Connections between Linear and Rotational Quantities

[Soniteflash]

Homework Statement

A wheel of radius R starts from rest and accelerates with a constant angular acceleration α about a fixed axis. At what time t will the centripetal and tangential acceleration of a point on the rim have the same magnitude?

Homework Equations

a_cp=r x ω²

a_t= r x α

ω= 2π / T → T=2π/ω

The Attempt at a Solution

The problem states that the centripetal and tangential acceleration will have the same magnitude at time t.
So I listed the equation for centripetal acceleration and tangential acceleration and thought that I can put them equal to each other since the magnitudes are the same.

R x ω² =R x α I canceled out the R

ω²= α Here I thought that since ω equals 2π/T I could substitute it for ω since it has T for period

which relates to time​

(2π/T)² = α I took the square root of that.

2π/T = √α Solving for T

T = 2π/(√α)

So would this be the time? The AP Physics Book (3rd Edition by James S. Walker doesn't give me a solution since the problem number is even (only gives for odd).

 

[Nathanael]

The period isn't important. You have ω²=α. You know that α is constant. You know that ω starts at zero. Can you then find ω as a function of time?

 

[Soniteflash]

Hmm. So would that be w(final)=w(initial)+ a(alpha) t ?
Only one that doees not invovle angular distance. Hmm so I could substitute w^2 for alpha in the equation.

 

[Nathanael]

And what do you get?

 

[Soniteflash]

I apologize for replying so late. An assignment turned out to be more time consuming than I thought it would be.
Anyways so if I plug in w²,

I get W(final)=W(initial) + W² * t W(initial) is zero and solving for t give me

t = W(final)/ W²

Would it be possible to cancel out the W to get 1/W ?

 

[Nathanael]

Would it be possible to cancel out the W to get 1/W ?

Yes that would be okay. But it may be better to write the final answer in terms of α, since α is a known quantity.

 

[Soniteflash]

How would write it in terms of alpha since I substitute it with w^2

 

[lightgrav]

substitute the alpha back in?