Determine d if a plank is not uniform

[skysunsand]

Homework Statement

See attachment!

A plank is 4m long and has a weight of 500 N. It is pivoted frictionlessly about a nail which is driven through the center of the plank. When a 200 N weight is hung as shown in the figure (attached), the plank is horizontal and in equilibrium.

Suppose the center of gravity is a distance d from the left end of the plank. Determine d.

Homework Equations

His work says

500*(d-2) = 160*2 = 320

d= 2+ 320/500 = 2.64

Where are those numbers coming from? He wrote into the diagram 160 and 120, near the 37 degree angle sign, but I don't know where those came from and why they factor into this problem.

He tends to skip...every step. So if someone could explain what he is doing and how this relates to rotational physics, angular physics, or SHM, that would be lovely...

The Attempt at a Solution

 

[ehild]

Try to solve the problem by yourself instead of trying to read his mind.

The plank is in equilibrium. What does it mean on the forces and torques acting on it?

ehild

 

[skysunsand]

Try to solve the problem by yourself instead of trying to read his mind.

The plank is in equilibrium. What does it mean on the forces and torques acting on it?

ehild

Well, it would mean everything has to equate out to 0.

With the forces, I think there's the force of tension + mg on the 200 weight, multiplied by two because there's two strings supporting the weight.
And then there's gravitational force on the 500 N plank.

So then for the forces, it would have to be

0= Ft + Mg weight +mg plank ?

 

[ehild]

There is only one string supporting the hanging weight.
As for the hanging weight, there is the tension of the string acting on it upward and gravity of 200 N acting downward. The sum T-200 has to be zero as the weight is in rest. What is tension then?
AS for the plank, the forces acting on it are the tension of the string T, its own weight and the force of the nail at the pivot. Take care, the tension force acts at an angle 37° with the vertical. You do not need the force at the pivot, so you can ignore this equation.
The third equation comes from the balance of torques. Find the torque around the pivot at the middle of the plank for both forces: the weight of the plank and the tension. The torque of the force at the pivot is zero (why?)

ehild