A cat rides a merry-go-round turning with uniform circular motion. At time t1 = 2.00 s, the
cat's velocity is (3.00 m/s) + (2.70 m/s) , measured on a horizontal xy coordinate system.
At t2 = 5.90 s, its velocity is (-3.00 m/s) + (-2.70) . 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 t2 â t1, which is less than a period of the motion?
To solve this problem, we need to understand and apply the concepts of uniform circular motion, centripetal acceleration, and average acceleration.
Uniform circular motion is the movement of an object in a circular path with constant speed. In this case, the cat is riding a merry-go-round with uniform circular motion.
Centripetal acceleration is the acceleration directed toward the center of the circular path. It is given by the formula:
ac = v^2 / r
where ac is centripetal acceleration, v is the velocity of the object, and r is the radius of the circular path.
Average acceleration is the change in velocity divided by the change in time. It is given by the formula:
average acceleration = (final velocity - initial velocity) / (final time - initial time)
Now let's solve the problem step by step:
(a) To find the magnitude of the cat's centripetal acceleration, we need to calculate the magnitude of its velocity at both time instances. Adding the given velocities vectorially, we get:
Initial velocity vector = (3.00 m/s) + (2.70 m/s) = (5.70 m/s)
Final velocity vector = (-3.00 m/s) + (-2.70 m/s) = (-5.70 m/s)
The magnitude of the cat's velocity (v) is given by the Pythagorean theorem:
v = sqrt((vx)^2 + (vy)^2)
Magnitude of initial velocity = sqrt((5.70 m/s)^2) = 5.70 m/s
Magnitude of final velocity = sqrt((-5.70 m/s)^2) = 5.70 m/s
Since the magnitude of the velocity is constant in uniform circular motion, the magnitude of the cat's centripetal acceleration (ac) is given by:
ac = v^2 / r
To calculate the centripetal acceleration, we need to know the radius of the circular path. Unfortunately, the problem statement does not provide this information. Without the radius, we cannot determine the magnitude of the centripetal acceleration.
(b) To find the magnitude of the cat's average acceleration during the time interval t2 - t1, we need to calculate the change in velocity and divide it by the change in time.
Change in velocity = final velocity - initial velocity = (-5.70 m/s) - (5.70 m/s) = -11.4 m/s
Change in time = t2 - t1 = 5.90 s - 2.00 s = 3.90 s
Average acceleration = (change in velocity) / (change in time) = -11.4 m/s / 3.90 s
Magnitude of average acceleration = |-11.4 m/s / 3.90 s| = 2.92 m/s^2
Therefore, the magnitude of the cat's average acceleration during the time interval t2 - t1 is 2.92 m/s^2.
In summary:
(a) The magnitude of the cat's centripetal acceleration cannot be determined without the radius of the circular path.
(b) The magnitude of the cat's average acceleration during the time interval t2 - t1 is 2.92 m/s^2.