Circular Motion with constant angular acceleration

[anhchangdeptra]

Homework Statement

An object travels counterclockwise on a circular path with radius R and constant angular
acceleration α, so that

vector r(t) = R cos(αt^2/2) i^+ R sin(αt^2/2) j^


Homework Equations

b. Find the time T when the object made a single revolution and returned to its
original position. Evaluate vectors r, v, and a at both t = 0 and t = T.
c. Show by computation that at t = T, the acceleration vector is the sum of
a part parallel to the velocity vector with magnitude dv/dt , and a part perpendicular to the
velocity vector with magnitude v^2/R

The Attempt at a Solution

I am calculating based on the fact that the object will travel a distance of 2πR at the time it made a revolution, but it doesn't work !

 

[voko]

A revolution brings the object where it was originally. Express that mathematically.

 

[anhchangdeptra]

A revolution brings the object where it was originally. Express that mathematically.

I am sorry I really don't know how to express that mathematically. I have just calculate its speed to be Rαt but I can not make an equation because the object has an increasing acceleration (because its angular accel is constant). This is quite new to me.

 

[SteamKing]

Well, take your equation for r(t) from the OP.

What are the coordinates for the object at time t = 0?

At time t = T, you will have these same coordinates. Knowing that sine and cosine are periodic functions, use this fact to figure out what T must be to return the object to its original position.

 

[voko]

r(t) that you were given is the position of the object. As one revolution brings the object where it started from, you should have r(0) = r(T).

 

[anhchangdeptra]

Oh thank you SteamKing and Voko I know how to do it now. My problem is that I was stuck with the idea that the magnitude of r(t) is always R so I thought I must use another equation rather than r(t). Thanks a ton!