Calculating Angular and Linear Acceleration in a Vertical Plane

[dvvz2006]

Can someone please help me with this problem:

You are swinging a mass at the ned of a string contained in a vertical plane. The length of the string is r. The mass moves just enough to complete hte circle. The motion is a constant total energy and we take the sense of rotation as positive.
a) Find the angular velocity at the points where A is right at the top at the tip of the circle, B is 90 degress down, C is 90 degrees further, and D 90 degress further than that.
b) Find the angular acceleration at these four points and include the sign with the answers.
c) Calculate hte magnitude of the linear acceleration of the mass at each of the four points. Note the mutually perpendicular vector components.

 

[Tide]

Someone may be willing to help -- provided you showed some of your own work first.

 

[quicknote]

To solve this problem, we can use the equation for angular velocity: ω = v/r, where ω is the angular velocity, v is the linear velocity, and r is the radius of the circle. In this case, the linear velocity is constant, so the angular velocity will also be constant.

a) At point A, the mass is at the top of the circle and the radius is equal to the length of the string, r. Therefore, the angular velocity at point A is ω = v/r = v/r.

At point B, the radius is now equal to r/2 (since the mass has moved 90 degrees down), so the angular velocity is ω = v/(r/2) = 2v/r.

At point C, the radius is now equal to 0 (since the mass has completed half a circle), so the angular velocity is ω = v/0 = undefined.

At point D, the radius is now equal to -r/2 (since the mass has moved 90 degrees further), so the angular velocity is ω = v/(-r/2) = -2v/r.

b) To find the angular acceleration, we can use the equation α = ω/t, where α is the angular acceleration and t is the time. Since the angular velocity is constant, the angular acceleration will be 0 at all four points.

c) To find the linear acceleration, we can use the equation a = ω^2r, where a is the linear acceleration. At point A, the linear acceleration is a = (v/r)^2r = v^2/r.

At point B, the linear acceleration is a = (2v/r)^2(r/2) = 2v^2/r.

At point C, the linear acceleration is a = (undefined)^2(0) = undefined.

At point D, the linear acceleration is a = (-2v/r)^2(-r/2) = 2v^2/r.

Note that the linear acceleration is always perpendicular to the radius, since it is directed towards the center of the circle.