# Solve Falling Plank Problem: Acceleration & Angle of Separation

• ViolinIsLife
In summary, the plank would separate from the floor at about 2/3 of the original height where the plank was leaning against the wall.
ViolinIsLife
Could someone help me with this problem? I appreciate it!

A uniform plank of mass M and length 2L is resting on a frictionless floor and leaning against a frictionless vertical wall. It is held steady by a massless string connecting the lower end of the plank to the base of the wall. The angle between the floor and the plank is theta. Calculate the acceleration of the upper end of the plank immediately after the string is cut, and the angle theta at which the upper end of the plank first separates from the wall.

My prof. said that the plank would separate from the floor at about 2/3 of the original height where the plank was leaning against the wall.

Thank you very much in advance!

ViolinIsLife said:
Could someone help me with this problem? I appreciate it!

A uniform plank of mass M and length 2L is resting on a frictionless floor and leaning against a frictionless vertical wall. It is held steady by a massless string connecting the lower end of the plank to the base of the wall. The angle between the floor and the plank is theta. Calculate the acceleration of the upper end of the plank immediately after the string is cut, and the angle theta at which the upper end of the plank first separates from the wall.

My prof. said that the plank would separate from the floor at about 2/3 of the original height where the plank was leaning against the wall.

Thank you very much in advance!
Please post your attempt at solving this problem. Draw a free body diagram with the forces acting and apply Newton's second law to the motion of the center of mass and its rotational analog to the angular motion.

I have attached a diagram for this problem. Sorry, don't know why the jpg file doesn't show the forces that I have drawn with MS Paint on the diagram.

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ViolinIsLife said:
I have attached a diagram for this problem. Sorry, don't know why the jpg file doesn't show the forces that I have drawn with MS Paint on the diagram.
We cannot yet see the diagram, and probably do not need it to see if you are on the right track. Do you have the equations related to the diagram?

## 1. What is the falling plank problem?

The falling plank problem is a classic physics problem that involves a plank of wood being held up at one end and then released to fall to the ground. The goal is to determine the acceleration of the plank and the angle at which it separates from the ground.

## 2. How is the acceleration of the plank calculated?

The acceleration of the plank can be calculated using the formula a = g * sin(theta), where g is the acceleration due to gravity (9.8 m/s²) and theta is the angle of separation. This formula assumes that there is no air resistance and the plank is released from rest.

## 3. How do you determine the angle of separation?

The angle of separation can be determined by using the formula theta = arcsin (a/g), where a is the acceleration of the plank and g is the acceleration due to gravity. This formula assumes that the plank is released from rest and there is no air resistance.

## 4. How does air resistance affect the solution to the falling plank problem?

Air resistance can significantly affect the solution to the falling plank problem. It can change the acceleration of the plank and the angle of separation, making the calculations more complex. In real-life scenarios, air resistance cannot be ignored and must be taken into account when solving the problem.

## 5. What are some real-world applications of the falling plank problem?

The falling plank problem has several real-world applications, such as determining the angle at which a plane's wing must be tilted to generate lift, or calculating the trajectory of a projectile launched at an angle. It is also relevant in engineering and architecture for determining the stability and strength of structures.

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