Centripetal and Tangential Acceleration of drill bit

[invadertak]

Homework Statement

An electric drill bit starts from rest and rotates with a constant angular acceleration. After the drill has rotated through a certain angle, the magnitude of the centripetal acceleration is twice the magnitude of the tangential acceleration. What is the angle?

Homework Equations

a_c=rw²
a_T=r.angular acc
angular acc=(w-w₀)/t-t₀
w=angular displacement/t

The Attempt at a Solution

a_c=2a_t
Therefore
rw²=2r.angular acc
The radius cancels and therefore
w²=2angular acc
=2(w-w₀)/t
Given that the drill bit starts from rest, w₀=0 and therefore
w²=2w/t
Divide throughout by w and we have
w=2/t
Sub w=angular displacment/t and we have
angular displacement/t=2/t
t cancels and we are left with
angular displacement=2 rads
This is not the answer given in the textbook, the answer is 1 rad
Please advise where I'm going wrong

 

[LowlyPion]

Consider that ω²r is your Centripetal acceleration.

And that your Tangential acceleration is a = α*r.

You know from kinematics that ω_f² = ω_i² + 2*α*θ or since there is no initial ω :

ω² = 2*α*θ

So ... for the given that

ω²*r = 2*α*r

Substitute for ω²

 

[nlightner]

in my solution.

Your solution is correct. The answer given in the textbook may be incorrect or there may be a typo. It is always important to double check your work and make sure it makes sense in the given context. In this case, your solution makes sense and is supported by the given information.