Answer:
r = 3519.55 m
Explanation:
We know that the acceleration of a particle in a circular motion is directed towards the center of the circumference and has magnitude:
F = rω^2
Where r is the radius of the circumference and ω is the angular velocity.
From the two acceleration vectors we find that their magnitude is
√(7^2+6^2) = √85
Therefore:
√85 m/s^2= rω^2
Now we need to calculate the angular velocity to obtain the radius. Since t2-t1 = 3s is less than one period we can be sure that the angular velocity is equals to the angle traveled between this time divided by 3 s.
The angle with respect to the x-axis for the particle at t1 and t2 is:
[tex]\theta 1 =\cos ^{-1}\left(\frac{7}{\sqrt{85}}\right)\\\theta 2 =\cos ^{-1}\left(\frac{6}{\sqrt{85}}\right)\\[/tex]
Therefore, the angular velocity ω is (in radians per second):
[tex]\omega = \frac{\theta2 - \theta1}{3 s} = 0.0511813 \frac{1}{s}[/tex]
Therefore:
r = √85 / (0.0511813)^2 = 3519.55 m
