Calculating the Upward Force Necessary to Balance a Plank

[gtaylor]

A uniform plank us supported by a hinge at one end. What is the magnitude (expressed as a multiple of the plank's weight Fg) of the upward force you have to exert on the other end to keep the plank at rest and level?

 

[kuruman]

The sum of all the forces must be zero and the sum of all the torques must be zero. Go for it.

 

[tomcue]

The magnitude of the upward force required to balance the plank will depend on several factors, such as the length and weight of the plank, the distance between the hinge and the point where the force is applied, and the angle at which the plank is held.

To calculate the exact magnitude of the upward force, we can use the principles of static equilibrium. This means that the sum of all forces acting on the plank must be equal to zero, and the sum of all torques (rotational forces) must also be equal to zero.

First, we can calculate the weight of the plank (Fg) by multiplying its mass by the acceleration due to gravity (g). Next, we need to consider the forces acting on the plank. The only two forces acting on the plank are the weight (Fg) and the upward force (F). Therefore, to keep the plank in equilibrium, the upward force must be equal in magnitude to the weight of the plank (F = Fg).

However, we also need to consider the torques acting on the plank. The hinge at one end of the plank creates a torque that is equal to zero, as it is balanced by the reaction force from the ground. The upward force (F) creates a clockwise torque, while the weight of the plank (Fg) creates a counterclockwise torque. To keep the plank in equilibrium, these two torques must be equal in magnitude. This can be expressed mathematically as F x d = Fg x L, where d is the distance between the point where the force is applied and the hinge, and L is the length of the plank.

Therefore, the magnitude of the upward force (F) can be calculated as F = (Fg x L)/d. This means that the magnitude of the upward force required to balance the plank will be directly proportional to the weight of the plank and its length, and inversely proportional to the distance between the point where the force is applied and the hinge.

In conclusion, the magnitude of the upward force needed to balance a plank supported by a hinge at one end can be calculated using the principles of static equilibrium. By considering both the forces and torques acting on the plank, we can determine that the upward force must be equal in magnitude to the weight of the plank, and the distance between the point where the force is applied and the hinge will also play a role in determining the required force.