How Far Can a Man Walk on a Plank Before It Tips Over?

[einstein18]

Homework Statement

A man, of m = 70 kg, walks along a thin rigid plank, of M = 210 kg. The plank sits on the roof of a 10 story building and is placed so that its center of mass sits directly on the edge of the roof.

Where, along the plank, can the man move before the plank tips over the edge? Assume that friction is strong enough to keep the plank from moving horizontally.

Homework Equations

Statics:
Y: (M+m)g=N(due to floor)
X: Not applicable
Torque=mgd=0

Rotational:
Torque=mgd=I*alpha

The Attempt at a Solution

I am confused on how to relate the statics to the rotational part. I understand that as long as the plank doesn't fall the equilibrium equations remain true, however as soon as it does it is no longer true and the ratational torque becomes the reality. Now I don't understand how to create and equation that relates the change from one to the other as a function of how far the man walks. Can someone please help me out?

 

[Doc Al]

In order to tip, you just need an unbalanced torque about the pivot point (the edge of the roof, in this case).

This problem is a bit odd since the plank's center is right at the edge of the roof. Where does the weight of the plank act? Will it exert any torque about the pivot point?

 

[nlightner]

The key to solving this problem is understanding the relationship between statics and rotational motion. In this scenario, the plank is in static equilibrium, meaning that all forces and torques acting on it are balanced. This is represented by the equations you have listed for the Y and X directions.

However, when the man starts to walk along the plank, he introduces a torque (mgd) that is not balanced by any other force or torque. This causes the plank to start rotating, and eventually, if the torque is large enough, it will tip over the edge.

To determine where the man can walk before the plank tips over, we need to find the point where the torque from the man's weight is equal to the maximum torque that the plank can withstand before tipping over. This maximum torque is determined by the moment of inertia (I) of the plank and the angular acceleration (alpha) it experiences.

The moment of inertia (I) can be calculated using the formula I = (1/3)ML^2, where M is the mass of the plank and L is the length of the plank. The angular acceleration (alpha) can be calculated using the formula alpha = (net torque)/(moment of inertia).

Once you have these values, you can set the torque from the man's weight (mgd) equal to the maximum torque and solve for the distance (d) that the man can walk before the plank tips over. This will give you the answer to the question.

I hope this helps clarify the relationship between statics and rotational motion in this scenario. Let me know if you have any further questions.