Question

An object moves in a circle of radius R at a constant speed with a period T. If you want to change only the period in order to cut the object's acceleration in half, the new period should be?
The answer my professor gave was radical2*T, but I don't know how to get to that answer.

Answer 1

the acceleration before the change is:
\[\Large a = {\omega ^2}R = \frac{{4{\pi ^2}}}{{{T^2}}}R\]
after the change the new acceleration is:
\[\Large {a_1} = {\omega ^2}R = \frac{{4{\pi ^2}}}{{{T_1}^2}}R\]
where T_1 isthe new period

Answer 2

now from the text of your proble, we have the subsequent condition:
\[\Large {a_1} = \frac{a}{2}\]

Answer 3

substituting in that condition, we can write:
\[\Large \frac{{4{\pi ^2}}}{{{T_1}^2}}R = \frac{1}{2}\frac{{4{\pi ^2}}}{{{T^2}}}R\]

Answer 4

after a simplification, we get:
\[\Large T_1^2 = 2{T^2}\]
now, please take the square root of both sides of that equation, and you will get the answer of your professor