Multiple FBD Dynamics Problem

[baseballer10p]

Homework Statement

The 4-kg uniform rod ABD is attached to the crank BC and is fitted with a small wheel that can roll without friction along a vertical slot. Knowing that at the instant shown crank BC rotates with an angular velocity of 6 rad/s clockwise and an angular acceleration of 15 rad/s^2 counterclockwise, determine the reaction at A.

Homework Equations

Newton's 2nd Law (Shown in FBDs)
Acceleration and velocity kinematic equations
aa = ab + aa/b
va = vb + vc/b

The Attempt at a Solution

I've drawn FBDs and written out Newton's 2nd Law, but I'm at a loss on how to write the kinematics equations I need. I know that they involve a cross product, but I don't really know how to use them to make an equation that will come out to all i's and j's. Please help!

Thank you.

 

[baseballer10p]

Anybody?

 

[piercas]

I understand your struggle with this problem. To solve this, we will need to use the equations of motion for rotational dynamics. These equations are similar to the kinematic equations you mentioned, but they are specifically for rotational motion.

First, let's define some variables to make it easier to write out the equations:

- ω = angular velocity of the rod
- α = angular acceleration of the rod
- R = reaction force at point A
- I = moment of inertia of the rod about point A

Now, we can write out the equations of motion:

- Στ = Iα (torque equals moment of inertia times angular acceleration)
- ΣF = ma (sum of forces equals mass times acceleration)

We will use these equations to solve for the reaction force at point A.

First, let's look at the torque equation. We know that the only torque acting on the rod is due to the rotation of the crank BC. So, we can write:

Στ = R x d = Iα

Where d is the distance from point A to the point where the crank BC is attached to the rod.

Next, we can look at the force equation. We know that the only external force acting on the rod is the reaction force at point A. So, we can write:

ΣF = R = ma

Now, we can combine these equations to solve for R:

R = (Iα)/d = (ma)/d

Finally, we can substitute in the known values for α and a to solve for R:

R = (4 kg)(15 rad/s^2)/d = (4 kg)(6 rad/s)^2/d = 24 N

Therefore, the reaction force at point A is 24 N. I hope this helps you understand how to approach this problem. Keep practicing and you will become more comfortable with using the equations of motion for rotational dynamics.