A particle rotates in a circle with centripetal acceleration a = 8.8 m/s2 . What is a if the radius is doubled without changing the particle's speed? What is a if the speed is doubled without changing the circle's radius?

To find the value of centripetal acceleration (a) when the radius is doubled without changing the particle's speed, we can use the formula for centripetal acceleration:

a = v^2 / r

where v is the velocity and r is the radius of the circle.

1. If the radius is doubled without changing the particle's speed, it means that the new radius (r') is equal to twice the original radius (r).

Let's substitute these values into the centripetal acceleration formula:

a' = v^2 / r'

Now, we know that the speed of the particle (v) remains constant, so we can rewrite the formula as:

a' = v^2 / (2r)

Since the velocity (v) is unchanged, we can say a' = a / 2

Therefore, when the radius is doubled without changing the particle's speed, the centripetal acceleration becomes half of its original value.

2. To find the value of centripetal acceleration (a') when the speed is doubled without changing the circle's radius, we can use the same formula:

a' = v'^2 / r

Again, let's substitute the given values into the formula:

a' = (2v)^2 / r

Simplifying this expression, we get:

a' = 4v^2 / r

Since the radius (r) remains constant, we can say a' = 4a

Therefore, when the speed is doubled without changing the circle's radius, the centripetal acceleration becomes four times its original value.