A cat rides a merry-go-round turning with uniform circular motion. At time t1 = 1.80 s, the cat's velocity is (3.60 m/s)i + (2.00 m/s)j , measured on a horizontal xy coordinate system. At t2 = 5.70 s, its velocity is (-3.60 m/s)i + (-2.00)j . What are (a) the magnitude of the cat's centripetal acceleration and (b) the magnitude of the cat's average acceleration during the time interval t_(2) – t_(1), which is less than a period of the motion?
To solve this problem, we need to first understand the concepts of centripetal acceleration and average acceleration.
Centripetal acceleration is the acceleration of an object moving in a circular path, directed towards the center of the circle. It can be calculated using the formula:
ac = v^2 / r
where ac is the centripetal acceleration, v is the velocity of the object, and r is the radius of the circular path.
Average acceleration, on the other hand, is the change in velocity divided by the time interval. It can be calculated using the formula:
av = (v2 - v1) / (t2 - t1)
where av is the average acceleration, v2 and v1 are the final and initial velocities respectively, and t2 and t1 are the final and initial times respectively.
Now let's solve the problem using these concepts:
(a) To calculate the magnitude of the cat's centripetal acceleration, we need to find the magnitude of its velocity and the radius of the circular path it is traveling on. Since the velocity vectors are given, we can simply take the magnitude of the velocity at either t1 or t2. Let's use t1.
v1 = √((3.60)^2 + (2.00)^2)
= √(12.96 + 4.00)
= √16.96 m/s
Next, we need to find the radius of the circular path. Unfortunately, the radius is not given directly. However, since the cat is moving with uniform circular motion, its path is a circle. Therefore, we need to find some other information that can help us determine the radius.
(b) To calculate the magnitude of the cat's average acceleration, we need to find the change in velocity during the time interval t2 - t1, which is 5.70 s - 1.80 s = 3.90 s.
Δv = v2 - v1
= (-3.60 m/s)i + (-2.00 m/s)j - (3.60 m/s)i + (2.00 m/s)j
= (-7.20 m/s)i - (4.00 m/s)j
Next, we need to find the magnitude of the change in velocity:
|Δv| = √((-7.20)^2 + (-4.00)^2)
= √(51.84 + 16.00)
= √67.84 m/s
Finally, we can calculate the magnitude of the average acceleration:
av = |Δv| / (t2 - t1)
= √67.84 m/s / 3.90 s
You can now use a calculator to find the numerical values of the centripetal acceleration and average acceleration.