Spring

A block hangs motionless at the end of a spring. How far has the spring been stretched?
 
Their Way
 
The force of gravity acting on the block is:

Fg = mg

The spring force acting on the block is:

Fs = −ky

The net force acting on the blocks is:

Fnet = Fg + Fs

When the block hangs motionless, the spring is stretched a length of Y.

When the block hangs motionless, the net force acting on the block is zero. We have:

0 = Fg + Fs

0 = mg − kY

kY = mg
 
 
My Way
 
The elastic potential energy of the spring is:

EPE = ½ ky ²

The gravitational potential energy of the block is:

GPE = GPEi − mgy

The total potential energy of the spring and the block is:

PE = ½ ky ² − mgy + GPEi

The following is a graph of this equation:




 
 
 

The graph is a parabola.

The derivative of potential energy is:

DPE = ky − mg

When the block hangs motionless, the spring is stretched a length of Y.

When the block hangs motionless, the spring and the block have the least amount of potential energy.

Potential energy is least when the derivative of potential energy is zero. We have:

kY − mg = 0

kY = mg
 
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Here are their two formulas for force:

Fg = mg

Fs = −ky

Both of these formulas are wrong. Here are the correct formulas:

DGPE= −mg

DEPE = ky

Force is a bunch of baloney.