How Do You Calculate Radial Acceleration and Velocity in Circular Motion?

[yitriana]

Homework Statement

A ball swings in a vertical circle at the end of a rope 1.50 m
long. When the ball is 36.9° past the lowest point on its
way up, its total acceleration is (-22.5i, 20.2j) m/s2. At
that instant, (a) determine the magnitude
of its radial acceleration, and (b) determine the speed and
velocity of the ball.

Homework Equations

a = v²/r

The Attempt at a Solution

Centripetal acceleration resolved into two components, in x direction, -22.5 and y direction, 20.2. I would think that in order to get the net acceleration, just (22.5^2 + 20.2^2)^0.5. However, this yielded an incorrect solution. Why doesn't this work and what is the correct approach?

 

[LowlyPion]

I think they give you the acceleration vector in x,y, but you also have a component from gravity in the vertical direction to take into account, don't you?

So you have 2 ways to figure it.

1) observe that tanθ = i/j and use the i component to resolve the unknown j or
2) add 9.8 to the j

 

[nlightner]

The equation you used, a = v2/r, is the correct equation for centripetal acceleration. However, the values given in the problem are not the actual centripetal acceleration, but rather the total acceleration of the ball. This includes both the centripetal acceleration and the tangential acceleration (the component of acceleration that is tangent to the circle). In order to find the magnitude of the radial acceleration, you will need to use the Pythagorean theorem to find the magnitude of the tangential acceleration and then subtract it from the total acceleration.

Once you have the magnitude of the centripetal acceleration, you can use the equation v = √(a*r) to find the speed of the ball. To find the velocity, you will need to use vector addition to find the resultant velocity vector at that instant. This can be done by adding the tangential velocity vector (which is perpendicular to the radius vector) to the radial velocity vector (which is parallel to the radius vector).

I hope this helps clarify the correct approach to solving this problem. Let me know if you have any further questions.