximately 6.40 seconds for the bit to reach its maximum speed of 7.85 × 10^4 rad/s, starting from rest.

To solve this problem, we can use the kinematic equation that relates angular acceleration, final angular velocity, initial angular velocity, and the angle turned. The equation is as follows:

ω_f^2 = ω_i^2 + 2αθ

Where:
- ω_f is the final angular velocity (7.85 x 10^4 rad/s)
- ω_i is the initial angular velocity (0 rad/s, since the bit starts from rest)
- α is the angular acceleration (which is assumed to be constant)
- θ is the angle turned (1.92 x 10^4 rad)

Rearranging the equation, we get:

α = (ω_f^2 - ω_i^2) / (2θ)

Let's substitute the given values into the equation to find the angular acceleration:

α = ( (7.85 x 10^4 rad/s)^2 - (0 rad/s)^2 ) / (2 x 1.92 x 10^4 rad)
= ( 6.15 x 10^9 rad^2/s^2 ) / (3.84 x 10^4 rad)
= 1.599 x 10^5 rad/s^2

Now that we have the angular acceleration, we can find the time it takes to reach the maximum speed using the equation:

ω_f = ω_i + αt

Since the initial angular velocity (ω_i) is zero, the equation simplifies to:

ω_f = αt

Rearranging the equation, we get:

t = ω_f / α
= (7.85 x 10^4 rad/s) / (1.599 x 10^5 rad/s^2)
= 0.491 seconds

Therefore, it would take approximately 0.491 seconds for the bit to reach its maximum speed of 7.85 x 10^4 rad/s, starting from rest.