A circular plate with a slot in its surface is rotating with angular velocity ω and angular acceleration α with respect to a fixed coordinate system, OXYZ as shown. A ball B moves along the slot in the direction shown with velocity, v, and acceleration, a, both measured relative to the plate. A body coordinate system Axyz is attached to the rotating plate as shown. The ball is slightly smaller than the sot and therefore only contacts one side of the slot at at time.
FOR IMAGE
Copy and Paste in Browser AND PUT A (.)BEFORE COM(it wouldn't accept any . com entries sorry for the in convince.)
i39.tinypiccom/nl4rqg.jpg
Questions:
GROUND FRAME VELOCITY (6.0/6.0 points)
For the instant of time shown in the figure, give the ball's velocity vB with respect to the fixed ground frame Oxyz. Express your answer using the body coordinates i^ and j^.
Please enter the quantity terms of a,v,α,ω, and/or L.
vB=
i^:
j^:
GROUND FRAME ACCELERATION (6.0/6.0 poin
For the instant of time shown in the figure, give the ball's acceleration aB with respect to the fixed ground frame Oxyz. Express your answer using the body coordinates i^ and j^.
Please enter the quantity terms of a,v,α,ω, and/or L.
aB=
i^:
j^:
Q1-2 ANGULAR VELOCITY (6 points possible)
What range of angular velocities is the ball in contact with the outer side of the slot - the side furthest away from the center of rotation?(the angular speed w you must rotate the circular plate in order for the ball to be touching the side of the slot furthest away from the center)
Please enter the quantity terms of a,v,α,ω, and/or L.
ω>
To find the velocity of the ball relative to the fixed ground frame Oxyz, we need to consider the velocity of the ball relative to the rotating plate and the velocity of the rotating plate relative to the fixed ground frame.
We know that the ball's velocity with respect to the plate is v (given in the question) and the plate's angular velocity is ω. The plate's velocity with respect to the fixed ground frame is given by the formula:
v_plate = R * ω
where R is the distance from the center of rotation to the ball (the radius of the circular plate).
The ball's velocity with respect to the fixed ground frame is the vector sum of its velocity relative to the plate and the plate's velocity relative to the ground frame:
v_B = v + v_plate
To express the answer in terms of the body coordinates i^ and j^, we need to resolve the velocity vectors v and v_plate into their components along these body coordinates.
v_B = v_i + v_j
where v_i and v_j are the components of the velocity vector along the i^ and j^ directions respectively.