How Do Linear and Angular Accelerations Affect Stick Balancing?

[saltine]

Homework Statement

Find the linear and angular accelerations given that FA is the applied force.
v is the current linear velocity and w is the current angular velocity.
r is the radius of the stick.
h is length of the stick.
d is angle from the vertical axis.

Homework Equations

a = dv/dt; linear acceleration
u = dw/dt; angular acceleration
I_center = m h²/12; moment of inertia of stick

The Attempt at a Solution

Sum of horizontal forces:
m a_x = -F_A
a_x = -F_A/m

Sum of vertical forces:
ma_y = -mg + F_N = 0

Sum of torque
T = I u = 0.5 h F_N sin(d) - 0.5 h F_A cos(d)
u = 0.5 h ( mg sin(d) - F_A cos(d) ) / ( m h²/12 )
u = ( 6 / m h )( mg sin(d) - F_A cos(d) )

Is this correct?

Shouldn't the vertical acceleration be non-zero? Because if w and u are non-zero, and the finger maintains at the same height, then the center of mass of the stick must be moving. What is wrong with the above?

- Thanks
And thank you for replying to my other thread. There are so many threads I thought I shouldn't bump the one that was solved.

 

[AvroArrow]

The vertical acceleration is zero because the stick is rotating at a constant height. The angular acceleration (u) is non-zero, but this is because the angular velocity (w) is changing, not because the stick is rising or falling. So the linear acceleration of the center of mass of the stick is zero, since the stick is rotating in place.

 

[thomate1]

I would like to point out that the equations and solution provided are correct and follow the principles of Newton's laws of motion and conservation of angular momentum. However, there may be some assumptions made that may not accurately represent the real-world scenario of stick balancing.

Firstly, the equations assume that the stick is a rigid body, which may not be the case in reality. The stick may have some flexibility or may bend under the applied force, which can affect the linear and angular accelerations.

Secondly, the equations also assume that the stick is perfectly balanced and that there are no external forces acting on it besides the applied force. In reality, there may be other forces such as air resistance, friction, or uneven ground that can affect the balance of the stick.

Lastly, the equations assume that the stick is being balanced by a finger at one end, which may not be the case in all scenarios. The position and force applied by the finger can also affect the balance of the stick.

In conclusion, while the provided solution is mathematically correct, it may not accurately represent the real-world scenario of stick balancing. Further factors and variables would need to be considered to fully understand and predict the behavior of a balancing stick.