What Is the Angular Speed of a Falling Meter Stick?

[gmiller4th]

Homework Statement

A meter stick is hinged at its lower end and allowed to fall from a vertical position. With what angular speed does it hit the table?

Homework Equations

V = rW (Not sure?)

The Attempt at a Solution

I have no clue where to even start with this problem, given the small amount of given data.

Obviously a meter stick is 1m long. But this doesn't get me very far. I know that gravity is also acting on the object so that would have to be included somewhere in my calculation. Basically I'm completely lost on this one.

Thanks for your help / guidance.

 

[tiny-tim]

Welcome to PF!

A meter stick is hinged at its lower end and allowed to fall from a vertical position. With what angular speed does it hit the table?

Hi gmiller4th! Welcome to PF!

Hint: use conservaton of energy.

 

[baba89]

First, let's define some variables that will help us solve this problem:

- W = angular speed (in radians per second)
- r = distance from the hinge to the center of mass of the meter stick (in meters)
- g = acceleration due to gravity (9.8 m/s^2)

Next, let's consider the initial and final states of the meter stick. At the initial state, the meter stick is at rest and at a vertical position. At the final state, the meter stick is rotating around the hinge and about to hit the table. Since the meter stick is falling under the influence of gravity, we can use the equation for angular acceleration:

W^2 = W0^2 + 2aΘ

where W0 is the initial angular speed (which is 0 in this case), a is the angular acceleration, and Θ is the angle through which the meter stick has rotated.

We know that the meter stick has rotated 90 degrees (π/2 radians) when it hits the table, so we can rearrange the equation to solve for the angular acceleration:

a = (W^2)/2Θ

Now, we need to find the value of W, the angular speed at which the meter stick hits the table. We can use the equation V = rW, where V is the linear speed (in meters per second) at the end of the meter stick and r is the distance from the hinge to the center of mass of the meter stick. Since the meter stick is falling under the influence of gravity, we can use the equation V^2 = V0^2 + 2ad, where V0 is the initial linear speed (which is 0 in this case), a is the acceleration due to gravity, and d is the distance through which the meter stick has fallen (which is equal to the length of the meter stick, 1m). Rearranging this equation, we get V = √(2gd).

Substituting this value of V into the equation V = rW, we get:

√(2gd) = rW

Solving for W, we get:

W = √(2gd)/r

Now, we can substitute this value of W into the equation for angular acceleration to get:

a = (2gd)/(2Θr) = gd/(Θr)

Finally, substituting the values of g (9.8 m