# Circular Motion with constant angular acceleration

## 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 !

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

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
Staff Emeritus
Homework Helper
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.

• 1 person
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).

• 1 person
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!