Force exerted by a stick on its axle?

[EstimatedEyes]

Homework Statement

A stick of uniform density with mass M = 8.6 kg and length L = 0.7 m is pivoted about an axle which is perpendicular to its length and located 0.24 m from one end. Ignore any friction between the stick and the axle. What is the magnitude of the vertical component of the force exerted by the stick on the axle when the stick passes through the vertical?

Homework Equations

The Attempt at a Solution

I already correctly calculated that the angular velocity when it is in the vertical position is 6.38 rad/s and that the center of mass is .11m below the axis of rotation when it is in the vertical position. From this I calculated v of the center of mass by multiplying the angular velocity to the radius. Once I got this answer I calculated the centripetal force by v^2/r and then added it to mg. This answer is not right, but I don't know where I went wrong. I'm sure it's something really simple. Thanks in advance for your help!

 

[kuruman]

The (net) centripetal force is mv²/r. Assuming that you did not forget to multiply by the mass, your method is correct. Redo the numbers - you may have forgotten to, say, square the speed.

 

[kerimek]

I would first like to commend you on your efforts to solve this problem and for seeking help when you encountered difficulties. This shows a great attitude towards problem-solving and a willingness to learn.

Now, moving on to the problem, it seems like you have correctly calculated the angular velocity and the position of the center of mass when the stick is in the vertical position. However, your approach to calculating the force exerted by the stick on the axle may not be entirely correct.

The force exerted by the stick on the axle can be broken down into two components- the radial force and the tangential force. The radial force is the component that acts towards the center of rotation and is responsible for keeping the stick in a circular motion. This is the force that you have calculated using the formula v^2/r.

However, the tangential force is the component that acts perpendicular to the radial force and is responsible for changing the direction of the stick's motion. This force is equal to the product of the mass, the linear velocity, and the angular velocity of the stick. In this case, the linear velocity of the center of mass is equal to the tangential velocity, which you have already calculated.

Therefore, to find the total force exerted by the stick on the axle, you need to add the radial force and the tangential force together. This will give you the correct value for the magnitude of the vertical component of the force exerted by the stick on the axle.

I hope this helps you in finding the correct answer. Keep up the good work!